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(p

//

Q(^3 ENGINEERING LIBRARY ^

MATHEMATICAL

AND

PHYSICAL PAPEKS.

Soiaion: C. J. CLAY, M.A. & SON,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,

17, PATBBMOflTEB RoW.

tl^

-,iV

CAMBRIDGE: DEIOIITON. BELL. AND CO.
LEIPZIG : F. A. BROCKIIAU8.

MATHEMATICAL

AND

PHYSICAL PAPERS

BY

GEORGE GABRIEL STOKES, M.A., D.C.L, LL.D, F.RS,

FELLOW OF PEMBROKE COLLEGE AND LUCA8IAN PBOFESSOR OF MATHEMATICS

IN THE UNIVERSITY OF CAMBRIDGE.

lUprhUed from tJie Original Journals and Transactions^
with Additional Notes hi/ the Author,

VOL. II.

CAMBRIDGE :

AT THE UNIVERSITY PRESS.

1883

[The rights of translation and reproduction are reserved.]

rRI5T£J> DT C. J. CLAY, M.A. AND HON.
AT TUE CNIVKUblTY PBEB8.

CONTENTS.

PAGE

Notes on Hydrodynamics. HI. On the Dynamical Equations ... 1

On the constitntion of the Luminiferoos Ether 8

On the Theory of certain Bands seen in the Spectrom .14

Section L Explanation of the Formation of the Bands on the imper-
fect Theoiy of Interferences. Mode of Calculating the Number
of Bands seen in a given part of the Spectrum .... 15

Section IL Investigation of the Intensity of the Light on the com-
plete Theoiy of Undulations, including the Explanation of the
apparent Polarity of the Bands 24

Notes on Hydrodynamics. IV. Demonstration of a Fundamental

Theorem 36

On a difficulty in the Theoiy of Sound 51

On the Formation of the Central Spot of Newton*s Rings beyond the

Critical Angle 56

On some points in the Beceived Theory of Sound 82

On the perfect Blackness of the Central Spot in Newton's Bings, and on
the Verification of Fresnel's Formulae for the intensities of Reflected
and Refracted Rays 89

On Attractions, and on Clairaut's Theorem 104

On the Variation of Gravity at the Surface of the Earth .131

On a Mode of Measuring the Astigmatism of a Defective Eye . .172

On the Determination of the Wave Length corresponding with any Point

of the Spectrum 176

Discussion of a Differential Equation relating to the Breaking of Railway

Bridges 178

Notes on Hydrodynamics. VI. On Waves 221

On the Dynamical Theory of Diffraction 243

Past L Theoretical Investigation.

Section I. Preliminary Analysis 250

Section II. Propagation of an Arbitrary Disturbance in an

Elastic Medium 257

Section m. Determination of the Law of the Disturbance in a

Secondary Wave of Light 280

Vlll CONTENTS.

PAOK

Part II. Experiments on the Rotation of tlie Plane of Polarization
of Diffracted Light.

Suction I. Description of the Experiments .... 290

Section II. Discassion of the Numerical Results of the Ex-
periments, with reference to Theory 307

On the Numerical Calculation of a class of Definite Integrals and

Infinite Series 329

On the Mode of Disappearance of Newton's Rings in passing the Angle of

Total Internal Reflection .358

On Metallic Reflection 360

On a Fictitious Displacement of Fringes of Interference .... 361

On Haidinger's BrusheR 362

Index 365

ERRATUM.
P. 221, in the Number of the Note. For IV. read VI.

MATHEMATICAL AND PHYSICAL PAPERS.

[From the Cambridge and Dublin Mathematical Journal, Vol. iil p. 121,

March, 1848.]

Notes on Hydrodynamics*.

III. — On the Dynamical Equations.

In reducing to calculation the motion of a system of rigid
bodies, or of material points, there are two sorts of equations with
which we are concerned ; the one expressing the geometrical con-
nexions of the bodies or particles with one another, or with curves
or surfaced external to the system, the other expressing the rela-
tions between the changes of motion which take place in the system
and the forces producing such changes. The equations belonging
to these two classes may be called respectively the geometrical, and
the dynamical equations. Precisely the same remarks apply to
the motion of fluids. The geometrical equations which occur in

• The series of •* notes on Hydrodynamics" which are printed in Vols, ii., in.
and rv. of the Cambridge and Dublin Mathematical Journal^ were written by agree-
ment between Sir William Thomson and myself mainly for the ase of Students. As
far as my own share in the series is concerned, there is little contained in the
"notes" which may not be foand elsewhere. Acting however apon the general
advice of my friends, I have included my share of the series in the present reprint.
It may be convenient to give here the references to the whole series.

I. On the Equation of Continuity (Thomson), Vol. ii. p. 282.

n. On the Equation of the Bounding Surface (Thomson), Vol. in. p. 80.

HL (Stokes) as above.

rV. Demonstration of a Fundamental Theorem (Stokes), Vol. lu. p. 209.

V. On the Vit Viva of a Liquid in motion (Thomson), Vol. rv. p. 90.

VI. On Waves (Stokes), Vol. rv. p. 219.

a II. 1

o

2 NOTES ON HYDRODYNAMICS.

Hydrodynamics have been already considered by Professor Thom-
son, in Notes I. and II. The object of the present Note is to form
the dynamical equations.

The fundamental hypothesis of Hydrostatics is, that the mutual
pressure of two contiguous portions of a fluid, separated by an
imaginary plane, is normal to the surface of separation. This
hypothesis forms in fact the mathematical definition of a fluid.
The equality of pressure in all directions is in reality not an inde-
pendent hypothesis, but a necess<ary consequence of the former.
A proof of this may be seen at the commencement of Prof. Miller's
Hydrostatics, The truth of our fundamental hypothesis, or at
least its extreme nearness to the truth, is fully established by
experiment. Some of the nicest processes in Physics depend upon
it ; for example, the determination of specific gravities, the use of
the level, the determination of the zenith by reflection from tlio
surface of mercury.

The same hypothesis is usually made in Hydrodynamics. If it
be assumed, the equality of pressure in all directions will follow as
a necessary consequence. This may be proved nearly as before,
the only difference being that now we have to take into account,
along with the impressed forces, forces equal and opposite to tlie
effective forces. The verification of our hypothesis is however
much more difficult in the case of motion, partly on account of the
mathematical difficulties of the subject, partly because the experi-
ments do not usually admit of great accuracy. Still, theory and
experiment have been in certain cases sufficiently compared to
shew that our hypothesis may be employed with very little error
in many important instances. There are however many pheno-
mena which point out the existence of a tangential force in fluids
in motion, analogous in some respects to friction in the case of
solids, but differing from it in this respect, that whereas in solids
friction is exerted at the surface, and between points which move
relatively to each other with a finite velocity, in fluids friction is
exerted throughout the mass, where the velocity varies continu-
ously from one point to another. Of course it is the same thing
to say that in such cases there is a tangential force along with a
normal pressure, as to say that the mutual pressure of two adjacent
elements of a fluid is no longer normal to their common surface.

ON THE DYNAMICAL EQUATIONS. 3

The subsidence of the motion in a cup of tea which has been
stirred may be mentioned as a familiar instance of friction, or,
which is the same, of a deviation from the law of normal pressure ;
and the absolute regularity of the surface when it comes to rest,
whatever may have been the nature of the previous disturbance,
may be considered as a proof that all tangential force vanishes
when the motion ceases.

It does not fall in with the object of this Note to enter into tlie
theory of the friction of fluids in motion*, and accordingly the
liypothcsis of normal pressure will be adopted. The usual nota-
tion will be employed, as in the preceding Notes. Consider the
elementary parallelepiped of fluid comprised between planes parallel
to the coordinate planes and passing through the points whose co-
ordinates are x, y, z, and x + dx, y-hdi/, z+ dz. Let X, F, Z be
the accelerating forces acting on the fluid at the point (x, y, z) ;
then, p and X being ultimately constant throughout the element,
the moving force parallel to x arising from the accelerating forces
which act on the element will be ultimately pXdxdy dz. The
diflference between the pressures, referred to a unit of surface, at
opposite points of the faces dy dz is ultimately dp/dx . cir, acting in
the direction of x negative, and therefore the diflference of the total
pressures on these faces is ultimately dp/dx . dx dy dz ; and the
pressures on the other faces act in a direction perpendicular to the
axis of X. The eflfective moving force parallel to x is ultimately
p . B'x/Df . dx dy dz, where, in order to prevent confusion, D is
used to denote diflferentiation when the independent variables are
supposed to be t, and three parameters which distinguish one
particle of the fluid from another, as for instance the initial coordi-
nates of the particle, while d is reserved to denote differentiation
when the independent variables are x, y, z, t We have therefore,
ultimately,

p -j^^dxdydz = (^pX - ^^j dxdydz,

* The reader who feels an interest in the subject may consult a memoir by
Navier, Mimoirea de VAcadimie^ tom. vi. p. 389 ; another by Poisson, Journal de
VEcoU Polytechniquf, Gahier xz. p. 139 ; an abstract of a memoir by M. de Saiut-
Venant, CompUs ReTidus, tom. zvii. (Nov. 1843) p. 1240; and a paper in the Cam-
bridge Philosophical Traruactiotit, Vol. viii. p. 287. [Ante, Vol. i. p. 75.]

1—2

4 NOTES ON HYDRODYNAMICS.

with similar equations for y and z. Dividing hy pdxdy dz, trans-
posing, and taking the limit, wo get

ldp_^_:^x ldp_.^_B^ l^^7^^ n\
pdx" De' pdy^ Uf' pdz"^ Lf ^ ^*

These are the dynamical equations which must be satisfied at
every point in the interior of the fluid mass ; but they arc not at
present in a convenient shape, inasmuch as they contain differen-
tial coefficients taken on two different suppositions. It will be
convenient to express them in terms of differential coefficients
taken on the second supposition, that is, that x, y, z, i are the
independent variables. Now DxjDt = u, and on the second suppo-
sition w is a function of <, x, y, z, each of which is a function of t
on the first supposition. We have, therefore, by Differential Cal-
culus,

Du Ifx du duDx duDy duDz

nr = I I- - -- ^^ A •

Dt JJt' dt^ dxDt^ dyDt^dzDt'

or, since by the definitions of w, v, w,

Dx_ Dy__ Dz_

. L^x du du du du

wehavc -^ = _+„_+„^^ + ,,_.

with similar equations for y and z.

Substituting in (1), we have

Idp _^ y. du du du ^ du
pdx dt dx dy dz

^<ip _Y__dv ^ dv dv dv
pdy" dt dx dy dz

I dp „ dw dw dw dw

- -j-= Z — V — u-^ v-j w-j-

p dz at ax ay dz

which is the usual form of the equations.

The equations (1) or (2), which are physically considered the
same, determine completely, so far as Dynamics alone are concerned,
the motion of each particle of the fluid. Hence any other purely
dynamical equation which we might set down would be identically
satisfied by (1) or (2). Thus, if we were to consider the fluid

(2),

ON THE DYNAMICAL EQUATIONS. 5

which at the time t is contained within a closed surface S, and set
down the last three equations of equilibrium of a rigid body be-
tween the pressures exerted on S, the moving forces due to the
accelerating forces acting on the contained fluid, and the effective
moving forces reversed, we should not thereby obtain any new
equation. The surface S inay be either finite or infinitesimal, as,
for example, the surface of the elementary parallelepiped with
which we started. Thus we should fall into error if we were to
set down these three equations for the parallelepiped, and think
that we had thereby obtained three new independent equations.

If the fluid considered be homogeneous and incompressible, p
is a constant. If it be heterogeneous and incompressible, p is a
function of x, y, z, t, and we have the additional equation Dp/Dt = 0,
or

P'i*4*«i">- ")•

which expresses the fact of the incompressibility. If the'^fluid be
elastic and homogeneous, and at the same temperature through-
out, and if moreover the change of temperature due to con-
densation and rarefaction be neglected, we shall have

p = kp(l+a0) (4),

where t is a given consent, depending on the nature of the gas,
and a a known constant which is the same for all gases [nearly].
The numerical value of a, as determined by experiment, is '00366,
being supposed to refer to the centigrade thermometer.

If the condensations and rarefactions of the fluid be rapid, we
may without inconsistency take account of the increase of tempe-
rature produced by compression, while we neglect the communica-
tion of heat from one part of the mass to another. The only
important problem coming under this class is that of sound. If we
suppose the changes in pressure and density small, and neglect tlio
squares of small quantities, we have, putting p^ , p^ for the values
of p, p in equilibrium,

p-p^ = kP:i£^ (5),

Pi Pi

JC being a constant which, as is well known, expresses the ratio of
the specific heat of the gas considered under a constant pressure

6 NOTES ON HYDRODYNAMICS.

to its specific heat when the vohime is constant. We are not,
however, obliged to consider specific heat at all; but we may if we
please regard K merely as the value of dlogjo/dlogp for p = p,,
p being that function of p which it is in the case of a mass of air
suddenly compressed or dilated. In whichever point of view we
regard if, the observation of the velocity of sound forms the best
mode of determining its numerical value.

It will be observed that in the proof given of equations (1) it
has been supposed that the pressure exerted by the fluid outside
the parallelepiped was exerted wholly on the fluid forming the
parallelepiped, and not partly on this portion of fluid and partly
on the fluid at the other side of the parallelepiped. Now, the
pressure arising directly from molecular forces, this imposes a re-
striction on the diminution of the parallelepiped, namely that its
.edges shall not become less than the radius of the sphere of activity
of the molecular forces. Consequently we cannot, mathematically
speaking, suppose the parallelepiped to be indefinitely diminished.
It is known, however, that the molecular forces are insensible at
sensible distances, so that we may suppose the parallelepiped to
become so small that the values of the forces, &c., for any point of
it, do not sensibly differ from their values for one of the corners,
and that all summations with respect to such elements may be
replaced without sensible error by integrations ; so that the values
of the several unknown quantities obtained from our equations by
diflbrentiation, integration, «&c. are sensibly correct, so far as this
ciiuse of error is concerned ; and that is all that we can ever attain
to in the mathematical expression of physical laws. The same
remarks apply as to the bearing on our reasoning of the supposition
of the existence of ultimate molecules, a question into which we
are not in the least called upon to enter.

There remains yet to be considered what may be called the
dynamical equation of the bounding surface.

Consider, first, the case of a fluid in contact with the surface of
a solid, which may be either at rest or in motion. Let P be a
point in the surface, about which the curvature is not infinitely
groat, 6) an element of the surface about P, PN a normal at P,
ilirected into the fluid, and let PN = h, Through N draw a plane
A perpendicular to PT, and project co on this plane by a circum-
scribing cylindrical surface. Suppose h greater than the radius r

ON THE DYNAMICAL EQUATIONS. 7

of the sphere of activity of the molecular forces, and likewise large
enough to allow the plane A not to cut the perimeter of (o. For
the reason already mentioned r will be neglected, and therefore no
restriction imposed on h on the first account. Let 11 be the pres-
sure sustained by the solid, referred to a unit of surface, 11 having
the value belonging to the point P, and let p be the pressure of
the fluid at N. Consider the element of fluid comprised between
G), its projection on the plane ^,and the projecting cylindrical sur-
face. The forces acting on this element are, first, the pressure of
the fluid on the base, which acts in the direction JVP, and is ulti-
mately equal to pen ; secondly, the pressure of the solid, which
ultimately acts along PiV and is equal to VL(o\ thirdly, the pressure
of the fluid on the cylindrical surface, which acts everywhere in a
direction perpendicular to PN ) and, lastly, the moving forces due
to the accelerating forces acting on the fluid ; and this whole sys-
tem of forces is in equilibrium with forces equal and opposite to
the efifective moving forces. Now the moving forces due to the
accelerating forces acting on the fluid, and the effective moving
forces, are both of the order (oh, and therefore, whatever may be
their directions, vanish in the limit compared with the force p'o),
if we suppose, as we may, that h vanishes in the limit. Hence we
get from the equation of the forces parallel to PN, passing to the
limit,

P = n (6),

p being the limiting value of p, or the result obtained by substi-
tuting in the general expression for the pressure the coordinates of
the point P for a?, y, z.

It should be observed that, in proving this equation, the forces
on which capillary phenomena depend have not been taken into
account And in fact it is only when such forces are neglected
that equation (6) is true.

In the case of a liquid with a free surface, or more generally in
the case of two fluids in contact, it may be proved, just as before,
that equation (6) holds good at any point in the surface, p, H being
the results obtained on substituting the coordinates of the point
considered for the general coordinates in the general expressions
for the pressure in the two fluids respectively. In this case, as
before, capillary attraction is supposed to be neglected.

[From the Philosophical Magazim^ VoL xxxii. p. 343, May, 1848.]

On the Constitution of the Luminifekous Ether.

The phenomeDon of aberration may be reconciled with the
undulatory theory of light, as I have already shown {Pldl. Mag.,
Vol. xxvii. p. 9*), without making the violent supposition that the
ether passes freely through the earth in its motion round the sun,
but supposing, on the contrary, that the ether close to the surface
of the earth is at rest relatively to the earth. This explanation
requires us to suppose the motion of the ether to be such, that the
expression usually denoted by udx -h vdu + wdz is an exact diffe-
rential. It becomes an interesting (luestion to inquire on what
physical properties of the ether this sort of motion can be explained.
Is it suflBcient to consider the ether as an ordinary fluid, or must
we have recourse to some property which does not exist in ordinary
fluids, or, to speak more correctly, the existence of which has not
been made manifest in such fluids by any phenomenon hitherto
observed ? I have already attempted to offer an explanation on
the latter supposition [PhiL Mag,, Vol. xxix. p. Gf).

In my paper last referred to, I have expressed my belief that
the motion for which ud^v + &c. is an exact differential, which
would take place if the ether were like an ordinary fluid, would be
unstable ; I now propose to prove the same mathematically, though
by an indirect method.

Even if we supposed light to arise from vibrations of the ether
accompanied by cimdensations and rare&ictions, analogous to the
vibrations of the air in the case of sound, since such vibrations
would be propjigated w ith about 10,000 times the velocity of the earth,

• .l7i/t', Vol. I. p. 131. t Antv, Vol. i. p. 153.

ON THE CONSTITUTION OF THE LUMINIFEBOUS ETHER. 9

we might without sensible error neglect the condensation of the
ether in the motion which we are considering. Suppose, then, a
sphere to be moving uniformly in a homogeneous incompressible
fluid, the motion being such that the square of the velocity may
be neglected. There are many obvious phenomena which clearly
point out the existence of a tangential force in fluids in motion,
analogous in many respects to friction in the case of solids. When
this force is taken into account, the equations of motions become
{Cambridge Philosophical Transactions, Vol. viil. p. 297*)

dp du fd'u d^u d'uK . .

dx ^d«+'*te + dp+rf?>l ^^^'

with similar equations for y and z. In these equations the square
of the velocity is omitted, according to the supposition made above,
p is considered constant, and the fluid is supposed not to be acted
on by external forces. We have also the equation of continuity

du dv dw ^^ .^v

dx dy dz

and' the conditions, (1) that the fluid at the surface of the spliere
shall be at rest relatively to the surface, (2) that the velocity shall
vanish at an infinite distance.

For my present purpose it is not requisite that the equations
such as (1) should be known to be true experimentally ; if they
were even known to be false they would be sufficient, for they may
be conceived to be true without mathematical absurdity. My
argument is this. If the motion for which udx-\- ,,,\^ an exact
differential, which would be obtained from the common equations,
v/ere stable, the motion which would be obtained from equations
(1) would approach indefinitely, as /x vanished, to one for which
ud^+ ... was an exact diff'erential, and therefore, for anything
proved to the contrary, the latter motion might be stable ; but if,
on the contrary, the motion obtained from (1) should turn out
totally different from one for which udx-\- ... is an exact diff*eren-
tial, the latter kind of motion must necessarily be unstable.

Conceive a velocity equal and opposite to that of the sphere
impressed both on the sphere and on the fluid. It is easy to prove

* Ante, Vol. i. p. 1)3.

10 ON THE CONSTTTUnOX OF THE LUinXIFEROUS ETHER.

that udx + . . . will or will not be an exact differential after the
velocity is impressed, according as it was or was not such before.
The sphere is thus reduced to rest, and the problem becomes one
of steady motion. The solution which I am about to give is
extracted from some researches in which I am engaged, but which
are not at present published. It would occupy far too much room
in this Magazine to enter into the mode of obtaining the solution :
but this is not necessary; for it will probably be allowed that
there is but one solution of the equations in the case proposed, as
indeed readily follows from physical considerations, so that it will
be suflBcient to give the result, which may be verified by differen-
tiation.

Let the centre of the sphere be taken for origin ; let the direc-
tion of the real motion of the sphere make with the axes angles
whose cosines are I, m, n, and let v be the real velocity of the
sphere ; so that when the problem is reduced to one of steady
motion, the fluid at a distance from the sphere is moving in the
opposite direction with a velocity v. Let a be the sphere's radius :
then we have to satisfy the general equations (1) and (2) with the
particular conditions

14 = 0, t; = 0, 1^= 0, when r = a (3) ;

u= — lv, t; = — mr, tt7 = — ni/, when r = oc (4),

r being the distance of the point considered from the centre of the
sphere. It will be found that all the equations are satisfied by
the following values,

3 a

p=U + 2^^ p {Ix + my + nz) ,

S /a a\ ,j , , /la* . 3a -\

with symmetrical expressions for v and w. 11 is here an arbitrary
constant, which evidently expresses the value of p at an infinite
distance. Now the motion defined by the above expressions does
not tend, as fi vanishes, to become one for which udx + ... is an
exact differential, and therefore the motion which would be
obtained by supposing ndx-h ... an exact differential, and applying
to the ether the common equations of hydrodynamics, would be

ON THE CONSTITUTION OF THE LUMINIFEROUS ETHER. 11

unstable. The proof supposes the motion in question to be steady ;
but such it may be proved to be, if the velocity of the earth be
regarded as uniform, and an equal and opposite velocity be con-
ceived impressed both on the earth and oh the ether. Hence the
8»ars ^ould appear to be displaced in a manner diflferent from that
expressed by the well-known law of aberration.

When, however, we take account of a tangential force in the
ether, depending, not on relative velocities, or at least not on rela-
tive velocities only, but on relative displacements, it then becomes
possible, as I have shewn (PhiL Mag., Vol. xxix. p. 6), to explain
not only the perfect regularity of the motion, but also the circum-
stance that iidx + . . . is an exact diflferential, at least for the ether
which occupies free space ; for as regards the motion of the ether
which penetrates the air, whether about the limits of the atmo-
sphere or elsewhere, I do not think it prudent, in the present
state of our knowledge, to enter into speculation ; I prefer resting
in the supposition that udx-^- ... w an exact differential. Accord-
ing to this explanation, any nascent irregularity of motion, any
nascent deviation from the motion for which udx 4- . . . is an exact
differential, is carried off into space, with the velocity of light, by
transversal vibrations, which as such are identical in their physical
nature with light, but which do not necessarily produce the sensa-
tion of light, either because they are too feeble, as they probably
would be, or because their lengths of wave, if the vibrations take
place in regular series, fall beyond the limits of the visible spec-
trum, or because they are discontinuous, and the sensation of light
may require the succession of a number of similar vibrations. It
is certainly curious that the astronomical phenomenon of the
aberration of light should afford an argument in support of the
theory of transversal vibrations.

Undoubtedly it does violence to the ideas that we should have
been likely to form d priori of the nature of the ether, to assert
that it must be regarded as an elastic solid in treating of the
vibrations of light. When, however, we consider the wonderful
simplicity of the explanations of the phenomena of polarization
when we adopt the theory of transversal vibrations, and the diffi-
culty, which to me at least appears quite insurmountable, of
explaining these phenomena by any vibrations due to the conden-

12 ON THE CONSTITUTION OF THE LUMINIFEROUS ETHER.

sation and rarefaction of an elastic fluid such as air, it seems
reasonable to suspend our judgement, and be content to learn from
phenomena the existence of forces which we should not beforehand
have expected. The explanations which I had in view are those
which belong to the geometrical part of the theory; but the
deduction, from dynamical calculations, of the laws which in the
geometrical theory take the place of observed facts must not be
overlooked, although here the evidence is of a much more compli-
cated character.

The following illustration is advanced, not so much as explain-
ing the real nature of the ether, as for the sake of offering a
plausible mode of conceiving how the apparently opposite proper-
ties of solidity and fluidity which we must attribute to the ether
may be reconciled.

Suppose a small quantity of glue dissolved in a little water, so
as to form a stiff jelly. This jelly forms in fact an elastic solid : it
may be constrained, and it will resist constraint, and return to its
original form when the constraining force is removed, by virtue of
its elasticity ; but if we constrain it too far it will break. Suppose
now the quantity of water in which the glue is dissolved to be
doubled, trebled, and so on, till at last we have a pint or a quart
of glue water. The jelly will thus become thinner and thinner,
and the amount of constraining force which it can bear without
being dislocated will become less and less. At last it will become
so far fluid as to mend itself again as soon as it is dislocated. Yet
there seems hardly sufficient reason for supposing that at a certain
stage of the dilution the tangential force whereby it resists con-
straint ceases all of a sudden. In order that the medium should
not be dislocated, and therefore should have to be treated as an
elastic solid, it is only necessary that the amount of constraint
should be very small. The medium would however be what we
should call a fluid, as regards the motion of solid bodies through it.
The velocity of propagation of normal vibrations in our medium
would be nearly the same as that of sound in water ; the velocity
of propagation of transversal vibrations, depending as it does on
the tangential elasticity, would become very small. Conceive now
a medium having similar properties, but incomparably rarer than
air, and wo have a niediuni such as we may conceive the ether to

ON THE CONSTITUTION OF THE LUMINIFEROUS ETHEK. 13

be, a fluid as regards the motion of the earth and planets 'through
it, an elastic solid as regards the small vibrations which constitute
light. Perhaps we should get nearer to the true nature of the
ether by conceiving a medium bearing the same relation to air
that thin jelly or glue water bears to pure water. The sluggish
transversal vibrations of our thin jelly are, in the case of the ether,
replaced by vibrations propagated with a velocity of nearly 200,000
miles in a second : we should expect, d priori, the velocity of
propagation of normal vibrations to be incomparably greater. This
is just the conclusion to which we are led quite independently,
from dynamical principles of the greatest generality, combined
with the observed phenomena of optics*.

* See the introdaction to an admirable memoir by Green, ** On the laws of the
Beflexion and Befraotion of Light at the common sarface of two non-crystallized
media.** Cambridge PhilosophiccU Traiuactions, Vol. vn. p. 1.

[From the Philosophical Transactions for 1848, p. 227.]

On the Theory of Certain Bands seen in the Spectrum.

[Read May 25, 1848.]

Some months ago Professor Powell communicated to me an
account of a new case of interference which he had discovered
in the course of some experiments on a fluid prism, requesting
at the same time my consideration of the theory. As the pheno-
menon is fully described in Professor Powell's memoir, and is
briefly noticed in Art. 1 of this paper, it is unnecessary here to
allude to it. It struck me that the theory of the phenomenon
was almost identical with that of the bands seen when a spectrum
is viewed by an eye, half the pupil of which is covered by a plate
of glass or mica. The latter phenomenon has formed the subject
of numerous experiments by Sir David Brewster, who has dis-
covered a very remarkable polarity, or apparent polarity, in the
banda The theory of these bands has been considered by the
Astronomer Royal in two memoirs " On the Theoretical Expla-
nation of an apparent new Polarity of Light,*' printed in the
Philosophical Transactions for 1840 (Part II.) and 1841 (Part I.).
In the latter of these Mr Airy has considered the case in which
the spectrum is viewed in focus, which is the most interesting
case, as being that in which the bands are best seen, and which is
likewise far simpler than the case in which the spectrum is viewed
out of focus. Indeed, from the mode of approximation adopted,
the former memoir can hardly be considered to belong to the
bands which formed the subject of Sir David Brewster's experi-
ments, although the memoir no doubt contains the theory of a
possible system of bands. On going over the theory of the bands
seen when the spectrum is viewed in focus, after the receipt of

BANDS SEEN IN THE SPECTRUM. 15

Professor Powell's letter, I was led to perceive that the intensity
of the light could be expressed in finite terms. This saves the
trouble of Mr Airy's quadratures, and aJlows the results to be
discussed with great facility. The law, too, of the variation of
the intensity with the thickness of the plate is very remarkable,
on account of its discontinuity. These reasons have induced me
to lay my investigation before the Royal Society, even though
the remarkable polarity of the bands has been already explained
by the Astronomer Royal. The observation of these bands seems
likely to become of great importance in the determination of the
refractive indices, and more especially the laws of dispersion, of
minerals and other substances which cannot be formed into prisms
which would exhibit the fixed lines of the spectrum.

Section I.

Explanation of the formation of the bands on the imperfect theory
of Interferences, Mode of calculating the number of bands
seen in a given part of the spectrum,

1. The phenomenon of which it is the principal object of the
following paper to investigate the theory, is briefly as follows.
Light introduced into a room through a horizontal slit is allowed
to pass through a hollow glass prism containing fluid, with its
refracting edge horizontal, and the spectrum is viewed through
a small telescope with its object-glass close to the prism. On
inserting into the fluid a transparent plate with its lower edge
horizontal, the spectrum is seen traversed from end to end by
very numerous dark bands, which are parallel to the fixed lines.
Under favourable circumstances the dark bands are intensely
black ; but in certain cases, to be considered presently, no bands
whatsoever are seen. When the plate is cut from a doubly re-
fracting crystal, there are in general two systems of bands seen
together; and when the light is analysed each system disappears
in turn at every quarter revolution of the analyser.

2. It is not difficult to see that the theory of these bands
must be almost identical with that of the bands described by
Sir David Brewster in the Report of the Seventh Meeting of the

16 ON THE THEORY OF CERTAIN

British Association, and elsewhere, and explained by Mr Airy in
the first part of the Philosophical Transactions for 1841. To
make this apparent, conceive an eye to view a spectrum through
a small glass vessel with parallel faces filled with fluid. The
vessel would not alter the appearance of the spectrum. Now con-
ceive a transparent plate bounded by parallel surfaces inserted
into the fluid, the plane of the plate being perpendicular to the
axis of the eye, and its edge parallel to the fixed lines of the
spectrum, and opposite to the centre of the pupil. Then w^e
should have bands of the same nature as those described by Sir
David Brewster, the only difiference being that in the present case
the retardation on which the existence of the bands depends is
the difference of the retardations due to the plate itself, and
to a plate of equal thickness of the fluid, instead of the ab-
solute retardation of the plate, or more strictly, the difiference
of retardations of the solid plate and of a plate of equal thick-
ness of air, contained between the produced parts of the bound-
ing planes of the solid plate. In Professor PowelFs experiment
the fluid fills the double ofiice of the fluid in the glass vessel and
of the prism producing the spectrum in the imaginary experiment
just described.

It might be expected that the remarkable polarity discovered
by Sir David Brewster in the bands which he has described, would
also be exhibited with Professor Poweirs apparatus. This anticipa-
tion is confirmed by experiment. With the arrangement of the
apparatus already mentioned, it was found that with certain
pairs of media, one being the fluid and the other the retarding
plate, no bands were visible. These media were made to exhibit
bands by using fluid enough to cover the plate to a certain
depth, and stopping by a screen the light which would otherwise
have passed through the thin end of the prism underneath the
plate.

3. Although the explanation of the polarity of the bands
depends on dififraction, it may be well to account for their for-
mation on the imperfect theory of interferences, in which it is
supposed that light consists of rays which follow the courses as-
signed to them by geometrical optics. It will thus readily appear
that the number of bands formed with a given plate and fluid,
and in a given part of the spectrum, has nothing to do with the

BANDS SEEN IN THE SPECTRUM. 17

form or magnitude of the aperture, whatever it be, which limits
the pencil that ultimately falls on the retina. Moreover, it seems
desirable to exhibit in its simplest shape the mode of calculating
the number of bands seen in any given case, more especially as
these calculations seem likely to be of importance in the deter-
mination of refractive indices.

4. Before the insertion of the plate, the wave of light be-
longing to a particular colour, and to a particular point of the slit,
or at least a certain portion of it limited by the boundaries of
the fluid, after being refracted at the two surfaces of the prism
enters the object-glass with an unbroken front. The front is here
called unbroken, because the modification which the wave suffers
at its edges is not contemplated. According to geometrical optics^
the light after entering the object-glass is brought to a point near
the principal focus, spherical aberration being neglected ; accord-
ing to the undulatory theory, it forms a small, but slightly dif-
fused image of the point from which it came. The succession of
these images due to the several points of the slit forms the image
of the slit for the colour considered, and the succession of coloured
images forms the spectrum, the waves for the different colours
covering almost exactly the same portion of the object-glass, but
differing from one another in direction.

Apart from aJl theory, it is certain that the image of a point or
line of homogeneous light seen with a small aperture is diSused.
As the aperture is gradually widened the extent of diffusion de-
creases continuously, and at last becomes insensible. The perfect
continuity, however, of the phenomenon shows that the true
and complete explanation, whatever it may be, of the narrow
image seen with a broad aperture, ought also to explain the dif-
fused image seen with a narrow aperture. The undulatory theory
explains perfectly both the one and the other, and even pre-
dicts the distribution of the illumination in the image seen
with an aperture of given form, which is what no other theory
has ever attempted.

As an instance of the effect of diffusion in an image, may
be mentioned the observed fact that the definition of a tele-
scope is impaired by contracting the apertura With a mode-
rate aperture, however, the diffusion is so slight as not to prevent

s. II. '^

18 ox THE THEORY OF CERTAIN

fine objects, such as the fixed lines of the spectrum, from being
well seen.

For the present, however, let us suppose the light entering
the telescope to consist of rays which are brought accurately to a
focus, but which nevertheless interfere. When the plate is in-
serted into the fluid the front of a wave entering the object-glass
will no longer be unbroken, but will present as it were Bt, fault, in
consequence of the retardation produced by the plate. Let R be
this retardation measured by actual length in air, p the retardation
measured by phase, M the retardation measured by the number of
waves' lengths, so that

then when 2f is an odd multiple of |, the vibrations produced by
the two streams, when brought to the same focus, will oppose
each other, and there will be a minimum of illumination; but
when JlT is an even multiple of i the two streams will combine,
and the illumination will be a maximum. Now M changes in
passing from one colour to another in consequence of the varia-
tions both of R and of X ; and since the different colours occupy
different angular positions in the field of view, the spectrum will
be seen traversed by dark and bright bands. It is nearly thus
that Mr Talbot has explained the bands seen when a spectrum
is viewed through a hole in a card which is half covered with a
plate of glass or mica, with its edge parallel to the fixed lines
of the spectrum. Mr Talbot however does not appear to have
noticed the polarity of the bands.

Let h, k he the breadths of the interfering streams; then
we may take

h sin -— vt, k sinl- vt — pj

to represent the vibrations produced at the focus by the two
streams respectively, which gives for the intensity /,

J=(A + icosp)" + (isinp)" = A" + A* + 2AA:cos/o (1),

which varies between the limits (h — k)* and (A + k)\

5. Although the preceding explanation is imperfect, for the
reason already mentioned, and does not account for the polarity,

BANDS SEEN IN THE SPECTRUM. 19

it is evident that if bands are formed at all in this way, the
number seen in a given part of the spectrum will be determined
correctly by the imperfect theory; for everything will recur, so
far as interference is concerned, when M is decreased or increased
by 1, and not before. This points out an easy mode of deter-
mining the number of bands seen in a given part of the spectrum.
For the sake of avoiding a multiplicity of cases, let an accelera-
tion be reckoned as a negative retardation, and suppose R positive
w^hen the stream which passes nearer to the edge of the prism is
retarded relatively to the other. From the known refractive
indices of the plate and fluid, and from the circumstances of the
experiment, calculate the values of R for each of the fixed lines

By C H of the spectrum, or for any of them that may be

selected, and thence the values of My by dividing by the known
values of X. Set down the results with their proper signs opposite
to the letters B, C ... denoting the rays to which they respectively
refer, and then form a table of differences by subtracting the
value of M for B from the value for C, the value for C from the
value for D, and so on. Let N be the number found in the table
of differences corresponding to any interval, as for example from
FtoO; then the numerical value of N, that is to say, i\r or — iV,
according as ^ is positive or negative, gives the number of bands
seen between F and O. For anything that appears from the
imperfect theory of the bands given in the preceding article, it
would seem that the sign of N was of no consequence. It will
presently be seen, however, that the sign is of great importance :
it will be found in fact that the sign + indicates that the second
arrangement mentioned in Art. 2 must be employed ; that is to
say, the plate must be made to intercept light from the thin end
of the prism, while the sign — indicates that the first arrange-
ment is required. It is hardly necessary to remark that, if N
should be fractional, we must, instead of the number of bands,
speak of the number of band-intervals and the fraction of an
interval.

Although the number of bands depends on nothing but the
values of N, the values of M are not without physical interest.
For M expresses, as we have seen, the number of waves' lengths
whereby one of the interfering streams is before or behind the
other. Mr Airy speaks of the formation of rings with the light cf

20 ON THE THEORY OF CERTAIN

a spirit-lamp when the retardation of one of the interfering
streams is as much as fifty or sixty waves' lengths. But in some
of Professor Powell's experiments, bands were seen which must
have been produced by retardations of several hundred waves'
lengtha This exalts our ideas of the regularity which must be
attributed to the undulations.

6. It appears then that the calculation of the number of
bands is reduced to that of the retardation R. As the calculation
of R is frequently required in physical optics, it will not be neces-
sary to enter into much detail on this point. The mode of per-
forming the calculation, according to the circumstances of the
experiment, will best be explained by a few examples.

Suppose the retarding plate to belong to an ordinary medium,
and to be placed so as to intercept light from the thin end of the
prism, and to have its plane equally inclined to the faces of the
prism. Suppose the prism turned till one of the fixed lines, as F^
is seen at a minimum deviation ; then the colours about F are
incident perpendicularly on the plate ; and all the colours may
without material error be supposed to be incident perpendicularly,
since the directions of the different colours are only separated by
the dispersion accompanying the first refraction into the fluid, and
near the normal a small change in the angle of incidence produces
only a very small change in the retardation. The dispersion
accompanying the first refraction into the fluid has been spoken of
as if the light were refracted from air directly into the fluid, which
is allowable, since the glass sides of the hollow prism, being
bounded by parallel surfaces, may be dispensed with in the expla-
nation. Let T be the thickness of the plate, /Li the refractive
index of the fluid, fi that of the plate ; then

B = {,i!-,.)T. (2).

If the plate had been placed so as to intercept light from the
thick end of the prism, we should have had —22= (/a' — /it) T,
which would have agreed with (2) if we had supposed T negative.
For the future T will be reckoned positive when the plate inter-
cepts light from the thin end of the prism, and negative when it
intercepts light from the thick end, so that the same formulse will
apply to both of the arrangements mentioned in Art. 2.

BANDS SEEN IN THE SPECTRUM. 21

If we put fi = l, the formula (2) will apply to the experiment
in which a plate of glass or mica is held so as to cover half the
pupil of the eye when viewing a spectrum formed in any manner,
the plate being held perpendicularly to the axis of the eye. The
eflfect of the small obliquity of incidence of some of the colours is
supposed to be neglected.

The number of bands which would be determined by means of
the formula (2) would not be absolutely exact, unless we suppose
the observation taken by receiving each fixed line in succession at
a perpendicular incidence. This may be eflfected in the following
manner. Suppose that we want to count the number of bands
between F and G, move the plate by turning it round a horizontal
axis till the bands about F are seen stationary ; then begin to
count from F, and before stopping at G incline the plate a little
till the bands about G are seen stationary, estimating the fractions
of an interval at F and G, if the bands are not too close. The
result will be strictly the number given by the formula (2). The
difference, however, between this result and that which would be
obtained by keeping the plate fixed would be barely sensible. If
the latter mode of observation should be thought easier or more
accurate, the exact formula which would replace (2) would be
easily obtained.

7. Suppose now the nearer face of the retarding plate made
to rest on the nearer inner face of the hollow prism, and suppose
one of the fixed lines, as F, to be viewed at a minimum deviation.
Let <f>, ^' be the angles of incidence and refraction at the first
surfiEkce of the fluid, t, i' those at the surface of the plate, 2e the
angle of the prism. Since the deviation of i'^ is a minimum, the
angle of refraction ^' ^ for F is equal to e, and the angle of inci-
dence ^ is given by sin <f> = fi^ sin <^'jp, and <f> is the angle of inci-
dence for all the colours, the incident light being supposed white.
The angle of refraction ^' for any fixed line is given by the equa-
tion sin <l> = l//jb . sin <^ = fi^/fi . sin e ; then i = 2e — <f)\ and i' is
known from the equation

fi smz =fi8uii (3).

The retardation is given by either of the formulae

i?=/r "°^*~.^"> (4).

smt ^ '

R= T (fi cos i' " fjL cost) (5).

22 ox THE THEORY OF CERTAIN

These formulae might be deduced from that given in Aiiy's
Tract, modified so as to suit the case in which the plate is im-
mersed in a fluid ; but either of them may be immediately proved
independently by referring everything to the wave's front and not
the ray.

By multiplying and dividing the second side of (5) by cos i,
and employing (3), we get

-B = 3rseci. {ji*' ^h)'^ T/Li' sec I versin (i — (6).

When the refractive indices of the plate and fluid are nearly
equal, the last term in this equation may be considered insensible,
so that it is not necessary to calculate % at all.

8. The formulas (2), (4), (5), (6) are of course applicable to the
ordinary ray of a plate cut from a uniaxal crystal. If the plate be
cut in a direction parallel to the axis, and if moreover the lower
edge be parallel to the axis, so that the axis is parallel to the
refracting edge of the prism, the formulae will apply to both rays.
If A^o* /t*, be the principal indices of refraction referring to the
ordinary and extraordinary rays respectively, fi in the case last
supposed must be replaced by fi^ for the bands polarized in a plane
perpendicular to the plane of incidence, and by fi^ for the bands
polarized in the plane of incidence. In the case of a plate cut
from a biaxal crystal in such a direction that one of the principal
axes, or axes of elasticity, is parallel to the refracting edge, the
same formulae will apply to that system of bands which is polarized
in the plane of incidence.

If the plate be cut from a biaxal crystal in a direction perpen-
dicular to one of the principal axes, and be held in the vertical
position, the formula (2) will apply to both systems of bands, if the
small effect of the obliquity be neglected. The formula would be
exact if the observations were taken by receiving each fixed line
in succession at a perpendicular incidence.

If the plate be cut from a uniaxal crystal in a direction per-
pendicular to the axis, and be held obliquely, we have for the
extraordinary bands, which are polarized in a plane perpendicular
to the plane of incidence,

It = t(^' J fi^^'"S^i--fi cost] (7),

BANDS SEEN IN THE SPECTRUM. 23

which is the same as the formula in Airy's Tract, only modified so
as to suit the case in which the plate is immersed in fluid, and
expressed in terms of refractive indices instead of velocities. If
we take a subsidiary angle j, determined by the equation

sinj = — sini (8),

the formula (7) becomes

-B= T (ji^ C08 J— fi cost) (9),

which is of the same form as (5), and may be adapted to logarith-
mic calculation if required by assuming fijfi = tan 0, The pre-
ceding formula will apply to the extraordinary bands formed by a
plate cut from a biaxal crystal perpendicular to a principal axis,
and inclined in a principal plane, the extraordinary bands being
understood to mean those which are polarized in a plane perpen-
dicular to the plane of incidence. In this application we must
take for /i„ /a, those two of the three principal indices of refraction
which are symmetrically related to the axis normal to the plate,
and to the axis parallel to the plate, and lying in the plane of
incidence, respectively; while in applying the formula (4), (5) or
(6) to the other system of bands, the third principal index must be
substituted for fi.

It is hardly necessary to consider the formula which would
apply to the general case, which would be rather complicated.

9. If a plate cut from a uniaxal crystal in a direction perpen-
dicular to the axis be placed in the fluid in an inclined position,
and be then gradually made to approach the vertical position, the
breadths of the bands belonging to the two systems will become
more and more nearly equal, and the two systems will at last
coalesce. This statement indeed is not absolutely exact, because
the whole spectrum cannot be viewed at once by light which
passes along the axis of the crystal, on account of the dispersion
accompanying the first refraction, but it is very nearly exact.
With quartz it is true there would be two systems of bands seen
even in the vertical position, on account of the peculiar optical
properties of that substance ; but the breadths of the bands
belonging to the two systems would be so nearly equal, that it
would require a plate of about one-fifth of an inch thickness to
give a diflference of one in the number of bands seen in the whole

24 ON THE THEORY OF CERTAIN

spectrum in the case of the two systems respectively. If the plate
shoTild be thick enough to exhibit both systems, the light would
of course have to be circularly analyzed to show one system by
itself.

Section II. — Investigation of the intensity of the light on the
complete theory of undulations, including the explanation of the
apparent polaHty of the hands,

10. The explanation of the formation of the bands on the im-
perfect theory of interferences considered in the preceding section
is essentially defective in this respect, that it supposes an annihi-
lation of light when two interfering streams are in opposition ;
whereas it is a most important principle that light is never lost by
interference. This statement may require a little explanation,
without which it might seem to contradict received ideas. It is
usual in fact to speak of light as destroyed by interference.
Although this is true, in the sense intended, the expression is
perhaps not very happily chosen. Suppose a portion of light
coming from a luminous point, and passing through a moderately
small aperture, to be allowed to fall on a screen. We know that
there would be no sensible illumination on the screen except
almost immediately in front of the aperture. Conceive now the
aperture divided into a great number of small elements, and
suppose the same quantity of light as before to pass through each
element, the only diflference being that now the vibrations in the
portions passing through the several elements are supposed to
have no relation to each other. The light would now be diffused
over a comparatively large portion of the screen, so that a point P
which was formerly in darkness might now be strongly illuminated.
The disturbance at P is in both cases the aggregate of the disturb-
ances due to the several elements of the aperture ; but in the first
case the aggregate is insensible on account of interference. It is
only in this sense that light is destroyed by interference, for the
total illumination on the screen is the same in the two cases ; the
effect of interference has been, not to annihilate any hght, but
only to alter the "distribution of the illumination," so that the
light, instead of being diffused over the screen, is concentrated in
front of the aperture.

BANDS SEEN IN THE SPECTRUM. 23

Now in the case of the bands considered in Section I., if we
suppose the plate extremely thin, the bands will be very broad ;
and the displacement of illumination due to the retardation being
small compared with the breadth of a band, it is evident, without
calculation, that at most only faint bands can be formed. This
particular example is sufficient to show the inadequacy of the im-
perfect theory, and the necessity of an exact investigation.

11. Suppose first that a point of homogeneous light is viewed
through a telescope. Suppose the object-glass limited by a screen
in which there is formed a rectangular aperture of length 21.
Suppose a portion of the incident light retarded, by passing
through a plate bounded by parallel surfaces, and having its edge
parallel to the length of the aperture. Suppose the unretarded
stream to occupy a breadth h of the aperture at one side, the re-
tarded stream to occupy a breadth k at the other, while an interval
of breadth 2g exists between the streams. In the apparatus men-
tioned in Section L, the object-glass is not limited by a screen, but
the interfering streams of light are limited by the dimensions of
the fluid prism, which comes to the same thing. The object of
supposing an interval to exist between the interfering streams, is
to examine the effect of the gap which exists between the streams
when the retarding plate is inclined. In the investigation the
effect of diffraction before the light reaches the object-glass of the
telescope is neglected.

Let be the image of the luminous point, as determined by
geometrical optics, / the focal length of the object-glass, or rather
the distance of from the object-glass, which will be a little greater
than the focal length when the luminous point is not very distant.
Let Che a, point in the object-glass, situated in the middle of the
interval between the two streams, and let the intensity be required
at a point Jf, near 0, situated in a plane passing through and
perpendicular to 00, The intensity at any point of this plane will
of course be sensibly the same as if the plane were drawn perpen-
dicular to the axis of the telescope instead of being perpendicular
to 00, Take 00 for the axis of z, the axes of x and y being
situated in the plane just mentioned, and that of y being parallel
to the length of the aperture. Let p, q be the co-ordinates of M ;
Xy y, z those of a point P in the front of a wave which has just
passed through the object-glass, and which forms part of a sphere

26 ON THE THEORY OF CERTAIN

with for its centre. Let c be the coefficient of vibration at the
distance of the object-glass; then we may take

^.prg-sin ^[vt-PM)dxdy. (a),

to represent the disturbance at M due to the element dxdy of the
aperture at P, P being supposed to be situated in the unretarded
stream, which will be supposed to lie at the negative side of the
axis of X, In the expression (a), it is assumed that the proper
multiplier of c/PM is l/X. This may be shown to be a necessary
consequence of the principle mentioned in the preceding article,
that light is never lost by interference ; and this principle follows
directly from the principle of vis viva. In proving that X"* is the
proper multiplier, it is not in the least necessary to enter into the
consideration of the law of the variation of intensity in a secondary
wave, as the angular distance from the normal to the primary wave
varies ; the result depends merely on the ajssumption that in the
immediate neighbourhood of the normal the intensity may be re-
garded as sensibly constant.

In the expression (a) we have

=/- -f (P^ + 9V)> nearly,

if we write / for V(/' +i^' + 2*)' It will be sufficient to replace
1/PM outside the circular function by 1/f, We may omit the con-
stant/under the circular function, which comes to the same thing
as changing the origin of t We thus get for the disturbance at il
due to the unretarded stream,

c f'ff r* 27r f 1 )

or on performing the integrations and reducing,

^^ IL sin M' 2£ sin -^ sin - (vt -P^- -^] (b)

For the retarded stream, the only difference is that we must
subtract li from vt, and that the limits of x are g and g + k. We
thus get for the disturbance at M due to this stream.

BANDS SEEN IN THE SPECTRUM. 2"

If we put for shortness t for the quantity under the last circular
function in (6), the expressions (6), (c) may be put under the forms
ttsin T, vsin (r — a), respectively ; and if / be the intensity, / will
be measured by the sum of the squares of the coefficients of sin t
and cos t in the expression

usmr + v sin (t — cr),

so that

I=u*-\-t^ -\-2uvcoQa, '

which becomes, on putting for u, v and a, their values, and putting

1^,'^'^-]' -'^ (•»)■

+ 2 sin^'' . sin '^-^ cos L - --^^ (4j + A + i) 1 L.(ll).

12. Suppose now that instead of a point we have a line of
homogeneous light, the line being parallel to the axis of y. The
luminous line is supposed to be a narrow slit, through which light
enters in all directions, and which is viewed in focus. Consequently
each element of the line must be regarded as an independent source
of light. Hence the illumination on the object-glass due to a por-
tion of the line which subtends the small angle ^ at the distance
of the object-glass varies as ^, and may be represented by A^.
Let the former origin be referred to a new origin 0' situated in
the plane a?y, and in the image of the line ; and let 97, q' be tho
ordinates of 0, M referred to 0', so that q = q' — v- ^^ order that
the luminous point considered in the last article may represent an
element of the luminous line considered in the present, we must
replace c' by Ad^ or Af~^drj; and in order to get the aggregate
illumination due to the whole line, we must integrate from a large
negative to a large positive value of 97, the largeness being esti-
mated by comparison with "Xf/L Now the angle 2'rrqlf\f changes
by TT when q changes by \//2l, which is therefore the breadth, in
the direction of y, of one of the diflfraction bands which would be
seen with a luminous point. Since I is supposed not to be ex-
tremely small, but on the contrary moderately large, the whole
system of diffraction bands would occupy but a very small portion
of the field of view in the direction of y, so that we may without

28 OK THE THEORY OF CEBTAIX

sensible error suppose the limits of 17 to be — x and + oo . We
have then

by taking the quantity under the circular function in place of 17 for
the independent variable. Now it is known that the value of the
last integral is tt, as will also presently appear, and therefore we
have for the intensity / at any point,

+ 2 sin'^j.mn'^ .cosL-^(4sr + A + Jfe) |....(12),
which is independent of q, as of course it ought to be.

13. Suppose now that instead of a line of homogeneous light
we have a line of white light, the component parts of which have
been separated, whether by refraction or by diffraction is imma-
terial, so that the different colours occupy different angular posi-
tions in the field of view. Let B^yfr be the illumination on the
object-glass due to a length of the line which subtends the small
angle fi, and to a portion of the spectrum which subtends the small
angle -^ at the centre of the object-glass. In the axis of x take a
new origin (7', and let f , p be the abscissae of 0\ M reckoned from
0", so that p=p' — ^^ In order that (12) may express the intensity
at M due to an elementary portion of the spectrum, we must
replace A by Bdy^, or B/'^d^; and in order to find the aggregate
illumination at M, we must integrate so as to include all values of
f which are sufficiently near to p' to contribute sensibly to the
illumination at M. It would not have been correct to integrate
using the displacement instead of the intensity, because the differ-
ent colours cannot interfere. Suppose the angular extent, in the
direction of x, of the system of diffraction bands which would be
seen with homogeneous light, or at least the angular extent of the
brighter part of the system, to be small compared with that of the
spectrum. Then we may neglect the variations of B and of X in
the integration, considering only those of f and p, and we may
suppose the changes of p proportional to those of ^ ; and we may
moreover suppose the limits of f to be — 00 and + 00 . Let p' be,
the value of p, and — tj that of dp/d^, when ^=p\ so that we may

BANDS SEEN IN THE SPECTRUM. 29

put p = p' + «- (p' — f ) ; and take p instead of f for the independent
variable. Then putting for shortness

^ = A,, ^ = ^/' ^'-^A^9-^h + k)^g^ (13),

we have for the intensity,
/= — ^^- 1 {sin* A^p+sin' k^p + 2 sin h,p . sin k^p . cos (/>' —ffp)] —^ .

Now I sin'A^p-^ = AJ sin'5-^ = 7rA^.

J -OP P J -09 C'

Similarly, I sin' k^p . -^ = Tri,.

J -00 J?

Moreover, if we replace

cos (p' — ff^p) by cos p' . cos g^p + sin p' . sin gj>,

the integral containing sin p will disappear, because the positive
and negative elements will destroy each other, and we have only to
find II?, where

w = I sin h^p . sin k^p . cos g^p . -^ .

J -00 _P

Now we get by differentiating under the integral sign,

dw f* . I -7 . dp

^ =s — I sm h^p . sm k^p . sm g^p . -^

1 r *

^ljJ^i^{ff. + K + k,)p + sm{g^--h^^k;)p

-sm{g, + h^''k)p-Bin(g^ + k,-h)p] -^ .

If

But it is well known that

r* sin w) ,

I -dp^ir^ or = — 7r,

according as « is positive or negative. If then we use F (a) to de-
note a discontinuous function of 8 which is equal to + 1 or — 1
according as « is positive or negative, we get

^'^[^(ff, + k, + k,)+F{g-h-k)-F(g,+h-K)-f'(g,+k-h,)\.

This equation gives

^-0, from fif, = -oo to g,=^-(.\ + k,)

30 ON THE THEORY OF CERTAIN

= 0, from5r^ = -(A,-*J to g^^ + ih^-^k)
= -2', from ff^ = h^^k, to ff,^h^ + k,

= 0, from g, = h^ + k^to g^=^(x>.

Now w vanishes when g^ is infinite, on account of the fluctuation
of the factor cos gjp under the integral sign, w^hence we get by
integrating the value of dw/dg, given above, and correcting the
integral so as to vanish for ^^ = — oo ,

w = 0, from 5r^= - 00 to 5^, = - (A^ + ij ;

w = I (A, + k, + g), from ff, = -{h, + k) to </, = - (A, ~ k) ;

w = "Trk^ or ■= irA,, (according as A, > &^ or A, < k^,)

from «/ = - (/t, - k,) tog, = + (/t, ~ k) ;

= J (^'/ + k,- g), from .7, = K - k, \ag, = h, + k,\
w; = 0, from flr, = A, + A-^ to ^r^ = co .

Substituting in the expression for the intensity, and putting
in (13) g^ = 'rrglXf, so that

^' = ^-45r-A-i (14),

wc get

I=j^{h + k) (15).

when the numerical value of ^r' exceeds h + k;

I-- P{h + k+{h + k'-Jg')oosp] (16),

when the numerical value of g' lies between h-^k and h'^k;

27?/ 9 7?/

I=-P(h + k'b2hQOsp\oT--~{h + k + 2kcosp') ...(17),

according as A or i is the smaller of the two, when the numerical
value of ^r' is less than h -^ k.

The discontinuity of the law of intensity is very remarkable.

w

BANDS SEEN IN THE SPECTRITM. 31

By supposing g^ = 0, A:^ = h^ in the expression for w^ and observ-
ing that these suppositions reduce V3 to

we get

a result already employed. This result would of course have been
obtained more readily by diflferentiating with respect to A^.

14. The preceding investigation will apply, with a very trifling
modification, to Sir David Brewster's experiment, in which the
retarding plate, instead of being placed in front of the object-glass
of a telescope, is held close to the eye. In this case the eye itself
takes the place of the telescope ; and if we suppose the whole
refraction to take place at the surface of the cornea, which will not
be far from the truth, we must replace / by the diameter of the
eye, and -^ by the angular extent of the portion of the spectrum
considered, diminished in the ratio of m to 1, 7M being the refrac-
tive index of the cornea. When a telescope is used in this experi-
ment, the retarding plate being still held close to the eye, it is
still the naked eye, and not the telescope, which must be assimi-
lated to the telescope considered in the investigation; the only
difference is that ^ must be taken to refer to the magnified, and
not the unmagnified spectrum.

Let the axis of a; be always reckoned positive in the direction
in which the blue end of the spectrum is seen, so that in the
image formed at the focus of the object-glass or on the retina,
according as the retarding plate is placed in front of the object-
glass or in front of the eye, the blue is to the negative side of the
red. Although the plate has been supposed at the positive side,
there will thus be no loss of generality, for should the plate be at
the negative side it will only be requisite to change the sign of p.

First, suppose p to decrease algebraically in passing from the
red to the blue. This will be the case in Sir David Brewster's
experiment when the retarding plate is held at the side on which
the red is seen. It will be the case in Professor Powell's experi-
ment when the first of the arrangements mentioned in Art. 2 is
employed, and the value, of ^ in the table of differences mentioned

\

32 ON THE THEORY OF CERTAIN

in Art. 5 is positive, or when the second arrangement is employed
and N is negative. In this case cr is negative, and therefore
g <" (A+A;), and therefore (15) is the expression for the inten-
sity. This expression indicates a uniform intensity, so that there
are no bands at all.

Secondly, suppose p to increase algebraically in passing from
the red to the blue. This will be the case in Sir David Brewster's
experiment when the retarding plate is held at the side on which
the blue is seen. It will be the case in Professor Powell's experi-
ment when the first arrangement is employed and N is negative,
or when the second arrangement is employed and N is positive.
In this case tj is positive ; and since vt varies as the thickness of
the plate, g' may be made to assume any value from — {^g + A + ifc)
to + 00 by altering the thickness of the plate. Hence, provided the
thickness lie within certain limits, the expression for the intensity
will be (16) or (17). Since these expressions have the same form
as (1), the magnitude only of the coefficient of cos p', as compared
with the constant term, being different, it is evident that the
number of bands and the places of the minima are given correctly
by the imperfect theory considered in Section I.

15. The plate being placed as in the preceding paragraph,
suppose first that the breadths ^, A; of the interfering streams are
equal, and that the streams are contiguous, so that ^r = 0. Then
the expression (17) may be dispensed with, since it only holds
good when ^ = 0, in which case it agrees with (16). Let T^ be
the value of the thickness T for which g = 0. Then T = corre-
sponds to 5r'=- (A +jfc), r= T; to 5r' = 0, and r=22; to ^' = A + i^;
and for values of T equidistant from T^, the values of g' are equal
in magnitude but of opposite signs. Hence, provided T be less
than 2r<„ there are dark and bright bands formed, the vividness of
the bands being so much the greater as T is more nearly equal to
y^, for which particular value the minima are absolutely black.

Secondly, suppose the breadths A, h of the two streams to be
equal as before, but suppose the streams separated by an interval
^g \ then the only difference is that 5^' = — (A + A:) corresponds to a
positive value, T, suppose, of T, If T be less than T^, or greater
than 2r^ — T,, there are no bands; but if T lie between T, and
2r^ — T, bands are formed, which are most vivid when T^T^/va
which case the minima are perfectly black.

BANDS SEEN IN THE SPECTRUM. 33

Thirdly, suppose the breadths h, h of the interfering streams
unequal, and suppose, as before, that the streams are separated by
an interval 2g ; then ^r' = — (A + A;) corresponds to a positive value,
T^ suppose, of T\ g' = ^(h-^k) corresponds to another positive
value, T^ suppose, of T, T^ lying between T, and T^, T^ being, as
before, the value of T which gives gr' = 0. As T increases from T^,
g becomes positive and increases from 0, and becomes equal to
h'-k when T= 22^ - T,, and to A + A when r= 2 7; - T,. When
T<T^ there are no bands. As T increases to T^ bands become
visible, and increase in vividness till T=T^, when the ratio of the
minimum intensity to the maximum becomes that of h — k to
h + Sk, or o{ k^h to k + 3A, according as A or A? is the greater of
the two, A, k. As T increases to 2T^— T^, the vividness of the
bands remains unchanged ; and as T increases from 2T^ — 2\ to
2T^ — T^, the vividness decreases by the same steps as it before in-
creased. When T = 2Tq^T^, the bands cease to exist, and no
bands are formed for a greater value of T,

Although in discussing the intensity of the bands the aperture
has been supposed to remain fixed, and the thickness of the plate
to alter, it is evident that we might have supposed the thickness
of the plate to remain the same and the aperture to alter. Since
«rx r, the vividness of the bands, as measured by the ratio of the
maximum to the minimum intensity, will remain the same when
T varies as the aperture. This consideration, combined with the
previous discussion, renders unnecessary the discussion of the effect
of altering the aperture. It will be observed that, as a general
rule, fine bands require a comparatively broad aperture in order
that they may be well formed, while broad bands require a narrow
aperture.

16. The particular thickness T^ may be conveniently called
the best thickness* This term is to a certain extent conventional,
since when h and k are unequal the thickness may range from T^
to 2T^ — T^ without any change being produced in the vividness of
the bands. The best thickness is determined by the equation

Now in passing from one band to its consecutive, p changes by 27r,
and { by e, if 6 be the linear breadth of a band; and for this small

s. II. ^

34 ox THE THEORY OF CERTAIN

change of f we may suppose the changes of p and f proportional,
or put — rfp/rf J = 27r/«. Hence the best aperture for a given thick-
ness is that for which

If ^ = and k = hf this equation becomes h = \//e.

The difference of distances of a point in the plane xy whose
coordinates are f, from the centres of the portions of the object-
glass which are covered by the interfering streams, is nearly

and if 8 be the change of f when this difference changes by X,

4gr + A + i = -^.

Hence, when the thickness of the plate is equal to the best thick-
ness, 6 =S, or the interval between the bands seen in the spectrum
is equal to the interval between the bands formed by the inter-
ference of two streams of light, of the colour considered, coming
from a luminous line seen in focus, and entering the object-glass
through two very narrow slits parallel to the axis of y, and situated
in the middle of the two interfering streams respectively. This
affords a ready mode of remembering and calculating the best
thickness of plate for a given aperture, or the best aperture for a
given thickness of plate.

17. According to the preceding explanation, no bands would
be formed in Sir David Brewster's experiment when the plate was
held on the side of the spectrum on which the red was seen. Mr
Airy has endeavoured to explain the existence of bands under such
circumstances*. Mr Airy appears to speak doubtfully of his ex-
planation, and in fact to offer it as little more than a conjecture to
account for an observed phenomenon. In the experiments of Mr
Talbot and Mr Airy, bands appear to have been seen when the
retarding plate was held at the red side of the spectrum; whereas
Sir David Brewster has stated that he has repeatedly looked for
the bands under these circumstances and has never been able to

^ Pkilosophieal Transaetiom for 1841, Part i. p. 6.

BANDS SEEN IN THE SPECTRUM. S3

find the least trace of them; and he considers the bands seen by
Mr Talbot and Mr Airy in this case to be of the nature of Newton's
rings. While so much uncertainty exists as to the experimental
circumstances under which the bands are seen when the retarding
plate is held at the red side of the spectrum, if indeed they are seen
at all, it does not seem to be desirable to enter into speculations as
to the cause of their existence.

[From the Cambridge and DMia Matktmaiioal Jourmadj VoL m. p. 209

{Xotember, 1&48}].

Notes ox Hydbodynamics.

IV. — Demonstration of a Fundamental Theorem.

THE0BE3C. Let the accelerating forces X, T, Z, acting on the
fluid, be such that Xdx + Ydy + Zdz is the exact differential d V
of a function of the coordinates. The function V mav also contain
the time t explicitly, but the differential is taken on the suppo-
sition that t is constant. Suppose the fluid to be either homo-
geneous and incompressible, or homogeneous and elastic, and of
the same temperature throughout, except in so far as the tem«
perature is altered by sudden condensation or rarefaction, so that
the pressure is a function of the density. Then if, either for the
whole fluid mass, or for a certain portion of it, the motion is at
occ instant such that vdx + vdy + wdz is an exact differential,
that expression 'will always remain an exact differential, in the
first case throughout the whole mass, in the second case throughout
the portion considered, a portion which will in general continually
change its position in space as the motion goes on. In particular,
the proposition is true when the motion begins from rest.

Two demonstrations of this important theorem will here be
given. The first is taken from a memoir by M. Cauchy, " M^-
moire sur la Th^orie des Ondes, &c." (M^m, des Savans Etran"
gere, Tom. i. (1827), p. 40). M. Cauchy has obtained three
first integrals of the equations of motion for the case in which
Xdx+ Ydy + Zdz is an exact differential, and in which the pres-
sure is a function of the density ; a case which embraces almost
all the problems of any interest in this subject M. Cauchy, it is

DEMONSTRATION OF A FUNDAMENTAL THEOREM. 37

true^ has only considered an incompressible fluid, in accordance
with the problem he had in hand, but his method applies to
the more general case in which the pressure is a function of the
density. The theorem considered follows as a particular conse-
quence from M. Cauchy's integrals. As however the equations
employed in obtaining these integrals are rather long, and the
integrals themselves do not seem to lead to any result of much
interest except the theorem enunciated at the beginning of this
article*, I have given another demonstration of the theorem,
which is taken from the Cambridge Philosophical Transactions
(Vol. VIII. p. 307 f). A new proof of the theorem for the case
of an incompressible fluid will be given by Professor Thomson in
this Journal.

First Demonstration. Let the time t and the initial co-
ordinates a, 6, c be taken for the independent variables ; and

let I — = P, p being by hypothesis a function of p. Since we

have, by the Differential Calculus,

dP^dPdx dPdy dPdz
da dx da dy da dz da'

with similar equations for b and c, we get from equations (1),
p. 124 (Notes on Hydrodynamics, No. ni.) [Ante, p. 4],

dV dP _ d^x dx d^y dy d^z dz^
da da df da d¥ da df da

d V dP _ d^x dx d^y dy d^z dz
li''ljb' dfdb^lfdb'^WdA

dV dP^d^xdx d^y dy d^z dz
dc dc df dc df dc df dc .

(1).

In these equations d^xjdf, dx/da, &c. have been written for L^xjDf,
DxjDa, &c., since the context will suflSciently explain the sense in
which the differential coefficients are taken. By differentiating
the first of equations (1) with respect to 6, the second with respect

* [See however the note at p. 47.]

t [Anttt Vol. I. p. 108. Although given abready in nearly the same form, the
demonstration is here retained, to avoid breaking the continuity of the present article.]

l^^

38 NOTES ON HTDRODTNAiaCS.

to a, and subtraciing, we get, after patting for dx/dt, dyldi^ dz/dt
their values u, v, vj,

d^u dx d^u dx d^v dy ^ dPv_ dy d^w dz
didbda''didadb^^cUt''cUdadb

d^fo dz ^ *^»

-^^=<> ^*)-

By treating the second and third, and then the third and first
of equations (1) as the first and second have been treated, we
should get two more equations, which with (2) would form a
symmetrical system. Now it is easily seen, on taking account of
the equations dxfdt = u, &c., that the first side of (2) is the dif-
ferential coe£5cient with respect to t of

dudx dudx dv dy ^ dv dy dw dz dw dz .^.

db da dadb dbda dadb db da da db

the differential coefficient in question being of course of the kind
denoted by D in No. ill. of these Notes. Hence the expression
(3) is constant for the same particle. Let u^, Vq, ir^ be the initial
velocities of the particle which at the time t is situated at the
point (a?, y, z) ; then if we observe that ar = a, y = 6, 5 = c, when
e = 0, we shall get from (2) and the two other equations of that
system,

du dx du dx dv dy dv dy dw dz dw dz du^ dv^

db da dadb db da da dh db da d^ db" db da

du dx du dx dv dy dv dy dw dz dw dz dv^ dw,,

dc db db dc dc db db dc dc db db do dc db

du dx ^du dx dv dy dv dy dw dz dw dz ^ dw^ du^

da dc dc da da do dc da da dc dc da da dc

....(4).

These are the three first integrals of the equations of motion
already mentioned. If we replace the differential coefficients
of Uy V and w, taken with respect to a, 6 and c, by differential
coefficients of the same quantities taken with respect to a?, y
and z, and differential coefficients of x, y and z taken with respect
to a, & and c, the first sides of equations (4) become

DEMONSTRATION OF A FUNDAMENTAL THEOREM.

39

•».

dz dy\ "

fdu dv\ (dy dx ^dy dx\ /dv ^ dw\ (dz dy __ dz dy\
\dy dx) \db da da dbj \dz dy) \db da da dbj

■^■i^

dvi\ /dx dz
\dx dz) \dh da

fdu dv\ (dy dx dy dx\ (dv
\dy dx) \dc db db dc) \dz

(dw ^ du\ (dx dz
\dx dz) \dc db

do\ (dy dx ^dy dx\ (dv
dx) \da dc dc da) \dz

(dw du\ (dx dz
\dx dz) \da dc

dx dz\
da db)

dw\ (dz dy
dy) \dc db

dxdz\
db dc)

(du
Ty

dw\ (dz dy ^ dz dy

dw\ (dz dy
dy ) \da dc

dx dz\
dc da)

dz^ dy\
dc da)

...(5).

Having put the first sides of equations (4) under the form (5),
we may solve the equations, regarding

d\i dv dv dw dw du
dy dx' dz dy* dx dz

as the unknown quantities. For this purpose multiply equations
(4) by dzldc, dz/da, dz/db, and add ; then the second and third
unknown quantities will disappear. Again, multiply by dx/dc,
dx/da, dx/db, and add; then the third and first will disappear.
Lastly, multiply by dy/dc, dy/da, dy/db, and add ; then the first
and second will disappear. Putting for shortness

dx dy dz dxdy dz dx dy dz dx dy dz
dadb dc dado db db dc da db da dc

dx dy dz ^ ^ ^ _ p /c\
dcdadb dc db da

we thus get

du dv

dy dx '

dv dw

dz dy

dw du

dx dz

1 {dz (d%
R\dc\db

1 (dx (du^
lR\dc\db

' B\ac\db

da)'^da\dc db)^db\da dc))

»

^,\ dx f dv^
da) da\dc

dwA dx fdw^ dwoM
db')'^db\da SI)

....(7).

dv^ ,dy(dv,_ dwA dy (dw, _
daJ'^daKdc dh)^db\da

'^)1j

40 NOTES ON HYDRODYNAMICS.

Consider the element of fluid which at first occupied the
rectangular parallelepiped formed by planes drawn parallel to
the coordinate planes through the points {a, b, c) and (a + da^
b + cB), c + dc). At the time t the element occupies a space
bounded by six curved surfaces, which in the limit becomes an
oblique-angled parallelepiped. The coordinates of the particle
which at first was situated at the point (a, b, c) are w, y, z at the
time i ; and the coordinates of the extremities of the three edges
of the oblique-angled parallelepiped which meet in the point
(a?, y, z) are

dx J , dy J dz J

s+^db, !f + %db, ''+%db;

dx t dy f dz 7

''+d-c^'' y+i'^'- '^dTo^-

Consequently, by a formula in analytical geometry, the volume
of the element which at first was dadbdc is R dadbdc at the
time t Hence if p^ ^ *t® initial density,

E = ^ (8).

P

From the mode in which this equation has been obtained, it is
evident that it can be no other than the equation of continuity
expressed in terms of a, 6, c and t as independent variables, and
integrated with respect to t

The preceding equations are true independently of any par-
ticular supposition respecting the motion. If the initial motion
be such that u^da + v^db + w^dc is an exact differential, and in
particular if the motion begin from rest, we shall have

du^ dv^ ^ dv^ duL ^ dto^ du^

db da ' dc db * da dc " *

and since by (8) jR cannot vanish, it follows from (7) that at any
time t

du dv ^^ dv dw __^ dw du __ ^
dy dx ^ dz dy ' dx dz ~ *

or udx + vdy + wdz is an exact differential.

DEMONSTRATION OF A FUNDAMENTAL THEOREM.

41

Since any instant may he taken for the origin of the time,
and t may be either negative or positive, it is evident that for
a given portion of the fluid udx + vdy + todz cannot cease to
be an exact differential if it is once such, and cannot become an
exact differential, not having been such previously.

Second Demonstration. The equations of motion in their
usual form are

1 dp ^ Y ^^ ^^ ^^ ^"
p dx dt dx dy dz

(9).

1 rfp _ ^ dv dv dv dv
p dy dt dx dy dz

\dp_y^dw ^ dw^ dw dw
p dz dt dx dy dz

Differentiating the first of these equations with respect to y and*
the second with respect to x, subtracting, and observing that by
hypothesis j? is a function of p, and Xdx + Ydy + Zdz is an exact
differential, we have

d d d
dx dy

d\ (du dv\ t^^^^t ^^ du
dz) \dy dx) dy dx dy dy

dw du du dv ^dv dv ^dw dv ^^
dy dz dxdx dx dy dx dz

(10).

According to the notation before employed,

d , d d , d
dt dx dy dz

means the same as DIDt Let

dy dz ^ * dz dx ' dx dy~ ^ ^'

then the last six terms of (10) become, on adding and subtracting
du dv*
dz dz '

c^du , ^dv „ o f^^ t ^A ''/
az az \ax ay J

• -r 3- would have done as well.
dx dy

42

NOTES ON HYDRODTNAMICS.

We thus get from (10), and the other two equations which would
be formed in a similar manner from (9),

-D©"' du , dv „ fdu

DT^Tz'' ^Tz""

dj,r

n

<D

dt

dv n , dw „,
dx ax

Deo"
Dt

dw „, . du
dy dy

fdu dt

\dx dy

/dv dw\ ,

\dy dz)

\dz dx)

(12).

Now the motion at any instant varying continuously from one
point of the fluid to another, the coefficients of «', ©", «'" on
the second sides of equations (12) cannot become infinite. Sup-
pose that when < = either there is no motion, or the motion
is such that udx + vdy + wdz is an exact differential This may
be the case either throughout the whole fluid mass or throughout
a limited portion of it. Then ©', oi", w"' vanish when ^ = 0. Let
£ be a superior limit to the numerical values of the coefficients
of 0)', tt)", ©"' on the second sides of equations (12) from the time
to the time t : then evidently g>', m\ ©'" cannot increase faster
than if they satisfied the equations

Dt

D<o
Dt

Da)
Dt

= 2/ (ft)' + ft>" + ft)'")

= L (ft)' + ft)" + 0)"') "

ft

—p^ = 2/ (ft)' + 0)" + ft)"')

(13).

instead of (12), vanishing in this case also when ^ = 0. By inte-
grating equations (13), and determining the arbitrary constants
by the conditions that ft)', ft)", ft)'" shall vanish when ^ = 0, we
should find the general values of ft)', ft)", and «'" to be zero.
We need not, however, take the trouble of integrating the
equations ; for, putting for shortness

ft) + ft) + ft) =11,

we get, by adding together the right and left-hand sides respect-
ively of equations (13),

DD.

Dt

=Ln.

DEMONSTRATION OF A FUNDAMENTAL THEOREM. 43

The integral of this equation is fl= (7€"; and since fl = when
t=sO, (7=0; therefore the general value of 12 is zero. But fl
is the sum of the three quantities ©', cd", o)'", which evidently
cannot be negative, and therefore the general values of o)', g>", g>'"
are each zero. Since, then, ©', ©", g>"' would have to be equal to
zero, even if they satisfied equations (13), they must a fortiori be
equal to zero in the actual case, since they satisfy equations (12),
which proves the theorem enunciated.

It is evident that it is for a given mass of fluid, not for the
fluid occupying a given poition of space, that the proposition is
true, since equations (12) contain the differential coefficients
Da/jDly &c. and not d(o jdt, &c. It is plain also that the same
demonstration will apply to negative values of t

If the motion should either be produced at first, or modified
during its course, by impulsive pressures applied to the surface
of the fluid, which of course can only be the case when the fluid
is incompressible, the proposition will still be true. In fact, the
change of motion produced by impulsive pressures is merely the
limit of the change of motion produced by finite pressures, when
the intensity of the pressures is supposed to increase and the
duration of their action to decrease indefinitely. The proposition
may however be proved directly in the case of impulsive forces
by using the equations of impulsive motion. If q be the impulsive
pressure, u^, v^y w^ the velocities just before, u, v, w the velocities
just after impact, it is very easy to prove that the equations of
impulsive motion are

No forces appear in these equations, because finite forces disappear
from equations of impulsive motion, and there are no forces which
bear to finite forces, like gravity, acting all over the mass, the
same relation that impulsive bear to finite pressures applied at
the surface; and the impulsive pressures applied at the sur-
face will appear, not in the general equations which hold good
throughout the mass, but in the particular equations which have
to be satisfied at the surface. The equations (14) are appli-
cable to a heterogeneous, as well as to a homogeneous liquid.
They must be combined with the equation of continuity of a
liquid, (equation (6), p. 286 of the preceding volume.) In the

44 NOTES ON HYDRODYNAMICS.

case under consideration, however, p is constant; and therefore
from (14)

(u — i/J At + (v - r J di/ + {w'- w^ dz

is an exact differential d{^qlp)\ and therefore if w^, v^j w^ be
zero, or if they be such that u^dx 4- v^dy + w^dz is an exact dif-
ferential dj>^y udx + vdy + wdz will also be an exact differential

d{(f>o-q/p)'

When udx + vdy + wdz is an exact differential dj>^ the expres-
sion for dP obtained from equations (9) is immediately integrable,
and we get

^-^-t-^m)<thm\ i'^).

supposing the arbitrary function of t introduced by integration
to be included in ^.

M. Cauchy's proof of the theorem just considered does not
seem to have attracted the attention which it deserves. It does
not even appear to have been present to Poisson's mind when
lie wrote his TraiU de Micaniqxie, The demonstration which
Poisson has given* is in fact liable to serious objections"!". Poisson
indeed was not satisfied as to the generality of the theorem. It
is not easy to understand the objections which he has raised J,
which after all do not apply to M. Cauchy's demonstration, in
which no expansions are employed. As Poisson gives no hint
where to find the "examples" in which he says the theorem
fails, if indeed he ever published them, we are left to conjecture.
In speaking of the developments of w, v, %o in infinite series of
exponentials or circular functions, suited to particular problems,
by which all the equations of the problem are satisfied, he re-
marks that one special character of such expansions is, not always
to satisfy the equations which are deduced from those of motion
by new differentiations. It is true that the equations which
would apparently be obtained by differentiation would not always
be satisfied ; for the differential coefficients of the expanded
functions cannot in general be obtained by direct differentiation,
that is by differentiating under the sign of summation, but must

♦ Trait€ de Micanique^ torn. ii. p. 688 (2nd edition).

t See Cambridge Philosophieal Trantactioru, Vol. Tni. p. 805. [Ante, Vol. i.p. 110.]

t Traits de Mieanique^ torn. ii. p. 690.

DEMONSTRATION OF A FUNDAMENTAL THEOREM. 45

be got from formulae applicable to the particular expansions*.
Poisson appears to have met with some contradiction, from
whence he concluded that the theorem was not universally true,
the contradiction probably having arisen from his having dif-
ferentiated under the sign of summation in a case in which it
is not allowable to do so.

It has been objected to the application of the theorem proved
in this note to the case in which the motion begins from rest,
that we are not at liberty to call udx + vdy -f wdz an exact dif-
ferential when w, v, and w vanish with t, unless it be proved that
if Wj, Vj, Wj be the results obtained by dividing w, v, w by the
lowest power of t occurring as a factor in u, v, w, and then putting
t = 0, u^dx + v^dy + w^dz is an exact differential. Whether we call
%idx + vdy + wdz in all cases an exact diflferential when w, v and w
vanish, is a matter of definition, although reasons might be as-
signed which would induce us to allow of the application of the
term in all such cases : the demonstration of the theorem is not
at all affected. Indeed, in enunciating and demonstrating the
theorem there is no occasion to employ the term exact differential
at all. The theorem might have been enunciated as follows.
If the three quantities du/dy — dv/dx, &c. are numerically equal
to zero when ^ = 0, they will remain numerically equal to zero
throughout the motion. This theorem having been established,
it follows as a result that when u, v, and w vanish with t,
Ujdx + v^dy + w^dz is an exact differential.

The theorem has been shewn to be a rigorous consequence
of the hypothesis of the absence of all tangential force in fluids
in motion. It now becomes a question, How far is the theorem
practically true, or nearly true ; or in what cases would it lead
to results altogether at variance with observation ?

As a general rule it may be answered that the theorem will
lead to results nearly agreeing with observation when the motion
of the particles which are moving is continually beginning from
rest, or nearly from rest, or is as good as if it were continually
beginning from rest ; while the theorem will practically fail when
the velocity of a given particle, or rather its velocity relatively

* See a paper '*0n the Critical Yalaes of the sums of Periodic Series," Cambridge
Philosophical TratuactioiUt Vol. txu. Part 5, [Ante, Vol. i. p. 236.]

44 NOTES ON HYDRODYNAMICS.

case under consideration, however, p is constant; and therefore
from (14)

(u — i/J At + (v — r J di/ + {w — w^ dz

is an exact differential d^—qjp)'^ and therefore if w^, v^j ^o ^®
zero, or if they be such that u^dx + vfiy + w^d« is an exact dif-
ferential d^^^ udx + vdy + wdz will also be an exact differential

d{^o-9/p)'

When ttdx + vdy + wdz is an exact differential d^, the expres-
sion for dP obtained from equations (9) is immediately integrable,
and we get

'-^-t-^mxho] (■=).

supposing the arbitrary function of t introduced by integration
to be included in ^.

M. Cauchy's proof of the theorem just considered does not
seem to have attracted the attention which it deserves. It does
not even appear to have been present to Poisson's mind when
he wrote his TraiM de Micaniqxie, The demonstration which
Poisson has given* is in fact liable to serious objections"!". Poisson
indeed was not satisfied as to the generality of the theorem. It
is not easy to understand the objections which he has raised J,
which after all do not apply to M. Cauchy's demonstration, in
which no expansions are employed. As Poisson gives no hint
where to find the "examples" in which he says the theorem
fails, if indeed he ever published them, we are left to conjecture.
In speaking of the developments of w, %\ w in infinite series of
exponentials or circular functions, suited to particular problems,
by which all the equations of the problem are satisfied, he re-
marks that one special character of such expansions is, not always
to satisfy the equations which are deduced from those of motion
by new differentiations. It is true that the equations which
would apparently be obtained by differentiation would not always
be satisfied ; for the differential coefficients of the expanded
functions cannot in general be obtained by direct differentiation,
that is by differentiating under the sign of summation, but must

• Train de Micanique^ torn. ii. p. 688 (2nd edition).

t See Cambridge Philofophieal Transaetioni, Vol. yixi. p. 805. [Ante^ Vol. i.p. 110.]

X Traits de Micanique, torn. ii. p. 690.

DEMONSTRATION OF A FUNDAMENTAL THEOREM. 45

be got from formulaB applicable to the particular expansions*.
Poisson appears to have met with some contradiction, from
whence he concluded that the theorem was not universally true,
the contradiction probably having arisen from his having dif-
ferentiated under the sign of summation in a case in which it
is not allowable to do so.

It has been objected to the application of the theorem proved
in this note to the case in which the motion begins from rest,
that we are not at liberty to call udx -f vdy + wdz an exact dif-
ferential when w, v, and w vanish with t, unless it be proved that
if Wj, Vj, w^ be the results obtained by dividing u, v, w by the
lowest power of t occurring as a factor in w, v, w, and then putting
t = 0, u^dx + v^dy + w^dz is an exact diflferential. Whether we call
udx + vdy + wdz in all cases an exact differential when u^ v and w
vanish, is a matter of definition, although reasons might be as-
signed which would induce us to allow of the application of the
term in all such cases : the demonstration of the theorem is not
at all affected. Indeed, in enunciating and demonstrating the
theorem there is no occasion to employ the term exdct differential
at all. The theorem might have been enunciated as follows.
If the three quantities du/dy — dv/dx, &c. are numerically equal
to zero when ^ = 0, they will remain numerically equal to zero
throughout the motion. This theorem having been established,
it follows as a result that when u, v, and w vanish with t,
Ujdx + v^dy + w^dz is an exact differential.

The theorem has been shewn to be a rigorous consequence
of the hypothesis of the absence of all tangential force in fluids
in motion. It now becomes a question, How far is the theorem
practically true, or nearly true ; or in what cases would it lead
to results altogether at variance with observation ?

As a general rule it may be answered that the theorem will
lead to results nearly agreeing w^ith observation when the motion
of the particles which are moving is continually beginning from
rest, or nearly from rest, or is as good as if it were continually
beginning from rest ; while the theorem will practically fail when
the velocity of a given particle, or rather its velocity relatively

* See a paper **0n the Critical Yalaes of the sums of Periodic Series," Cambridge
Philosophical Tratuactioiu, Vol. txu. Part 5. [AnUt Vol. i. p. 236.]

46 NOTES ON HYDBODTNAMICS.

to other particles, takes place for a long continuance in one
direction.

Thus, when a wave of sound is propagated through air, a new
set of particles is continually coming into motion ; or the motion,
considered with reference to the individual particles, is continually
beginning from rest. When a wave is propagated along the
surface of water, although the motion of the water at a distance
from the wave is not mathematically zero, it is insensible, so that
the set of particles which have got any sensible motion is con-
tinually changing. When a series of waves of sound is propa-
gated in air, as for example the series of waves coming from
a musical instrument, or when a series of waves is propagated
along the surface of water, it is true that the motion is not
continually beginning from rest, but it is as good as if it were
continually beginning from rest. For if at any instant the dis-
turbing cause were to cease for a little, and then go on again,
the particles would be reduced to rest, or nearly to rest, when
the first series of waves had passed over them, and they would
begin to move afresh when the second series reached them. Again,
in the case of the simultaneous small oscillations of solids and
fluids, when the forward and backward oscillations are alike, equal
velocities in opposite directions are continually impressed on the
particles at intervals of time separated by half the time of a com-
plete oscillation. In such cases the theorem would generally lead
to results agreeing nearly with observation.

If however water coming from a reservoir where it was sen-
sibly at rest were to flow down a long canal, or through a long
pipe, the tendency of friction being always the same way, the
motion would soon altogether differ from one for which
udx + vdy + wdz was an exact differential. The same would
be the case when a solid moves continually onwards in a fluid.
Even in the case of an oscillating solid, when the forward and
backward oscillations are not similar, as for example when a
cone oscillates in the direction of its axis, it may be con-
ceived that the tendency of friction to alter the motion of
the fluid in the forward oscillation may not be compensated in
the backward oscillation; so that, even if the internal friction
be very small, the motion of the fluid after several oscillations
may differ widely from what it would have been had there been

DEMONSTRATION OF A FUNDAMENTAL THEOREM, 47

absolutely no friction. I do not expect that there would be this
wide difiference; but still the actual motion would probably not
agree so well with the theoretical, as in those cases in which
the forward and backward oscillations are alike. By the theo-
retical motion is of course meant that which would be obtained
from the common theory, in which friction is not taken into
account.

It appears from experiments on pendulums that the eflfect
of the internal friction in air and other gases is greater than
might have been anticipated. In Dubuat's experiments on spheres
oscillating in air the spheres were large, and the alteration in
the time of oscillation due to the resistance of the air, as de-
termined by his experiments, agrees very nearly with the result
obtained from the common theory. Other philosophers, however,
having operated on smaller spheres, have found a considerable
discrepancy, which is so much the greater as the sphere employed
is smaller. It appears, moreover, from the experiments of Colonel
Sabine, that the resistance depends materially upon the nature
of the gas. Thus it is much greater, in proportion to the density,
in hydrogen than in air.

Note referred to at p. 37.

[It may be noticed that two of Helmholtz's fundamental pro-
positions respecting vortex motion* follow immediately from
Cauchy's integrals ; or rather, two propositions the same as those
of Helmholtz merely generalized so as to include elastic fluids
follow from Cauchy's equations similarly generalized.

On substituting in (7) for It the expression given by (8), and
introducing the notation of angular velocities, as in (11), equa-
tions (7) become

, p fdx , , dx „ . dx ,„\

6)

,„ p (dz , dz if.dz „\

♦ Grelle*B JowmaX^ Vol. lv. p. 25.

(a).

48 MOTES ON HTDRODTNAMICS.

We see at once from these equations that if

da db dc

/- //- n, (b),

^0 ^Q ^0

, dx dy dz , .

then — , = -^, = --, (c),

o> ca o>

but (6) are the differential equations of the system of vortex lines
at the time 0, and (c), as being of the form

dx _^dy _ dz

are the differential equations of the loci of the particles at the
time t which at the time formed the vortex lines respectively.
But when we further take account of the values of P, Q, iJ, as
exhibited in (6), we see that (c) are also the differential equations
of the system of vortex lin^ at the time t Therefore the same
loci of particles which at one moment are vortex lines remain
vortex lines throughout the motion.

Let n^ be the resultant angular velocity at the time of a
particle P^ which at the time f is at P, and has O for its angular
velocity ; let ds^ drawn from P^ be an element of the vortex line
at time passing through P,,, and ds the element of the vortex
line passing through P at the time t which consists of the same
set of particles. Then each member of equations (6) is equal
to dsJCl^ and each member of equations (c) equal to ds/n. Hence
we get from any one of equations (a)

pds ^ p^ds^
« iK '
Let Aq be the area of a perpendicular section, at P^, of a vortex
thread containing the vortex line passing through P^ at the time 0,
a vortex thread meaning the portion of fluid contained within
an elementary tube made up of vortex lines ; then by what pre-
cedes the same set of particles will at the time t constitute a
vortex thread passing through P; let J. be a perpendicular section
of it passing through P at the time t, and draw two other per-
pendicular sections passing respectively through the other ex-
tremities of the elements ds^ and ds. Then if we suppose, as
we are at liberty to do, that the linear dimensions of A^ are
indefinitely small compared with the length ds^, we see at once
that the elements of volume comprised between the tube and

DEMONSTRATION OF A FUNDAMENTAL THEOREM. 49

the pair of sections at the time and at the time t respectively
contain ultimately the same particles, and therefore

pAds = poA^ds^,

whence ilA^^il^A^,

or the angular velocity of any given particle varies inversely
as the area of a perpendicular section through it of the vortex
thread to which it belongs, and that, whether the fluid be incom-
pressible or elastic.

When these results are deduced from Cauchy's integrals,
the state of the fluid at any time is compared directly with its
state at any other time; in Helmholtz's method the state at
the time t is compared with the state at the time t + dt, and
so on step by step.

A remaining proposition of Helmholtz's, that along a vortex
line the angular velocity varies at any given time inversely as
the perpendicular section of the vortex thread, has no immediate
relation to Cauchy's integrals, inasmuch as it relates to a com-
parison of the state of the fluid at different points at the same
moment. It may however be convenient to the reader that the
demonstration, which is very brief, should be reproduced here.

We have at once from (11)

do)' cZg)" d<o" __ ^

dx dy dz *

and consequently

where the integration extends over any arbitrary portion of the
fluid. This equation gives

where the double integrals extend over the surface of the space
in question. The latter equation again becomes by a well-known
transformation

//

ftcos^dfif=0,

where dS is an element of the surface of the space, and the

S.' II. ^

50 NOTES ON HYDRODYNAMICS.

angle between the instantaneous axis and the normal to the
surface drawn outwards.

Let now the space considered be the portion of a vortex thread
comprised between any two perpendicular sections, of which let
A and A' denote the areas. All along the side of the tube 5= 90°,
and at the two ends = 180* and == 0^ respectively, and therefore
if XI' denotes the angular velocity at the second extremity of the
portion of the vortex thread considered

ilA = n'A\

which proves the theorem.]

[From the Philosophical Ilagasine, VoL xxxin., p. 349 (^Xovemhery 1848.)]

On a difficulty in the Theory of Sound.

The theoretical determination of the velocity of sound has
recently been the occasion of a discussion between Professor
Challis and the Astronomer Royal. It is not my intention to
enter into the controversy, but merely to consider a very re-
markable difficulty which Professor Challis has noticed in con-
nexion with a known first integral of the accurate equations of
motion for the case of plane waves.

The difficulty alluded to is to be found at page 496 of the
preceding volume of this Magazine*. In what follows I shall use
Professor Challis's notktion.

* [The followmg qnotation wiU enffioe to pnt the reader in possession of the
apparent contradiction discovered by Professor ChaUis. It should be stated that
the inyestigation relates to plane waves, propagated in the direction of z, and that
the pressure is supposed to vary as the density.

** The function / being quite arbitrary, we may give it a particular form. Let,

therefore,

2ir

tr =w sin — {z - (a + 1£?) /}.

A

This equation shows that at any time t^ we shall have tr = at points on the axis
of z, for which

z-(a + tr)fi=--,

or z=afi+-2-.

At the same time ir will have the value stm at points of the axis for whicli

z-(a+m)fi=^^-i^X,

or '=a^+-2-"*" »"4*

ifc— ^

42

NOTES ON HTDRODTNAMICS.

We thus get from (10), and tlie other two equations which would
be formed in a similar manner from (9),

D<o^ __du , dv „ /du dv\ ,„
JDt " dz dz \dx dy)

dw\ ,
Jo,

D(o ^dv ,, . dw ,„
dt dx dx

-(

dv

dy dz

Bi"

f(o

__ dw ,„ du , /dw du
Dt dy dy \dz dx

y\

(12).

Now the motion at any instant varying continuously from one
point of the fluid to another, the coefficients of ©', «", m" on
the second sides of equations (12) cannot become infinite. Sup-
pose that when ^ = either there is no motion, or the motion
is such that udx + vdy + wdz is an exact differential. This may
be the case either throughout the whole fluid mass or throughout
a limited portion of it. Then w', a>", w"' vanish when ^ = 0. Let
Z be a superior limit to the numerical values of the coefficients
of a>', ft)", ft)"' on the second sides of equations (12) from the time
to the time t : then evidently ft)', ft)", ft)"' cannot increase faster
than if they satisfied the equations

i)ft)"'
Z)ft)'

m

Jft)
Dt

-j^ = L{co + (o + ft) )

— y^ - =/y(ft)+ft)+ft)).

ff

= L {(o' + ft)" + ft)"')

(13).

instead of (12), vanishing in this case also when ^ = 0. By inte-
grating equations (13), and determining the arbitrary constants
by the conditions that ft)', ft)", co" shall vanish when < = 0, we
should find the general values of o)', ft)", and o)'" to be zero.
We need not, however, take the trouble of integrating the
equations ; for, putting for shortness

ft) -f ft) -f ft) =12,

we get, by adding together the right and left-hand sides respect-
ively of equations (13),

Dt " ^"-

DEMONSTRATION OF A FUNDAMENTAL THEOREM. 43

The integral of this equation is Xl= Ce"; and since 11 = when
t = 0, (7=0; therefore the general value of ft is zero. But ft
is the sum of the three quantities ©', o)", ©"', which evidently
cannot be negative, and therefore the general values of ©', ©", ©"'
are each zero. Since, then, w', ©", ©"' would have to be equal to
zero, even if they satisfied equations (13), they must a fortiori be
equal to zero in the actual case, since they satisfy equations (12),
which proves the theorem enunciated.

It is evident that it is for a given mass of fluid, not for the
fluid occupying a given portion of space, that the proposition is
true, since equations (12) contain the differential coefficients
Ba/Dt, &c. and not dto'/dt, &c. It is plain also that the same
demonstration will apply to negative values of t

If the motion should either be produced at first, or modified
during its course, by impulsive pressures applied to the surface
of the fluid, which of course can only be the case when the fluid
is incompressible, the proposition will still be true. In fact, the
change of motion produced by impulsive pressures is merely the
limit of the change of motion produced by finite pressures, when
the intensity of the pressures is supposed to increase and the
duration of their action to decrease indefinitely. The proposition
may however be proved directly in the case of impulsive forces
by using the equations of impulsive motion. If q be the impulsive
pressure, u^, t?^, w^^ the velocities just before, w, v, w the velocities
just after impact, it is very easy to prove that the equations of
impulsive motion are

No forces appear in these equations, because finite forces disappear
from equations of impulsive motion, and there are no forces which
bear to finite forces, like gravity, acting all over the mass, the
same relation that impulsive bear to finite pressures applied at
the surface; and the impulsive pressures applied at the sur-
face will appear, not in the general equations which hold good
throughout the mass, but in the particular equations which have
to be satisfied at the surface. The equations (14) are appli-
cable to a heterogeneous, as well as to a homogeneous liquid.
They must be combined with the equation of continuity of a
liquid, (equation (G), p. 286 of the preceding volume.) In the

44 NOTES ON HYDRODYNAMICS.

case under consideration, however, p is constant ; and therefore
from (14)

(u — Wq) tic + (v — t'o) dy + (w — Wo) ^^

is an exact diflferential d{'-q/p)\ and therefore if u^, v^, w^ be
zero, or if they be such that u^dx 4- v^dy + w^dz is an exact dif-
ferential d(f}^, udx + vdy + wdz will also be an exact differential

d{(f>o''q/p)'

When udx + vdy + wdz is an exact differential d^, the expres-
sion for dP obtained from equations (9) is immediately integrable,
and we get

supposing the arbitrary function of t introduced by integration
to be included in ^.

M. Cauchy's proof of the theorem just considered does not
seem to have attracted the attention which it deserves. It does
not even appear to have been present to Poisson's mind when
he wrote his Traits de iUcanique, The demonstration which
Poisson has given* is in fact liable to serious objections'f'. Poisson
indeed was not satisfied as to the generality of the theorem. It
is not easy to understand the objections which he has raised J,
which after all do not apply to M. Cauchy's demonstration, in
which no expansions are employed. As Poisson gives no hint
where to find the "examples" in which he says the theorem
fails, if indeed he ever published them, we are left to conjecture.
In speaking of the developments of w, v, w in infinite series of
exponentials or circular functions, suited to particular problems,
by which all the equations of the problem are satisfied, he re-
marks that one special character of such expansions is, not always
to satisfy the equations which are deduced from those of motion
by new differentiations. It is true that the equations which
would apparently be obtained by differentiation would not always
be satisfied ; for the differential coeflScients of the expanded
functions cannot in general be obtained by direct differentiation,
that is by differentiating under the sign of summation, but must

♦ Traits de MScaniquty torn. n. p. 688 (2nd edition).

t See Cambridge Philosophical Transactions, Vol. Tin. p. 806. [Ante, Vol. i.p. 110.]

X Traiti de Micanique, torn. n. p. 690.

DEMONSTRATION OF A FUNDAMENTAL THEOREM. 45

be got from formulae applicable to the particular expansions*.
Poisson appears to have met with some contradiction, from
whence he concluded that the theorem was not universally true,
the contradiction probably having arisen from his having dif-
ferentiated under the sign of summation in a case in which it
is not allowable to do so.

It has been objected to the application of the theorem proved
in this note to the case in which the motion begins from rest,
that we are not at liberty to call udx + vdy + wdz an exact dif-
ferential when w, r, and w vanish with t, unless it be proved that
if Wj, Vj, w^ be the results obtained by dividing u, t?, w by the
lowest power of t occurring as a factor in u, v, Wy and then putting
^ = 0, u^dx + v^dy + w^dz is an exact diflferential. Whether we call
udx 4- vdy + wdz in all cases an exact differential when u, v and w
vanish, is a matter of definition, although reasons might be as-
signed which would induce us to allow of the application of the
term in all such cases : the demonstration of the theorem is not
at all affected. Indeed, in enunciating and demonstrating the
theorem there is no occasion to employ the term exdct differential
at all. The theorem might have been enunciated as follows.
If the three quantities du/dy — dv/dx, &c. are numerically equal
to zero when ^ = 0, they will remain numerically equal to zero
throughout the motion. This theorem having been established,
it follows as a result that when u, v, and w vanish with t,
u^dx-^-v^dy-k-w^dz is an exact differential.

The theorem has been shewn to be a rigorous consequence
of the hypothesis of the absence of all tangential force in fluids
in motion. It now becomes a question, How far is the theorem
practically true, or nearly true ; or in what cases would it lead
to results altogether at variance with observation ?

As a general rule it may be answered that the theorem will
lead to results nearly agreeing with observation when the motion
of the particles which are moving is continually beginning from
rest, or nearly from rest, or is as good as if it were continually
beginning from rest ; while the theorem will practically fail when
the velocity of a given particle, or rather its velocity relatively

* See a paper ''On the Critical Valu^ of the sums of Periodic Series," Cambridge
Philosophical Transactions^ Vol. tiu. Part 5. [Ante^ Vol. i. p. 236.]

[From the Transactions of the Cambridge Philosophical Society^

Vol. VIII. p. 642.]

On the Formation of the Central Spot of Newton's
Rings beyond the Critical Angle.

[Read Deceinher 11, 1848.]

When Newton's Rings are formed between the under surface
of a prism and the upper surface of a lens, or of another prism
with a slightly convex face, there is no diflBcultj in increasing the
angle of incidence on the under surface of the first prism till it
exceeds the critical angle. On viewing the rings formed in this
manner, it is found that they disappear on passing the critical
angle, but that the central black spot remains. The most obvious
way of accounting for the formation of the spot under these cir-
cumstances is, perhaps, to suppose that the forces which the
material particles exert on the ether extend to a small, but sen-
sible distance from the surface of a refracting medium ; so that in
the case under consideration the two pieces of glass are, in the
immediate neighbourhood of the point of contact, as good as a
single uninterrupted medium, and therefore no reflection takes
place at the surfaces. This mode of explanation is however liable
to one serious objection. So long as the angle of incidence falls
short of the critical angle, the central spot is perfectly explained,
along with the rest of the system of which it forms a part, by
ordinary reflection and refraction. As the angle of incidence
gradually increases, passing through the critical angle, the ap-
pearance of the central spot changes gradually, and but slightly.
To account then for the existence of this spot by ordinary re-
flection and refraction so long as the angle of incidence falls short

FORMATION OF THE CENTRAL SPOT OF NEWTON'S RINGS, &C. 57

of the critical angle, but by the finite extent of the sphere of
action of the molecular forces when the angle of incidence exceeds
the critical angle, would be to give a discontinuous explanation to
a continuous phenomenon. If we adopt the latter mode of expla-
nation in the one case we must adopt it in the other, and thus
separate the theory of the central spot from that of the rings,
which to all appearance belong to the same system ; although the
admitted theory of the rings fully accounts likewise for the exist-
ence of the spot, and not only for its existence, but also for
some remarkable modifications which it undergoes in certain cir-
cumstances*.

Accordingly the existence of the central spot beyond the criti-
cal angle has been attributed by Dr Lloyd, without hesitation, to
the disturbance in the second medium which takes the place of
that which, when the angle of incidence is less than the critical
angle, constitutes the refracted Hghtf. The expression for the in-
tensity of the light, whether reflected or transmitted, has not how-
ever been hitherto given, so far as I am aware. The object of the
present paper is to supply this deficiency.

In explaining on dynamical principles the total internal reflec-
tion of light, mathematicians have been led to an expression for
the disturbance in the second medium involving an exponential,
which contains in its index the perpendicular distance of the point
considered from the surface. It follows from this expression that
the disturbance is insensible at the distance of a small multiple of
the length of a wave from the surface. This circumstance is all that
need be attended to, so far as the refracted light is concerned, in
explaining total internal reflection ; but in considering the theory
of the central spot in Newton's Rings, it is precisely the super-
ficial disturbance just mentioned that must be taken into account.
In the present paper I have not adopted any special dynamical
theory : I have preferred deducing my results from Fresnel's for-
muke for the intensities of reflected and refracted polarized light,
which in the case considered became imaginary, interpreting these
imaginary expressions, as has been done by Professor O'Brien J,

* I aUude especiaUy to the phenomena described by Mr Airy in a paper printed
in the fourth volume of the Cambridge Philosophical Transactions^ p. 409.

t Report on the present state of Physical Optics. Ueports of the British
Association^ Vol. ni. p. 810.

X Cambridge Philosophical Transactions^ Vol. viu. p. 20.

58 ox THE FORMATION OF THE CENTRAL SPOT OF

in the way in which general dynamical considerations show that
they ought to be interpreted.

By means of these expressions, it is easy to calculate the in-
tensity of the central spot. I have only considered the case in
which the first and third media are of the same nature : the
more general case does not seem to be of any particular interest.
Some conclusions follow from the expression for the intensity,
relative to a slight tinge of colour about the edge of the spot,
and to a difference in the size of the spot according as it is seen by
light polarized in, or by light polarized perpendicularly to the plane
of incidence, which agree with experiment.

1. Let a plane wave of light be incident, either externally or
internally, on the surface of an ordinary refracting medium, sup-
pose glass. Regard the surface as plane, and take it for tlie plane
xy ; and refer the media to the rectangular axes of x, y, z, the
positive part of the last being situated in the second medium,
or that into which the refraction takes place. Let I, m, n be the
cosines of the angles at which the normal to the incident wave,
measured in the direction of propagation, is inclined to the
axes; so that m»0 if we take, as we are at liberty to do,
the axis of y parallel to the trace of the incident wave on the
reflecting surface. Let F, F], V denote the incident, reflected,
and refracted vibrations, estimated either by displacements or
by velocities, it does not signify which; and let a, a^, a' denote
the coefficients of vibration. Then we have the following possible
system of vibrations :

V « a cos — (vf — te — m)y

At

V"^ = a^cos-— (v^ — te + nz), \ (A).

V «= a' cos —7- {vt — tx — nz),

In these expressions v, v are the velocities of propagation, and
X, V the lengths of wave, in the first and second media; so
that Vj v\ and the velocity of propagation in vacuum, are propor-
tional to X, \', and the length of wave in vacuum : I is the sine,
and n the cosine of the angle of incidence, V the sine, and n! the

Newton's rings beyond the critical angle. 59

cosine of the angle of refraction, these quantities being connected

by the equations

I y

— = —/, n-Ji-'P, n'^Jl — l* (1).

!2. The system of vibrations (A) is supposed to satisfy certain
linear differential equations of motion belonging to the two media,
and likewise certain linear equations of condition at the surface of
fieparation, for which « = 0. These equations lead to certain
relations between a, a^, and a, by virtue of which the ratios
of a, and a to a are certain functions of I, v, and v\ and it
might be also of \. The equations, being satisfied identically,
will continue to be satisfied when V becomes greater than 1, and
consequently rC imaginary, which may happen, provided v' > v ;
but the interpretation before given to the equations (A) and
(1) fails.

When n becomes imaginary, and equal to v V(— 1)> ^' being
equal to tJ{P — 1), z instead of appearing under a circular func-
tion in the third of equations (A), appears in one of the expo-
nentials €**'*'', k' being equal to 27r/\'. By changing the sign of
V(— 1) we should get a second system of equations (A), satisfying,
like the first system, all the equations of the problem ; and we
should get two new systems by writing vt + \/4 for vt By com-
bining these four systems by addition and subtraction, which is
allowable on account of the linearity of our equations, we should
be able to get rid of the imaginary quantities, and likewise of the
exponential €+*'•''', which does not correspond to the problem,
inasmuch as it relates to a disturbance which increases inde-
finitely in going from the surface of separation into the second
medium, and which could only be produced by a disturbing
cause existing in the second medium, whereas none such is sup-
posed to exist.

3. The analytical process will be a good deal simplified by
replacing the expressions (A) by the following symbolical ex-
pressions for the disturbance, where k is put for 27r/X, so that
kv = k'v ; _

F;«a/<''-«*+«'>^^»[ (B).

60 ON THE FORMATION OF THE CENTRAL SPOT OF

In these expressions, if each exponential of the form €^^^~^^ be re-
placed by cos P+ i^(— 1) sin P, the real part of the expressions
will agree with (A), and therefore will satisfy the equations of the
problem. The coeflScients of V(— 1) ^^ the imaginary part will be
derived from the real part by writing t + \/4t; for t, and therefore
will form a system satisfying the same equations, since the form of
these equations is supposed in no way to depend on the origin of
the time ; and since the equations are linear they will be satisfied
by the complete expressions (B).

Suppose now t to become greater than 1, so that n' becomes
± v' V(-" !)• Whichever sign we take, the real and imaginary
parts of the expressions (B), which must separately satisfy the
equations of motion and the equations of condition, will represent
two possible systems of waves ; but the upper sign does not corre-
spond to the problem, for the reason already mentioned, so that we
must use the lower sign. At the same time that n becomes
V V(""l)» l^t a, a^, a' become

pgtfVTT^ ^y.VTi^ ^'c** ^^^, respectively :

then we have the symbolical system

F^ = p^€-«'^l.€*(^-^+'«>^^, [ (C),

of which the real part

V — p coa [k (vt —Ix — nz) — 0], 1

V^^p^cos{k(vt-lx + nz)-0}, [ (D)

7'= p'e-^-'' cos {&' {vt - Vx) - 0'}, )

forms the system required.

As I shall frequently have occasion to allude to a disturbance
of the kind expressed by the last of equations (D), it will be con-
venient to have a name for it, and I shall accordingly call it a
superficial undulation.

4. The interpretation of our results is not yet complete, inas-
much as it remains to consider what is meant by V. When the
vibrations are perpendicular to the plane of incidence there is no
difficulty. In this case, whether the angle of incidence be greater
or less than the critical angle, V denotes a displacement, or

Newton's rings beyond the critical angle. 61

else a velocity, perpendicular to the plane of incidence. When
the vibrations are in the plane of incidence, and the angle of
incidence is less than the critical angle, V denotes a displacement
or velocity in the direction of a line lying in the plane xz, and
inclined at angles tt — i\ — (Jtt — i) to the axes of x, z, % being
the angle of refraction. But when the angle of incidence
exceeds the critical angle there is no such thing as an angle of
refraction, and the preceding interpretation fails. Instead there-
fore of considering the whole vibration F', consider its resolved
parts F/, V^ in the direction of the axes of a?, z. Then when the
angle of incidence is less than the critical angle, we have

F;=-nT' = -cosi'. T; F.' = rF' = sini'. F',

V being given by (A), and being reckoned positive in that direc-
tion which makes an acute angle with the positive part of the
axis of z. When the angle of incidence exceeds the critical angle,
we must first replace the coeflBcient of F' in F/, namely — n , by

vV' ^^^, and then, retaining v' for the coeflBcient, add \ir to the
phase, according to what was explained in the preceding article.

Hence, when the vibrations take place in the plane of inci-
dence, and the angle of incidence exceeds the critical angle, F'
in (D) must be interpreted to mean an expression from which the
vibrations in the directions of ar, z may be obtained by multiplying
by v\ V respectively, and increasing the phase in the former case
by Jtt. Consequently, so far as depends on the third of equations
(D), the particles of ether in the second medium describe small
ellipses lying in the plane of incidence, the semi-axes of the
ellipses being in the directions of x, r, and being proportional to
v\ Vy and the direction of revolution being the same as that in
which the incident ray would have to revolve in order to diminish
the angle of incidence.

Although the elliptic paths of the particles lie in the plane of
incidence, that does not prevent the superficial vibration just con-
sidered from being of the nature of transversal vibrations. For it
is easy to see that the equation

dx dz
is satisfied ; and this equation expresses the condition that there

62 ON THE FORMATION OF THE CENTRAL SPOT OF

is no change of density, which is the distinguishing characteristic
of transversal vibrations.

5. When the vibrations of the incident light take place in the
plane of incidence, it appears from investigation that the equa-
tions of condition relative to the surface of separation of the two
media cannot be satisfied by means of a system of incident, re-
flected and refracted waves, in which the vibrations are trans-
versal. If the media be capable of transmitting normal vibrations
with velocities comparable with those of transversal vibrations,
there will be produced, in addition to the waves already men-
tioned, a series of reflected and a series of refracted waves in
which the vibrations are normal, provided the angle of incidence
be less than either of the two critical angles corresponding to the
reflected and refracted normal vibrations respectively. It has
been shown however by Green, in a most satisfactory manner, that
it is necessary to suppose the velocities of propagation of normal
vibrations to be incomparably greater than those of transversal
vibrations, which comes to the same thing as regarding the ether
as sensibly incompressible ; so that the two critical angles men-
tioned above must be considered evanescent*. Consequently the
reflected and refracted normal waves are replaced by undulations
of the kind which I have called superficial. Now the existence of
these superficial undulations does not aflect the interpretation
which has been given to the expressions (A) when the angle of
incidence becomes greater than the critical angle corresponding to
the refracted tiunsversal wave ; in fact, so far as regards that
interpretation, it is immaterial whether the expressions (A) satisfy
the linear equations of motion and condition alone, or in con-
junction with other terms referring to the normal waves, or
rather to the superficial undulations which are their represen-
tatives. The expressions (D) however will not represent the
whole of the disturbance in the two media^ but only that part
of it which relates to the transversal waves, and to the superficial
undulation which is the representative of the refracted tranversal
wave.

6. Suppose now that in the expressions (A) n becomes imagi-
nary, n' remaining real, or that n and n' both become imaginary.

* Cambridge Philotophieal TransaetionSt Vol. yn. p. 2.

Newton's rings beyond the critical angle. 03

The former case occurs in the theory of Newton's Rings when
the angle of incidence on the surface of the second medium be-
comes greater than the critical angle, and we are considering the
superficial undulation incident on the third medium : the latter
case would occur if the third medium as well as the second were of
lower refractive power than the first, and the angle of incidence on
the surface of the second were greater than either of the critical
angles corresponding to refraction out of the first into the second,
or out of the first into the third. Consider the case in which n
becomes imaginary, v! remaining real ; and let V(^ — 1) = i/. Then
it may be shown as before that we must put — v\/(— 1), ai^d not

V V(— 1), for n ; and using />, Q in the same sense as before, we get
the symbolical system,

V = pe-* ^^. 6"*** . €*<*^-^>^-i,
F;=p^e-«'^^i.6*^.6*e<-to)^^, \...: (E),

to which corresponds the real system

V = pe-^^ cos [h {vt - Ix) - 6],
V,^p^^' QOs[k{vt-lx)-e],\ (F).

F' = p'cos{i'(v«-Z'^-w'^)-^},.

When the vibrations take place in the plane of incidence,

V and V^ in these expressions must be interpreted in the same
way as before. As far as regards the incident and reflected super-
ficial undulations, the particles of ether in the first medium will
describe small ellipses lying in the plane of incidence. The ellipses
will be similar and similarly situated in the two cases ; but the
direction of revolution will be in the case of the incident undula-
tion the same as that in wHich the refracted ray would have to
turn in order to diminish the angle of refraction, whereas in the
reflected undulation it will be the opposite.

It is unnecessary to write down the formulae which apply to
the case in which n and n both become imaginary.

7. If we choose to employ real expressions, such as (D) and
(F), we have this general rule. When any one of the undula-
tions, incident, reflected, or refiracted, becomes superficial, remove
z from under the circular function, and insert the exponential

G4 ON THE F0R31ATI0N OF THE CENTRAL SPOT OF

6"*^*, €*^*, or c"*^"'*, according as the incident, reflected, or re-
fracted undulation is considered. At the same time put the
coefficients, which become imaginary, under the form

p {cos ^ ± V (- 1) sin 0],

the double sign corresponding to the substitution of

± V V ( -" 1)> or ± I/' ^ (— 1) for n or n,

retain the modulus p for coefficient, and subtract from the
phase.

It will however be far more convenient to employ symbolical
expressions such as (B). These expressions will remain applicable
without any change when n or n' becomes imaginary : it will only
be necessary to observe to take

± 1/ V ( - 1)> or ± v V ( - 1)

with the negative sign. If we had chosen to employ the expres-
sions (B) with the opposite sign in the index, which would have
done equally well, it would then have been necessary to take the
positive sign.

8. We are now prepared to enter on the regular calculation of
the intensity of the central spot ; but before doing so it will be
proper to consider how far we are justified in omitting the
consideration of the superficial undulations which, when the vibra-
tions are in the plane of incidence, are the representatives of normal
vibrations. These undulations may conveniently be called normal
superficial undulations, to distinguish them from the superficial
undulations expressed by the third of equations (D), or the first
and second of equations (F), which may be called transversal.
The former name however might, without warning, be calculated
to carry a false impression ; for the undulations spoken of are not
propagated by way of condensation and rarefaction ; the disturb-
ance is in fact precisely the same as that which exists near the
surface of deep water when a series of oscillatory waves is propa-
gated along it, although the cause of the propagation is extremely
different in the two cases.

Now in the ordinary theory of Newton's Rings, no account is
taken of the normal superficial undulations which may be sup-
posed to exist ; and the result so obtained from theory agrees very

kewton's rings beyond the critical angle. 65

well with observation. When the angle of incidence passes through
the critical angle, although a material change takes place in the
nature of the refracted transversal undulation, no such change
takes place in the case of the normal superficial undulations : the
critical angle is in fact nothing particular as regards these undu-
lations. Consequently, we should expect the result obtained from
theory when the normal superficial undulations are left out of con-
sideration to agree as well with experiment beyond the critical
angle as within it.

9. It is however one thing to sliow why we are justified in
expecting a near accordance between the simplified theory and
experiment, beyond the critical angle, in consequence of the
observed accordance within that angle ; it is another thing to show
why a near accordance ought to be expected both in the one case
and in the other. The following considerations will show that the
effect of the normal superficial undulations on the observed
phenomena is most probably very slight.

At the point of contact of the first and third media, the reflec-
tion and refraction will take place as if the second medium were
removed, so that the first and third were in contact throughout.
Now Fresnel's expressions satisfy the condition of giving the same
intensity for the refiected and refracted light whether we suppose
the refraction to take place directly out of the first medium into
the third, or take into account the infinite number of reflections
which take place when the second medium is interposed, and then
suppose the thickness of the interposed medium to vanish. Conse-
quently the expression we shall obtain for the intensity by neg-
lecting the normal superficial undulations will be strictly correct
for the point of contact, Fresners expressions being supposed cor-
rect, and of course will be sensibly correct for some distance round
that point Again, the expression for the refracted normal su-
perficial undulation will contain in the index of the exponential
— klz, in place of '-kj{l^'-v^/ v'*) z, which occurs in the expres-
sion for the refracted transversal superficial undulation; and there-
fore the former kind of undulation will decrease much more rapidly,
in receding from the surface, than the latter, so that the effect
of the former will be insensible at a distance from the point of
contact at which the effect of the latter is still important. If we
cembine these two considerations, we can hardly suppose the
a n. &

66 ON THE FORMATION OF THE CENTRAL SPOT OP

effect of the normal superficial undulations at intermediate points
to be of any material importance.

10. The phenomenon of Newton's Rings is the only one in
which I see at present any chance of rendering these undulations
sensible in experiment : for the only way in which I can conceive
them to be rendered sensible is, by their again producing trans-
versal vibrations; and in consequence of the rapid diminution of
the disturbance on receding from the surface, this can only happen
when there exists a second reflecting surface in close proximity
with the first. It is not my intention to pursue the subject further
at present, but merely to do for angles of incidence greater than
the critical angle what has long ago been done for smaller angles,
in which case light is refracted in the ordinary way. Before quitting
the subject however I would observe that, for the reasons already
mentioned, the near accordance of observation with the expression
for the intensity obtained when the normal superficial undula-
tions are not taken into consideration cannot be regarded as any
valid argument against the existence of such undulations.

11. Let Newton's Rings be formed between a prism and a
lens, or a second prism, of the same kind of glass. Suppose the
incident light polarized, either in the plane of incidence, or in a
plane perpendicular to the plane of incidence. Let the coeflBcient
of vibration in the incident light be taken for unity; and, accord-
ing to the notation employed in Airy's Tracts let the coefiicient be
multiplied by 6 for reflection and by c for refraction when light
passes from glass into air, and by e for reflection and / for refrac-
tion when light passes from air into glass. In the case contem-
plated 6, c, e,/ become imaginary, but that will be taken into ac-
count further on. Then the incident vibration will be represented
symbolically by

according to the notation already employed ; and the reflected and
refracted vibrations will be represented by

Q^-Vv^z ^ ^k'Wt-Vx) V-T

Consider a point at which the distance of the pieces of glass is
D\ and, as in the usual investigation, regard the plate of air about
that point as bounded by parallel planes. When the superficial

Newton's rings beyond the critical angle. 67

undulation represented by the last of the preceding expressions is
incident on the second surface, the coefficient of vibration will be-
come cy, q being put for shortness in place of e"*'''-^; and the re-
flected and refracted vibrations will be represented by

z being now measured from the lower surface. It is evident that
each time that the undulation passes from one surface to the other
the coefficient of vibration will be multiplied by q, while the phase
will remain the same. Taking account of the infinite series of
reflections, we get for the symbolical expression for the reflected
vibration

{6 + ce/q^ (1 + cV + ^V+ -Ol €*(«*-^+«')v^.
Summing the geometric series, we get for the coefficient of the

exponential

ce/V»

b +

,a^a*

1 - 6 Y
Now it follows from Fresnel's expressions that

These substitutions being made in the coefficient, we get for the
symbolical expression for the reflected vibration

Q—lSJ _ ^kivt-lx+mW-T (Q\

l-g«6« ^ ^^^•

Let the coefficient, which is imaginary, be put under the form
p[cos'^ + V(— l)sin'^}; then the real part of the whole expres-
sion, namely

p cos {k {vt — /a? + nz) + -^J,

will represent the vibration in the reflected light, so that p' is the
intensity, and yjr the acceleration of phase.

12. Let i be the angle of incidence on the first surface of the
plate of air, fi the refractive index of glass; and let X now denote
the length of wave in air. Then in the expression for q

lev = \- >//!* sin* i — 1.

* I hare proved these equations in a veiy simple manner, without any reference
(o Fresnel's foimuls, in a paper ^hich wiU appear in the next number of tl>e
Comhridge and Dublin Mathematical Jcvmal [p. 89 of the present Tolume].

68 ON THE FORMATION OF THE CENTRAL SPOT OF

In the expression for b we must, according to Art. 2, take the
imaginary expression for cos i' with the negative sign. We thus
get for light polarized in the plane of incidence (Airy's Tract,

p. 362, 2nd edition*), changing the sign of ^ — 1,

b = cos 29 + J^^ sin 20,
where

taag = ^^'""'\-^ (2).

/A cost

Putting C for the coeflScient in the expression (Q), we have

c^ 1-g' ^ Lui!

6"* - q'b (1 - j«) cos 2d - ^/^ (1 + ?') sin 20

_ (l-q*){(l-q')co320 + J^(l-{'^6m2e] -

(l-2«)«4.4^sin*2^
whence

tan 1^ = ^^ tan 2^ (3),

g -gV /4N

P •"(l-.(f)« + 4(f 8in*2d ^ ^'

where

2irD ,
? = € (5).

If we take p positive, as it will be convenient to do, we must
take '^ so that cos'^ and cos 20 may have the same sign. Hence
from (3) sin yft must be positive, since sin 20 is positive, inasmuch
as lies between and ^tt. Hence, of the two angles lying be- '
tween — tt and tt which satisfy (2), we must take that which lies
between and tt.

For light polarized perpendicularly to the plane of incidence,
we have merely to substitute ^ for in the equations (3) and (4),
where

^ mVa^'si^;?^ ^

^ cost

The value of q does not depend* on the nature of the polarization.

* Mr Airy speaks of *'Tibrations perpendicular to the plane of incidence," and
** vibrations parallel to the plane of incidence,*' adopting the theory of Fresnel ; bnt
there is nothing in this paper which requires us to enter into the question whether
the vibrations in plane polarized light are in or perpendicular to the plane of
polarization.

Newton's rings beyond the critical angle* 69

13. For the transmitted light we have an expression similar
to (Q), with — nz in place of nz, and a different coefficient (7^,
where

When the light is polarized in the plane of incidence we have

^ - V^^ . 2q sin 29

' "" (1 - 2^) cos 25- V -1 (1 + q') sin 25

_ 2q sin 20 {(1 + g') sin 29 - /^l ( 1 - g') cos 25} ^.

(l_g«)«4.42«sin'25 ^^^'

so that if '^^ and p^ refer to the transmitted light we have

tan^,==-^?^cot25 (8),

'^ ' 1 4- 2*

s

p. =

(l-.g«)«4.4g»sin"25

If we take p^ positive, as it will be supposed to be, we must
take yjr^ such that cos yfr^ may be positive ; and therefore, of the
two angles lying between — tt and tt which satisfy (8), we must
choose that which lies between — ^tt and + ^tt. Hence, since from
(3) and (8) yfr^ is of the form -^ + Jtt + nir, n being an integer, we
must take -^^ = -^ — ^tt.

For light polarized perpendicularly to the plane of incidence
we have only to put ^ for d. It follows from (4) and (9) that the
sum of the intensities of the reflected and transmitted light is
equal to unity, as of course ought to be the case. This renders it
unnecessary to discuss the expression for the intensity of the trans-
mitted light.

14. Taking the expression (4) for the intensity of the reflected
light, consider first how it varies on receding from the point of
contact.

As the point of contact i? = 0, and therefore from (5) 2 = 1, and
therefore p* = 0, or there is absolute darkness. On receding from
the point of contact q decreases, but slowly at first, inasmuch as D
varies as r*, r being the distance from the point of contact. It
follows from (4) that the intensity p* varies ultimately as /, so

70 ON THE FORMATION OF THE CENTRAL SPOT OF

that it increases at first with extreme slowness. Consequently
the darkness is, as far as sense can decide, perfect for some
distance round the point of contact. Further on q decreases more
rapidly, and soon becomes insensible. Consequently the intensity
decreases, at first rapidly, and then slowly again as it approaches
its limiting value 1, to which it soon becomes sensibly equaL All
this agrees with observation.

15. Consider next the variation of intensity as depending on
the colour. The change in and (f> in passing from one colour to
another is but small, and need not here be taken into account :
the quantity whose variation it is important to consider is q. Now
it follows from (5) that q changes the more rapidly in receding
from the point of contact tHe emaller be X. Consequently the
spot must be smaller for blue light than for red ; and therefore
towards the edge of the spot seen by reflection, that is beyond the
edge of the central portion of it, which is black, there is a pre-
dominance of the colours at the blue end of the spectrum ; and
towards the edge of the bright spot seen by transmission the
colours at the red end predominate. The tint is more conspicuous
in the transmitted, than in the reflected light, in consequence of
the quantity of white light reflected about the edge of the spot.
The separation of colours is however but slight, compared with
what takes place in dispersion or diffraction, for two reasons.
First, the point of minimum intensity is the same for all the
colours, and the only reason why there is any tint produced is,
that the intensity approaches more rapidly to its limiting value 1
in the case of the blue than in the case of the red. Secondly, the
same fraction of the incident light is reflected at points for which
i? X X, and therefore ?• oc a/\ ; and therefore, on this account also,
the separation of colours is less than in diffraction, where the
colours are arranged according to the values of \, or in dispersion,
where they are arranged according to values of \~* nearly. These
conclusions agree with observation. A faint blueish tint may be
perceived about the dark spot seen by reflection ; and the fainter
]X)rtions of the bright spot seen by transmission are of a decided
reddish brown.

16. Let us now consider the dependance of the size of the
spot on the nature of the polarization. Let 8 be the ratio of the

NEWTON'S RINGS BEYOND THE CRITICAJi ANGLE. 71

intensity of the transmitted light to that of the reflected; 8^, 8^,
the particular values of s belonging to light polarized in the plane
of incidence and to light polarized perpendicularly to the plane of
incidence respectively; then

_ iq^m[29 _ 4 g'sin' 2</i

s^ Vsi

sin 2^Y^ j^^,_^^^ sin*t - !}» (10).

,sin 2(f)/

Now according as s is greater or less, the spot is more or less
conspicuous ; that is, conspicuous in regard to extent, and intensity
at some distance from the point of contact ; for in the immediate
neighbourhood of that point the light is in all cases wholly trans-
mitted. Very near the critical angle we have from (10) 8^^fi*8^,
and therefore the spot is much more conspicuous for light polarized
perpendicularly to the plane of incidence than for light polarized
in that plane. As i increases the spots seen in the two cases
become more and more nearly equal in magnitude : they become
exactly alike when i = t, where

2

sin c =

1+m'"

When i becomes greater than t the order of magnitude is
reversed; and the spots become more and more unequal as %
increases. When i = 90'* we have s^=^ fi\, so that the inequality
becomes very great. This however must be understood with
reference to relative, not absolute magnitude ; for when the angle
of incidence becomes very great both spots become very small.

I have verified these conclusions by viewing the spot through
a rhomb of Iceland spar, with its principal plane either parallel or
perpendicular to the plane of incidence, as well as by using a
doubly refracting prism ; but I have not attempted to determine
experimentally the angle of incidence at which the spots are
exactly equal. Indeed, it could not be determined in this way
-with any precision, because the difierence between the spots is
insensible through a considerable range of incidence.

17. It is worthy of remark that the angle of incidence t at
which the spots are equal, is exactly that at which the difference
of acceleration of phase of the oppositely polarized pencils, which
arises from total internal reflection, is a maximum.

72 ON THE FORMATION OF THE CENTRAL SPOT OF

When t SB i we have

Bin2^ = 8iii2^ = -j-^ ; whence cot ^ = tan ^=5/1 (11) ;

and .^ (l + Myq-g*)'

where 3 = 6 ^ (12).

If we determine in succession the angles 0, (^, 17 from the equa-
tions cot d^ II, tan (T— ?i tan 17 = sin 2d tan 2$",
we have />/ = 1 — p' = ^ versin 217.

The expression for the intensity may be adapted to numerical
computation in the same way for any angle of incidence, except
that 6 or (f> must be determined by (2) or (6) instead of (11), and
q by (5) instead of (12).

18. When light is incident at the critical angle, which I shall
denote by 7, the expression for the intensity takes the form 0/0.
Putting for shortness V(/** sin* t — 1) = w, we have ultimately

and we get in the limit

according as the light is polarized in or perpendicularly to the
plane of incidence. The same formulas may be obtained from the
expression given at page 304 of Airy's Tract, which gives the
intensity when i<y, and which like (4) takes the form 0/0 when %
becomes equal to y, in which case e becomes equal to — 1.

19. When t becomes equal to 7, the infinite series of Art 11
ceases to be convergent: in fact, its several terms become ulti-
mately equal to each other, while at the same time the coefficient
by which the series is multiplied vanishes, so that the whole takes
the form x x . The same remark applies to the series at page

Newton's rings beyond the critical anglk 73

303 of Airy's Tract If we had included the coeflScient in eaxsh
term of the series^ we should have got series which ceased to be
convergent at the same time that their several terms vanished.
Now the sum of such a series may depend altogether on the point
of view in which it is regarded as a limit. Take for example the
convergent infinite series

/{x,y) =ir8iny + ^ic'sin3y + ia;*sin5y + ... = i tan"* f- ,

where x is less than 1, and may be supposed positive. When x
becomes 1 and y vanishes f {x, y) becomes indeterminate, and its
limiting value depends altogether upon the order in which we
suppose X and y to receive their limiting values, or more generally
upon the arbitrary relation which we conceive imposed upon the
otherwise independent variables x and y as they approach their
limiting values together. Thus, if we suppose y first to vanish,
and then x to become 1, we have /(a?, y) = 0; but if we suppose x
first to become 1, and then y to vanish, /(a?, y) becomes ± 7r/4,
+ or — according as y vanishes positively or negatively. Hence in
the case of such a series a mode of approximating to the value of
X or y, which in general was perfectly legitimate, might become
inadmissible in the extreme case in which a? = 1, or nearly = 1.
Consequently, in the case of Newton's Rings when 1^7 is
extremely small, it is no longer safe to neglect the defect of paral-
lelism of the surfaces. Nevertheless, inasmuch as the expression
(4), which applies to the case in which i>7, and the ordinary
expression which applies when i < 7, alter continuously as i alters,
and agree with (13) when 1 = 7, we may employ the latter expres-
sion in so far as the phenomenon to be explained alters continu-
ously as i alters. Consequently we may apply the expression (13)
to the central spot when 1 = 7, or nearly =7, at least if we do not
push the expression beyond values of D corresponding to the limits
of the central spot as seen at other angles of incidence. To explain
however the precise mode of disappearance of the rings, and to
determine their greatest dilatation, we should have to enter on a
special investigation in which the inclination of the surfaces should
be taken into account.

20. I have calculated the following Table of the intensity of
the transmitted light, taking the intensity of the incident light at
100. The Table is calculated for values of D increasing by X/4,

74

ON THE FORMATION OF THE CENTRAL SPOT OF

and for three angles of incidence, namely, the critical angle, the
angle l hefore mentioned, and a considerable angle, for which I
have taken 60°. I have supposed /Lt = l-63, which is about the
refractive index for the brightest part of the spectrum in the case
of flint glass. This value of fju gives 7 = 37' 51', l = 42' 18'. The
numerals I., II. refer to light polarized in and perpendicularly to
the plane of incidence respectively.

4Z>

•

1=

■y

•

•

t=

60'>

"X

I.

11.

I. and II.

■

'I-

II.

100

100

100

100 1

100

1

49

87

33

16 i

6

2

20

63

6

1

3

10

43

1

4

6

30

;

6

4

22

1

6

3

16

7

2

12

' 1

8

2

10

(

.

9

8

1

10

6

11

6

1

12

5

1

1

13

•

4

•

1

1

i

•

26

•
•

1

1

27

1

1

21. A Table such as this would enable us to draw the curve
of intensity, or the curve in which the abscissa is proportional to
the distance of the point considered from the point of contact, and
the ordinate proportional to the intensity. For this puqjoso it
would only be requisite to lay down on the axis of the abscissae,
on the positive and negative sides of the origin, distances propor-
tional to the square roots of the numbers in the first column, and
to take for ordinates lengths proportional to the numbers in one of
the succeeding columns. To draw the curve of intensity for 1 = 4
or for 1 = 60^ the table ought to have been calculated with smaller
intervals between the values of D ; but the law of the decrease of
the intensity cannot be accurately observed.

22. From the expression (13) compared with (4), it will be
seen that the intensity decreases much more rapidly, at some

NEWTON'S BINQS BEYOND THE CBITICAL ANGLE. 75

distance from the point of contact, when i is considerably greater
than 7 than when 1 = 7 nearly. This agrees with observation.
What may be called the ragged edge of the bright spot seen by
transmission is in fact much broader in the latter case than in the
former.

When I becomes equal to 90° there is no particular change in
the value of q, but the angles d and (f> become equal to 90°, and
therefore sin 29 and sin 2<f> vanish, so that the spot vanishes.
Observation shows that the spot becomes very small when i
becomes nearly equal to 90°.

23. Suppose the incident light to be polarized in a plane
making an angle a with the plane of incidence. Then at the
point of contact the light, being transmitted as if the first and
third media formed one uninterrupted medium, will be plane
polarized, the plane of polarization being the same as at first.
At a suflBcient distance from the point of contact there is no
sensible quantity of light transmitted. At intermediate distances
the transmitted light is in general elliptically polarized, since
it follows from (8) and the expression thence derived by writing
if> for d that the two streams of light, polarized in and perpen-
dicularly to the plane of incidence respectively, into which the
incident light may be conceived to be decomposed, are unequally
accelerated or retarded. At the point of contact, where J = 1,
tliese two expressions agree in giving '^^= 0. Suppose now
that the transmitted light is analyzed, so as to extinguish
the light which passes through close to the point of contact
Then the centre of the spot will be dark, and beyond a certain
distance all round there will be darkness, because no sensible
quantity of light was incident on the analyzer ; but at interme-
diate distances a portion of the light incident on the analyzer will
be visible. Consequently the appearance will be that of a lumi-
nous ring with a perfectly dark centre.

24. Let the coefficient of vibration in the incident light be
taken for unity ; then the incident vibration may be resolved into
two, whose coefficients are cos a, sin a, belonging to light polarized
in and perpendicularly to the plane of incidence respectively. The
phases of vibration will be accelerated by the angles yfr^, 1^^^, and
the coefficients of vibration will be multiplied by p,, p,,, if yfr^^, p^^

81D a cos a

76 ON THE FORMATION OF THE CENTRAL SPOT OF

are what yjt^, p^ in Art. (13) become when ^ is put for 0, Hence
we may take

p.cosa.cosj— (yt'-fix) + '^},

p,, sin a . cos | ^^ (vt - /Aa?') +-^1 ,

to represent the vibrations which compounded together make up
the transmitted light, x being measured in the direction of propa-
gation. The light being analyzed in the way above mentioned, it
is only the resolved parts of these vibrations in a direction perpen-
dicular to that of the vibrations in the incident light which are
preserved. We thus get, to express the vibration with which we
are concerned,

r cos (~ {vt - /ix') + 1^,) -/>,,cos ^-J (vt-fix') +^,,j| ,

which gives for the intensity (/) at any point of the ring
/ = J sin* 2a { (p, cos i^^ - p,, cos ylr^y+ (p^ sin ^,- p,, sin yjrj^} . . . (14),
= i sin»2a {p; + p,; - 2p^p,, cos (^,,- f)}.

Let P^, Qe be respectively the real part of the expression at the
second side of (7) and the coeflScient of \/(— 1), and let P^, Q^ be
what Ptf, Qb become when ^ is put for 6, Then we may if we
please replace (14) by

/=isin»2a{(P,-P^)' + (Q,-Q^)«} (15).

The ring is brightest, for a given angle of incidence, when
a = 45^ When % = t, the two kinds of polarized light are trans-
mitted in the same proportion; but it does not therefore follow
that the ring vanishes, inasmuch as the change of phase is different
in the two cases. In fact, in this case the angles (f>, d are comple-
mentary ; so that cot 2^, cot 2fl are equal in magnitude but oppo-
site in sign, and therefore from (8) the phase in the one case is
accelerated and in the other case retarded by the angle

.n-.(l^.t.«),orU.-.(l^'ti).

It follows from (14) that the ring cannot vanish unless
p^ cos -^^ = p^^ cos -^^^j and p^ sin -^^ = p,^ sin '^^^. This requires

-I

Newton's rings beyond the critical angle. 77

that p^' = p^^*, which is satisfied only when i = t, in which case
as we have seen the ring does not vanish. Consequently a
ring is formed at all angles of incidence; but it should be
remembered that the spot, and consequently the ring, vanishes
when i becomes 90^

25. When t = y, the expressions for P^, Q^, take the form 0/0,
and we find, putting for shortness irDfK =p.

If we take two subsidiary angles ^, a>, determined by the
equations

-^ V/** — 1 = tan X = /** t*° «*»

we get

P^ = cos* x» ^* = ^^8* <**»

Ctf = — sin ;^ cos ;^, Q^ = — sin co cos ©.

Substituting in (15) and reducing we get, supposing a = 45^

/ = Jversin(2x-2a>) (16).

When % = t, cos 2^ = — cos 25, sin 2<f> = sin 25 ; and therefore
P^ = Ptf , 0^ = - C* , which when a = 45' reduces (15) to / = Q9\

If we determine the angle «r from the equation

1 — g* = 2g sin 25 tan «r, or tan «r = cot 2f . cosec 25,

we get

/= 1 sin* 2tjr . cos* 25 (17).

In these equations

log.tanr=-?^y^^,-^J, cot^ = M.

26. The following Table gives the intensity of the ring for
the two angles of incidence t = 7 and t = t, and for values of D
increasing by X/10. The intensity is calculated by the formulaB
(16) and (17). The intensity of the incident polarized light is
taken at 100, and /x is supposed equal to 1*63, as before.

78

ON THE FOBMATION OF THE CENTRAL SPOT OF

D

/

/

D

/

X

l=y

t=i

X

t=y

•0

0-0

0-0

1 -6

1 -4

• 1

1 -3

3-2

1 -7

1 •

2

•2

3-5

51

1-8

1 •

1

•3

4-8

3-6

1 -9

1 •

•4

6-1

1 -9

2-0

9

•5

4-9

•9

2-1

9

•6

4-5

•4

2 -2

•8

•7

4-0

•2

2-3

'7

•8

3-6

•1

2-4

'7

•9

3-1

•0

2-5

'6

1 -0

2-8

2-6

•6

1 1

2-4

2-7

•5

1 -2

2-1

2-8

•5

1 -3

1 -9

2 -9

•5

1 -4

1 -7

30

•4

1 -5

1 -6

The column for 1 = 7 may be continued with suflScient
accuracy, by taking / to vary inversely as the square of the num-
ber in the first column.

27. I have seen the ring very distinctly by viewing the light
transmitted at an angle of incidence a little greater than the
critical angle. In what follows, in speaking of angles of position, I
shall consider those positive which are measured in the direction
of motion of the hands of a watch, to a person looking at the
light. The plane of incidence being about 45® to the positive side
of the plane of primitive polarization, the appearance presented as
the analyzer (a Nicol's prism) was turned, in the positive direction,
through the position in which the light from the centre was extin-
guished, was as follows. On approaching that position, in addi-
tion to the general darkening of the spot, a dark ring was observed
to separate itself from the dark field about the spot, and to move
towards the centre, where it formed a broad dark patch, sur-
rounded by a rather faint ring of light. On continuing to turn,
the ring got brighter, and the central patch ceased to be quite
black. The light transmitted near the centre increased in intensity
till the dark patch disappeared : the patch did not break up into
a dark ring travelling outwards.

On making the analyzer revolve in the contrary direction, the
same appearances were of course repeated in a reverse order : a

Newton's rings beyond the ckitical angle. 79

dull central patch was seen, which became darker and darker till
it appeared quite black, after which it broke up into a dark ring
which travelled outwards till it was lost in the dark field surround-
ing the spot. The appearance was a good deal disturbed by the
imperfect annealing of the prisms. When the plane of incidence
was inclined at an angle of about — 45° to the plane of primitive
polarization, the same appearance as before was presented on
reversing the direction of rotation of the analyzer.

28. Although the complete theoretical investigation of the
moving dark ring would require a great deal of numerical calcu-
lation, a general explanation may very easily be given. At the
point of contact the transmitted light is plane polarized, the plane
of polarization being the same as at first*. At some distance
from the point of contact, although strictly speaking the light is
elliptically polarized, it may be represented in a general way by
plane polarized light with its plane of polarization further removed
than at first from the plane of incidence, in consequence of the
larger proportion in which light polarized perpendicularly to the
plane of incidence is transmitted, than light polarized in that
plane. Consequently the transmitted light may be represented
in a general way by plane polarized, with its plane of polarization
receding from the plane of incidence on going from the centre
outwards. If therefore we suppose the position of the plane of
incidence, and the direction of rotation of the analyzer, to be those
first mentioned, the plane of polarization of light transmitted by
the analyzer will become perpendicular to the plane of polarization
of the transmitted light of the spot sooner towards the edge of the
spot than in the middle. The locus of the point where the two
planes are perpendicular to each other will in fact be a circle, whose
radius will contract as the analyzer turns round. When the
analyzer has passed the position in which its plane of polarization
is perpendicular to that of the light at the centre of the spot, the
inclination of the planes of polarization of the analyzer and of the
transmitted light of the spot decreases, for a given position of the
analyzer, in passing from the centre outwards ; and therefore there
is formed, not a dark ring travelling outwards as the analyzer turns
round, but a dark patch, darkest in the centre, and becoming

* The rotation of the plane of polarization dne to the refraction at the surfaces
at which the light enters the first prism and quits the second is not here mentioned,
as it has nothing to do with the phenomenon disonsped.

80 ON THE FORMATION OF THE CENTRAL SPOT OF

brighter, and therefore less and less conspicuous, as the analyzer
turns round. The appearance will of course be the same when the
plane of incidence is turned through 90®, so as to be equally in-
clined to the plane of polarization on the opposite side, provided
the direction of rotation of the analyzer be reversed.

29. The investigation of the intensity of the spot formed
beyond the critical angle when the third medium is of a
diflFerent nature from the first, does not seem likely to lead to
results of any particular interest. Perhaps the most remark-
able case is that in which the second and third media are both
of lower refractive power than the first, and the angle of inci-
dence is greater than either of the critical angles for refraction
out of the first medium into the second, or out of the first
into the third. In this case the light must be wholly reflected;
but the acceleration of phase due to the total internal reflection
will alter in the neighbourhood of the point of contact. At that
point it will be the same as if the third medium occupied the
place of the second as well as its own ; at a distance suflicient to
render the influence of the third medium insensible, it will be the
same as if the second medium occupied the place of the third as
well as its own. The law of the variation of the acceleration from
the one to the other of its extreme values, as the distance from the
point of contact varies, would result from the investigation. This
law could be put to the test of experiment by examining the
nature of the elliptic polarization of the light reflected in the
neighbourhood of the point of contact when the incident light is
polarized at an azimuth of 45^ or thereabouts. The theoretical
investigation does not present the slightest difficulty in principle,
but would lead to rather long expressions ; and as the experiment
would be difficult, and is not likely to be performed, there is no
occasion to go into the investigation.

30. In viewing the spot formed between a prism and a
lens, I was struck with the sudden, or nearly sudden disappearance
of the spot at a considerable angle of incidence. The cause of
the disappearance no doubt was that the lens was of lower re-
fractive power than the prism, and that the critical angle was
reached which belongs to refraction out of the prism into the lens.
Before disappearing, the spot became of a bright sky blue, which

kewton's rings beyond the critical angle. 81

shows that the ratio of the refractive index of the prism to that of
the lens was greater for the blue rays than for the red. As the
disappearance of the spot can be observed with a good deal of
precision, it may be possible to determine in this way the refrac-
tive index of a substance of which only a very minute quantity
can be obtained. The examination of the refractive index of the
globule obtained from a small fragment of a fusible mineral might
afford the mineralogist a means of discriminating between one
mineral and another. For this purpose a plate, which is what a
prism becomes when each base angle becomes 90^ would probably
be more convenient than a prism. Of course the observation is
possible only when the refractive index of the substance to be
examined is less than that of the prism or plate.

B. Jt. <i

[From the PkUoBaphical Magazine^ VoL xxxiv. p. 52, (January, 1849.)]

On some Points in the Received Theory of Sound*.

I proceed now to notice the apparent contradiction at which
Professor Challis has arrived by considering spherical waves, a
contradiction which it is the chief object of this communication to
consider. The only reason why I took no notice of it in a former
communication was, that it was expressed with such brevity by
Professor Challis (VoL xxxii. p. 497), that I did not perceive how
the conclusion that the condensation varies inversely as the square
of the distance was arrived at. On mentioning this circumstance
to Professor Challis, he kindly explained to me his reasoning,
which he has since stated in detail (Vol. xxxiii. p. 463) f.

* The beginning and end of this Paper are omitted, as being merely contro-
versial, and of ephemeral interest.

t The objection is put in two slightly different forms in the two Papers. The
substance of it may be placed before the reader in a few words.

Conceive a wave of sound of small disturbance to be travelling outwards from
a centre, the disturbance being alike in all directions round the centre. Then
according to the received theory the condensation is expressed by equation (1),
where r is the distance from the centre, and 8 the condensation. It follows from
this equation that any phase of the wave is carried outwards with the velocity
of propagation a, and that the condensation varies inversely as the distance from
the centre. But if we consider the shell of infinitesimal thickness a comprised
between spherical surfaces of radii r and r + a corresponding to given phases, so
that these surfaces travel outwards with the velocity a, the excess of matter in the
shell over the quantity corresponding to the undisturbed density will vary as the
condensation multiplied by the volume, and therefore as t^s *, and as the constancy
of mass requires that this excess should be constant, s must vary inversely as
r* not r.

Or instead of considering only an infinitesimal shell, consider the whole of an
outward travelling wave, and for simplicity's sake suppose it to have travelled
80 far that its thickness is small compared with its mean radius r or at, t being

ON SOME POINTS IN THE RECEIVED THEORY OF SOUND. 83

The whole force of the reasoning rests on the tacit supposition
that when a wave is propagated from the centre outwards, any
arbitrary portion of the wave, bounded by spherical surfaces con-
centric with the bounding surfaces of the wave, may be isolated,
the rest of the wave being replaced by quiescent fluid ; and that
being so isolated, it will continue to be propagated outwards as
before, all the fluid except the successive portions which form the
wave in its successive positions being at rest. At first sight it
might seem as if this assumption were merely an application of
the principle of the coexistence of small motions, but it is in
reality extremely different. The equations are competent to decide
whether the isolation be possible or not. The subject may be
considered in different ways ; they will all be found to lead to the
same result.

1. We may evidently without absurdity conceive an outward
travelling wave to exist already, without entering into the question
of its original generation ; and we may suppose the condensation
to be given arbitrarily throughout this wave. By an outward
travelling wave, I mean one for which the quantity usually denoted
by (f> contains a function of r — at, unaccompanied by a function of
r 4- at, in which case the expressions for v and s will likewise con-
tain functions of r — at only. Let

■ a.=-^^--°^> (1).

r

We are at liberty to suppose f (z) = 0, except from « = 6 to
^ = c, where 6 and c are supposed positive; and we may take/^ (z)
to denote any arbitrary function for which the portion from z = b

the time of trayelling from the origin to the distance r. Then assnming the
expression (1), and patting the factor r outside the sign of integration, as we are at
liberty to do in consequence of the supposition made above as to the distance
the wave has travelled, we have for the quantity of matter existing -at any time in
the wave beyond what would oocupy the same space in the quiescent state of the
fluid,

4r . aH^xp If (r - at) dr-r-aU

veiy nearly, or 4rp^t, putting A for the value of the integral \f{r-at)dr taken from

the inner to the outer boundaiy of the wave. Hence the matter increases in
quantity with the time.

Si ON SOME POINTS IN THE

to ^ = c has been isolated, the rest having been suppressed. Equa-
tion (1) gives

^^-a'jsdt=^~^^^^+ylrir), (2),

^|r (r) being an arbitrary function of r, to determine which we must
substitute the value of <f> given by (2) in the equation which ^ has
to satisfy, namely

dt''"'' "W ^•^^•

This equation gives -^ (r) = C + Djr, C and D being arbitrary
constants, whence

^ ^ # ^f'ir-at) _ fir ~ at) _ D

Now the function f{z) is merely defined as an integral of
/' (z) dz, and we may suppose the integral so chosen as to
vanish when z = b, and therefore when z has any smaller value.
Consequently we get from (4), for every point within the sphere
which forms the inner boundary of the wave of condensation,

v = --i (5)-

Again, if we 'pni f(c)=^A, so that f{z)^A when z>c, we
have for any point outside the wave of condensation,

^ = ;:r- (6).

The velocities expressed by (5) and (6) are evidently such
as could take place in an incompressible fluid. Now Professor
Challiss reasoning requires that the fluid be at rest beyond the
limits of the wave of condensation, since otherwise the conclusion
cannot be drawn that the matter increases with the time. Conse-
quently we must have i) = 0, J. = ; but if J. = the reasoning at
p. 463 evidently falls to the ground.

2. We may if we please consider an outward travelling wave
which arose from a disturbance originally confined to a sphere of
radius €. At p. 463 Professor Challis has referred to Poisson's
expressions relating to this case. It should be observed that
Poisson's expressions at page 706 of the TraiU de MAxinique
(second edition) do not apply to the whole wave from r=^at^e

RECEIVED THEORY OF SOUXD. 85

to r = at + €, but only to the portion from r — at — e to r==at; the
expressions which apply to the remainder are those given near the
bottom of page 705. We may of course represent the condensa-
tion 9 by a single function 1/ar. j^ (r — at), where

z being positive ; and we shall have

Now Poisson has proved, and moreover expressly stated at
I)age 706, that the functions -F,/ vanish at the limits of the wave ;
so that/(€) = 0, 2^(6) = 0. Also Poisson's equations (6) give in
the limiting case for which « = 0, /(O) + F(0) = 0, so that ^ =
as before.

3. We may evidently without absurdity conceive the velocity
and condensation to be both given arbitrarily for the instant at
which we begin to consider the motion; but then we must take
the complete integral of (3), and determine the two arbitrary
functions which it contains. We are at liberty, for example, to
8uppose the condensation and velocity when < = given by the
equations

from r = 6 to r = c, and to suppose them equal to zero for all other
values of r ; but we are not therefore at liberty to suppress the
second arbitrary function in the integral of (3). The problem is
only a particular case of that considered by Poisson, and the
arbitrary functions are determined by his equations (6) and (8),
where, however, it must be observed, that the arbitrary functions
which Poisson denotes by /, F must not be confounded with the
given function here denoted by f, which latter will appear at the
right-hand side of equations (8). The solution presents no diflS-
culty in principle, but it is tedious from the great number of cases
to be considered, since the form of one of the functions which
enter into the result changes whenever the value of r-\- at or of
r^at passes through either b or c, or when that of r—at passes
through zero. It would be found that unless/ (6) = 0, a backward
wave sets out from the inner surface of the spherical shell contain-

86 ON SOME POINTS IN THE

ing the disturbed portion of the fluid ; and unless /(c) = 0, a
similar wave starts from the outer surface. Hence, whenever the
disturbance can be propagated in the positive direction only, we
must have -4, or /(c) — /(6), equal to zero. When a backward
wave is formed, it first approaches the centre, which in due time
it reaches, and then begins to diverge outwards, so that after the
time cja there is nothing left but an outward travelling wave, of
breadth 2c, in which the fluid is partly rarefied and partly con-
densed, in such a manner that /r^dr taken throughout the wave,
or -4, is equal to zero.

It appears, then, that for any outward travelling wave, or for
any portion of such a wave which can be isolated, the quantity A
is necessarily equal to zero. Consequently the conclusion arrived
at, that the mean condensation in such a wave or portion of a wave
varies ultimately inversely as the distance from the centre, proves
not to be true. It is true, as commonly stated, that the conden-
sation at corresponding points in such a wave in its successive
positions varies ultimately inversely as the distance from the
centre; it is likewise true, as Professor Challis has argued, that
the mean condensation in any portion of the wave which may be
isolated varies ultimately inversely as the square of the distance ;
but these conclusions do not in the slightest degree militate
against each other.

If we suppose h to increase indefinitely, the condensation or
rarefaction in the wave which travels towards the centre will be a
small quantity, of the order 6'*, compared with that in the shell.
In the limiting case, in which 6 = oo , the condensation or rarefac-
tion in the backward travelling wave vanishes. If in the equations
of paragraph 3 we write 6 + a; for r, ha {x) for/' (r), and then sup-
pose h to become infinite, we shall get a5 = <r [x), t; = <r(a?). Con-
sequently a plane wave in which the relation v = as is satisfied
will be propagated- in the positive direction only, no matter
whether / a {x) dx taken from the beginning to the end of the
wave be or be not equal to zero; and therefore any arbitrary
portion of such a wave may be conceived to be isolated, and being
isolated, will continue to travel in the positive direction only,
without sending back any wave which will be propagated in the
negative direction. This result follows at once from the equations
which apply directly to plane waves ; I mean, of course, the approxi-

RECEIVED THEORY OF SOUND. 87

mate equations obtained by neglecting the squares of small quan-
tities. It may be observed, however, that it appears from what
has been proved, that it is a property of every plane wave which is
the limit of a spherical wave, to have its mean condensation equal
to zero ; although there is no absurdity in conceiving a plane wave
in which that is not the case as already existing, and inquiring in
what manner such a wave will be propagated.

There is another way of putting the apparent contradiction
arrived at in the case of spherical weaves, which Professor Challis
has mentioned to me, and has given me permission to publish.
Conceive an elastic spherical envelope to exist in an infinite mass
of air which is at rest, and conceive it to expand for a certain time,
and then to come to rest again, preserving its spherical form and
the position of its centre during expansion. We should apparently
have a wave consisting of condensation only, without rarefaction,
travelling outwards, in which case the conclusion would follow,
that the quantity of matter altered with the time.

Now in this or any similar case we have a perfectly definite
problem, and our equations are competent to lead to the complete
solution, and so make known whether or not a wave will be propa-
gated outwards leaving the fluid about the envelope at rest, and if
such a wave be formed, whether it will consist of condensation
only, or of condensation accompanied by rarefaction : that conden-
sation will on the whole prevail is evident beforehand, because a
certain portion of space which was occupied by the fluid is now
occupied by the envelope.

In order to simplify as much as possible the analysis, instead
of an expanding envelope, suppose that we have a sphere, of a
constant radius b, at the surface of which fluid is supplied in such
a manner as to produce a constant velocity V from the centre out-
wards, the supply lasting from the time to the time t, and then
ceasing. This problem is evidently just as good as the former for
the purpose intended, and it has the advantage of leading to a
result which may be more easily worked out. On account of the
length to which the present article has already run, I am unwiUing
to go into the detail of the solution ; I will merely indicate the
process, and state the nature of the result.

Since we have no reason to suspect the existence of a function
of the form F {r+at) in the value of (f> which belongs to the

88 ON SOME POINTS IN THE RECEIVED THEORY OF SOUND.

present case, we need not burden our equations with this fanction,
but we may assume as the expression for (f>

^J-^ (7).

For we can always, if need be, fall back on the complete integral
of (3) ; and if we find that the particular integral (7) enables us to
satisfy all the conditions of the problem, we are certain that we
should have arrived at the same result had we used the complete
integral all along. These conditions are

^ = when ^ = 0, from r = 6 to r= oo (8);

for ^ must be equal to a constant, since there is neither condensa-
tion nor velocity, and that constant we are at liberty to suppose
equal to zero ;

^= 7whenr = 6, from« = Oto< = T (9);

-? = when r = 6, fromf = t to f = oo (10).

(8) determines /(-^) from « = 6 to « = oo ; (9) determines /(-?) from
« = J to z^h-^ar) and (10) determines f{z) from « = 6 — ar to
« = — 00 , and thus the motion is completely determined.

It appears from the result that if we consider any particular
value of r there is no condensation till at^r^h, when it suddenly
commences. The condensation lasts during the time t, when it is
suddenly exchanged for rarefaction, which decreases indefinitely,
tending to as its limit as t tends to oo . The sudden commence-
ment of the condensation, and its sudden change into rarefaction,
depend of course on the sudden commencement and cessation of
the supply of fluid at the surface of the sphere, and have nothing
to do with the object for which the problem was investigated.
Since there is no isolated wave of condensation travelling outwards,
the complete solution of the problem leads to no contradiction, as
might have been confidently anticipated.

[From the CantJbridge <md DtMin Mathematical Journal^ VoL nr. p. 1,

{February, 1849.)]

On the Perfect Blackness of the Central Spot in Newton's
Rings, and on the Verification of Fresnel's Formulae
FOR the Intensities of Reflected and Refracted Rays.

When Newton's rings are fonned between two glasses of the
same kind, the central spot in the reflected rings is observed to be
perfectly black. This result is completely at variance with the
theory of emissions, according to which the central spot ought to
be half as bright as the brightest part of the bright rings, supposing
the incident light homogeneous. On the theory of undulations,
the intensity of the light reflected at the middle point depends
entirely on the proportions in which light is reflected and refracted
at the two surfaces of the plate of air, or other interposed medium,
whatever it may be. The perfect blackness of the central spot
was first explained by Poisson, in the case of a perpendicular
incidence, who shewed that when the infinite series of reflections
and refractions is taken into account, the expression for the inten-
sity at the centre vanishes, the formula for the intensity of light
reflected at a perpendicular incidence first given by Dr Young
being assumed* Fresnel extended this conclusion to all incidences
by means of a law discovered experimentally by M. Arago, that
light is reflected in the same proportions at the first and second
surfaces of a transparent plate*. I have thought of a very simple
mode of obtaining M. Arago's law from theory, and at the same

* See Dr Lloyd's Report on Physical Optics.— i{«port« of the British Auoeiatian,
Vol. ni. p. 844.

00 ON THE PERFECT BLACKNESS OF THE

time establishing theoretically the loss of half an undulation in
internal, or else in external reflection.

This method rests on what may be called the principle of rever-
sion, a principle which may be enunciated as follows.

If any material system, in which the forces acting depend only
on the positions of the particles, be in motion, if at any instant the
velocities of the particles be reversed, the previous motion will be
repeated in a reverse order. In other words, whatever were the
positions of the particles at the time t before the instant of rever-
sion, the same will they be at an equal interval of time t after
reversion; from whence it follows that the velocities of the par-
ticles in the two cases will be equal in magnitude and opposite in
direction.

Let S be the surface of separation of two media which are both
transparent, homogeneous, and uncrystallized. For the present
purpose S may be supposed a plane. Let ^ be a point in the
surface S where a ray is incident along I A in the first medium.
Let AR^ AFhQ the directions of the reflected and refracted rays,
AR the direction of the reflected ray for a ray incident along FA,
and therefore also the direction of the refracted ray for a ray inci-
dent along RA. Suppose the vibrations in the incident ray to be
either parallel or perpendicular to the plane of incidence. Then
the vibrations in the reflected and refracted rays will be in the
first case parallel and in the second case perpendicular to the plane
of incidence, since everything is symmetrical with respect to that
plane. The direction of vibration being determined, it remains to
determine the alteration of the coefficient of vibration. Let the
maximum vibration in the incident light be t.aken for unity, and,
according to the notation employed in Airy's Tract, let the coeffi-
cient of vibration be multiplied by 6 for reflection and by c for
refraction at the surface S, and by e for reflection and /for refrac-
tion at a parallel surface separating the second medium from a
third, of the same nature as the first.

Let X be measured from A negatively backwards along AI,
and positively forwards along AR or AF, and let it denote the
distance from A of the particle considered multiplied by the refrac-
tive index of the medium in which the particle is situated, so that
it expresses an equivalent length of path in vacuum. Let X be the

CENTRAL SPOT IN NEWTON'S RINGS. 91

length of a wave, and v the velocity of propagation in vacuum ;
and for shortness sake let

~ (vt — x) = X.

Then sin X, b sin X, c sin X may be taken to represent respec-
tively the incident, reflected, and refracted rays ; and it follows
from the principle of reversion, if we suppose it applicable to light,
that the reflected and refracted rays reversed will produce the
incident ray reversed. Now if in the reversed rays we measure x
positively along -4/ or AR, and negatively along AR or AF, the
reflected ray reversed will give rise to the rays represented by

b* sin X, reflected along A I;
be sin X, refracted along AR' ;*

and the refracted ray reversed will give rise to

cf sin X, refracted along AT]
ce sin X, reflected along AR\

The two rays along AR' superposed must destroy each other, and
the two along AI must give a ray represented by sin X. We have
therefore

6c + ce = 0, 6' + c/=l;

* It does not at once appear whether on reyersing a ray we ought or ought not
to change the sign of the coefficient ; hut the following considerations will shew
that we most leave the sign unaltered. Let the portion of a wave, in which the
displacement of the ether is in the direction which is considered positiye, be called
the positive portion, and the remaining part the negative portion; and let the
points of separation he called nodes. There are evidently two sorts of nodes : the
nodes of one sort, which may be called positive nodes, being situated in front of the
positive portions of the waves, and the nodes of the other sort, which may be called
negative nodes, being situated behind the positive portions or in front of the
negative, the terms in front and behind referring to the direction of propagation.
Now when the angle X vanishes, the particle considered is in a node ; and since, at
the same time, the expression for the velocity of the particle is positive, the co-
efficient of sin Z being supposed positive, the node in question is a positive node.
When a ray is reversed, we must in the first instance change the sign of the
coefficient, since the velocity is reversed ; but since the nodes which in the direct
ray were positive are negative in the reversed ray, and vice versd, we must more-
over add dbr to the phase, which comes to the same thing as changing the sign
back again. Thus we must take l^ sin X, as in the text, and not ~ 6^ sin X, to
represent the ray reflected along AI, and so in other cases.

92 ON THE PERFECT BLACKNESS OF THE

and therefore, since c is not zero,

6 = -6 (1),

c/=l-6' = l-6« (2).

Equation (1) contains at the same time M. Arago's law and the
loss of half an undulation; and equations (1) and (2) together
explain the perfect blackness of the centre of Newton's rings.
(See Airy's Tract)

If the incident light be common light, or polarized light, of any
kind except plane polarized for which the plane of polarization
either coincides with the plane of incidence or is perpendicular to
it, we can resolve the vibrations in and perpendicular to the plane
of incidence, and consider the two parts separately.

It may be observed that the principle of reversion is just as
applicable to the theory of emissions as to the theory of undula-
tions ; and thus the emissionists are called on to explain how two
rays incident along It A, FA respectively can fail to produce a ray
along AR\ In truth this is not so much a new difficulty as an old
difficulty in a new shape ; for if any mode could be conceived of
explaining interference on the theory of emissions, it would pro-
bably explain the non-existence of the ray along AR\

Although the principle of reversion applies to the theory of
emissions, it does not lead, on that theory, to the law of intensity
resulting from equations (1) and (2). For the formation of these
equations involves the additional principle of superposition, which
on the theory of undulations is merely a general dynamical
principle applied to the fundamental hypotheses, but which does
not apply to the theory of emissions, or at best must be assumed,
on that theory, as the expression of a property which we are
compelled to attribute to light, although it appears inexplicable.

In forming equations (1) and (2) it has been tacitly assumed
that the reflections and refractions were unaccompanied by any
change of phase, except the loss of half an undulation, which may
be regarded indiflferently as a change of phase of 180**, or a change
of sign of the coefficient of vibration. In very highly refracting
substances, however, such as diamond, it appears that when the
incident light is polarized in a plane perpendicular to the plane
of incidence, the reflected light does not wholly vanish at the

CENTRAL SPOT IN NEWTOX's KINGS. 93

polarizing angle ; but as the angle of incidence passes through the
polarizing angle, the intensity of the reflected light passes through
a small minimum value, and the phase changes rapidly through
an angle of nearly 180^ Suppose, for the sake of perfect gener-
ality, that all the reflections and refractions are accompanied by
changes of phase. While the coefficient of vibration is multiplied
by h, c, e, or/, according to the previous notation, let the phase of
vibration be accelerated by the angle )8, 7, e, or <^, a retardation
being reckoned as a negative acceleration. Then, if we still take
sin X to represent the incident ray, we must take b sin {X + fi),
csin(-X'+7) to represent respectively the reflected and the re-
fracted rays. After reversion we must change the signs of )3 and
% because, whatever distance a given phase of vibration has
receded from A in consequence of the acceleration accompanying
reflection or refraction, the same additional distance will it have to
get over in returning to A after reversion. We have therefore
bsin{X—^), csin(Z — 7) to represent the rays incident along
BA, FA, which together produce the ray sinX along AL Now
the ray along RA alone would produce the rays

V sin X along AT, 6c sin (Z - /9 + 7) along AR ;

and the ray along FA alone would produce the rays

cf sin (X— 7 + ^) along AI, ce sin (Z — 7 + c) along AR,

We have therefore in the same way as before

c/sm (Z-7 + <^) = (1- J') sinX,
b sin (X-/9 + 7) + e sin (X-7 + e) = 0.

Now each of these equations has to hold good for general
values of X, and therefore, as may very easily be proved, the
angles added to X in the two terms must either be equal or must
differ by a multiple of 180^ But the addition of any multiple of
360* to the angle in question leaves everything the same as before,
and the addition of 180® comes to the same thing as changing the
sign of c or / in the first equation, or of J or e in the second. We
are therefore at liberty to take

^ = 7 (^)'

/3 + € = 27 (4);

and the relations between 6, c, e, and / will be the same as before.

94 ON THE PERFECT BLACKNESS OF TUE

Hence M. Arago's law holds good even when reflection and re-
fraction are accompanied by a change of phase.

Equations (3) and (4) express the following laws with refer-
ence to the changes of phase. The sum of ike accelerations of
pJuLse at the two reflections is equal to the sum of the accelerations
at the two refractions ; and the accelerations at the two refractions
are equal to each other. It will be observed that the accelerations
are here supposed to be so measured as to give like signs to c and
f, and unlike to b and e.

If we suppose the reflections and refractions accompanied by
changes of phase, it is easy to prove, from equations (3) and (4),
that when Newton's rings are formed between two transparent
media of the same kind, the intensities of the light in the re-
flected and transmitted systems are given by the same formulae as
when there are no changes of phase, provided only we replace the
retardation 27rV/\ (according to the notation in Airy's Tract)
by 27rV/\ — 2e, or replace D, the distance of the media, by
D - X€/27r cos )S.

Let us now consider some circumstances which might at first
sight be conceived to affect the conclusions arrived at.

When the vibrations of the incident light take place in the
plane of incidence, it appears from investigation that the condi-
tions at the surface of separation cannot all be satisfied by means
of an incident, reflected, and refracted wave, each consisting of
vibrations which take place in the plane of incidence. If the
media could transmit normal vibrations with velocities com-
parable to those with which they transmit tranversal vibrations,
the incident wave would occasion two reflected and two refracted
waves, one of each consisting of normal, and the other of trans-
versal vibrations, provided the angle of incidence were less than
the smallest of the three critical angles (when such exist), cor-
responding to the refracted transversal vibrations and to the re-
flected and refracted normal vibrations respectively. There appear
however the strongest reasons for regarding the ether as sensibly
incompressible, so that the velocity of propagation of normal vibi-a-
tions is incomparably greater than that of transversal vibrations.
On this supposition the two critical angles for the normal vibra-
tions vanish, so that there are no normal vibrations transmitted in
the regular way whatever be the angle of incidence. Instead of

CENTRAL SPOT IN NEWTON's RINGS. 95

such vibrations there is a sort of superficial undulation in each
medium, in which the disturbance is insensible at the distance of
a small multiple of X from the surface : the expressions for these
disturbances involve in fact an exponential with a negative index,
which contains in its numerator the distance of the point consi-
dered from the common surface of the media. It is easy to see
that the existence of the superficial undulations above mentioned
does not affect the truth of equations (1), (2), (3), (4) ; for, to ob-
tain these equations, it is sufficient to consider points in the media
whose distances from the surface are greater than that for which
the superficial undulations are sensible.

No notice has hitherto been taken of a possible motion of the
material molecules, which we might conceive to be produced by the
vibrations of the ether. If the vibrations of the molecules take
place in the same period as those of the ether, and if moreover
they are not propagated in the body either regularly, with a
velocity of propagation of their own, or in an irregular manner, the
material molecules and the ether form a single vibrating system ;
they are in fact as good as a single medium, and the principle of
reversion will apply.

In either of the excepted cases, however, the principle would
not apply, for the same reason that it might lead to false results if
there were normal vibrations produced as well as transversal, and
the normal vibrations were not taken into account. In the case
of transparent media, in which there appears to be no sensible loss
of light by absorption for the small thicknesses of the media with
which we are concerned in considering the laws of reflection and
refraction, we are led to suppose, either that the material mole-
cules are not sensibly influenced by the vibrations of the ether, or
that they form with the ether a single vibrating system; and
consequently the principle of reversion may be applied. In the
case of opaque bodies, however, it seems likely that the labour-
ing force brought by the incident luminous vibrations is partly
consumed in producing an irregular motion among the molecules
themselves.

When a convex lens is merely laid on a piece of glass, the
central black spot is not usually seen ; the centre is occupied by
the colour belonging to a ring of some order. It requires the
exertion of a considerable amount of pressure to bring the glasses

96 ox THE PERFECT BLACKNESS OF THE

into sufficiently intimate contact to allow of the perfect formation
of the central spot.

Suppose that we deemed the glasses to be in contact when
they were really separated by a certain interv^al A, and for simpli-
city suppose the reflections and refractions unaccompanied by any
change of phase, except the loss of half an undulation. It evidently
comes to the same thing to suppose the reflections and refractions
to take place at the surfaces at which they do actually take place,
as to suppose them to take place at a surface midway between the
glasses, and to be accompanied by certain changes of phase ; and
these changes ought to satisfy equations (3) and (4). This may
be easily verified. In fact, putting fjt, fi for the refractive
indices of the first and second media, i, f for the angles of in-
cidence and refraction, we easily find, by calculating the retarda-
tions, that

from which we get, by interchanging i and i\ fi and fju, and chang-
ing the signs, since for the first reflection and refraction the true
surfjEice comes before the supposed, but for the second the supposed
surface comes before the true,

27rA , ., ttA fi . ,., ^

€ = r — a cos I, = -r .^— . Sm U — I) I

\ ^ X sin 1 ^ '

and these values satisfy equations (3) and (4), as was foreseen.

Hitherto the common surface of the media has been spoken of
as if the media were separated by a perfectly definite surface, up to
which they possessed the same properties respectively as at a
distance from the surface. It may be observed, however, that the
application of the principle of reversion requires no such restriction.
We are at liberty to suppose the nature of the media to change in
any manner in approaching the common surface ; we may even sup-
pose them to fade insensibly into each other ; and these changes
may take place within a distance which need not be small in com-
parison with X.

It may appear to some to be superfluous to deduce particular
results from hypotheses of great generality, when these results may
be obtained, along with many others which equaDy agree with
observation, from more refined theories which start with more

CENTRAL SPOT IN NEWTON's RINGS. 97

particular hypotheses. And indeed, if the only object of theories
were to group together observed facts, or even to allow us to pre-
dict the results of observation in cases not very different from
those already observed, and grouped together by the theory, such
a view might be correct. But theories have a higher aim than
this. A well*established theory is not a mere aid to the memory,
but it professes to make us acquainted with the real processes of
nature in producing observed phenomena. The evidence in favour
of a particular theory may become so strong that the fundamental
hypotheses of the theory are hardly less certain than observed
facts. The probability of the truth of the hypotheses, however,
cannot be greater than the improbability that another set of
equally simple hypotheses should be conceivable, which should
equally well explain all the phenomena. When the hypotheses
are of a general and simple character, the improbability in ques-
tion may become extremely strong ; but it diminishes in propor-
tion as the hypotheses become more particular. In sifting the
evidence for the truth of any set of hypotheses, it becomes of great
importance to consider whether the phenomena explained, or some
of them, are explicable on more simple and general hypotheses, or
whether they appear absolutely to require the more particular
restrictions adopted. To take an illustration from the case in
hand, we may suppose that some theorist, starting with some par-
ticular views as to the cause of the diminished velocity of light
in refracting media, and supposing that the transition from one
medium to another takes place, if not abruptly, at least in a space
which is very small compared with X, has obtained as the result of
his analysis M. Arago's law and the loss of half an undulation. We
may conceive our theorist pointing triumphantly to these laws as
an evidence of the correctness of his particular views. Yet, as wo
have seen, if these were the only laws obtained, the theorist would
have absolutely no solid evidence of the truth of the particular
hypotheses with which he started.

This fictitious example leads to the consideration of the ex-
perimental evidence for Fresners expressions for the intensity of
reflected and refracted polarized light

There are three particular angles of incidence, namely the
polarizing angle, the angle of 90**, and the angle 0°, for which
special results are deducible from Fresnel's formulsa, which admit
of being put, and which have been put, to the test of experiment.

an. T

98 ON THE VERIFICATIOK OF FRESNEL'S FORMUUE FOB

The accordance of the results with theory is sometimes adduced as
evidence of the truth of the formulae: but this point will require
consideration.

In the first place, it follows from Fresnel's formula for the
intensity of reflected light which is polarized in a plane perpen-
dicular to the plane of incidence, that at a certain angle of inci-
dence the reflected light vanishes; and this angle is precisely that
determined by experiment. This result is certainly very remark-
able. For Fresnel's expressions are not mere empirical formula),
chosen so as to satisfy the more remarkable results of experiment*
On the contrary, they were obtained by him fit)m dynamical con-
siderations and analogies, which, though occasionally somewhat
vague, are su£Scient to lead us to regard the formulse as having a
dynamical foundation, as probably true under circumstances which
without dynamical absurdity might be conceived to exist; though
whether those circumstances agree with the actual state of reflect-
ing transparent media is another question. Consequently we
should a priori expect the formulae to be either true or very nearly
true, the diflFerence being attributable to some modifying cause
left out of consideration, or else to be altogether false: and there-
fore the verification of the formulae in a remarkable, though a
particular case, may be looked on as no inconsiderable evidence of
their general truth. It will be observed that the truth of the
formulae is here spoken of, not the truth of the hypotheses con-
cerned in obtaining them from theory.

Nevertheless, even the complete establishment of the formula
for the reflection of light polarized in a plane perpendicular to the
plane of incidence would not establish the formula for light pola-
rized in the plane of incidence, although it would no doubt increase
the probability of its truth, inasmuch as the two formulae were
obtained in the same sort of way. But, besides this, the simplicity
of the law, that the reflected ray vanishes when its direction be-
comes perpendicular to that of the refracted ray, is such as to lead
us to regard it as not improbable that difierent formulae, corre-
sponding to difierent hypotheses, should agree in this point And
in fact the investigation shews that when sound is reflected at the
common surface of two gases, the reflected sound vanishes when
the angle of incidence becomes equal to what may be called, from
the analogy of light, the polarizing angle. It is true that the
formula for the intensity of the reflected sound agrees with the

THE INTENSITIES OF REFLEC5TED AND REFRACTED RAYS. 99

formula for the intensity of reflected light when the light is pola-
rized in a plane perpendicular to the plane of incidence, and that
it is the truth of the formulae, not that of the hypotheses, which is
under consideration. Nevertheless the formulae require further
confirmation.

When the angle of incidence becomes 90**, it follows from
Fresnel's expressions that, whether the incident light is polarized
in or perpendicularly to the plane of incidence, the intensity of the
reflected light becomes equal to that of the incident, and conse-
quently the same is true for common light. This result has been
compared with experiment, and the completeness of the reflection
at an incidence of 90** has been established*. The evidence, how-
ever, for the truth of Fresners formulae which results from this
experiment is but feeble: for the result follows in theory from the
principle of vis viva, provided we suppose none of the labouring
force brought by the incident light to be expended in producing
among the molecules of the reflecting body a disturbance which is
propagated into the interior, as appears to be the case with opaque
bodies. Accordingly a great variety of difierent particular hypo-
theses, leading to formulae dififering from one another, and from
Fresnel's, would agree in giving a perfect reflection at an incidence
of 90**. Thus for example the formula which Green has givenffor
the intensity of the reflected light, when the incident light is pola-
rized in a plane perpendicular to the plane of incidence, gives the
intensities of the incident and reflected light equal when the angle
of incidence becomes 90**, although the formula in question differs
from Fresnel's, with which it only agrees to a first approximation
when fi is supposed not to differ much from 1. It appeared in
the experiment last mentioned that the sign of the reflected vibra-
tion was in accordance with Fresnel's formulae, and that there was
no change of phase. Still it is probable that a variety of formulae
would agree in these respects.

When the angle of incidence vanishes, it follows from Fresnel's
expressions, combined with the fundamental hypotheses of the
theory of transversal vibrations, that if the incident light be circu-
larly polarized, the reflected light will be also circularly polarized,
but of the opposite kind, the one being right-handed, and the other

• Transactions of the Royal Irish Academy, vol. XTii. p. 171.

t Transactions of the Cambridge Philosophical Society y vol. yn. p. 22.

7—2

100 ON THE VERIFICATION OF FRESNEL's FORMULA FOR

left-handed*. The experiment has been performed, at least per-
formed for a small angle of incidence^, from whence the result
which would have been observed at an angle of incidence 0® may
be inferred ; and theory has proved to be in complete accordance
with experiment. Yet this experiment, although confirming the
theory of transversal vibrations, offers absolutely no confiimation of
Fresnel's formulse. For when the angle of incidence vanishes,
there ceases to be any distinction between light polarized in, and
light polarized perpendicularly to the plane of incidence: be the
intensity of the reflected light what it may, it must be the same in
the two cases; and this is all that is necessary to assume in de-
ducing the result from theory. The result would necessarily be
the same in the case of metallic reflection, although Fresnel's for-
mula do not apply to metals.

By the fundamental hypotheses of the theory of transverse
vibrations, are here meant the suppositions, flrst, that the vibrations,
at least in vacuum and in ordinary media, take place in the front
of the wave; and secondly, that the vibrations in the case of plane
polarized light are, like all the phenomena presented by such light,
symmetrical with respect to the plane of polarization, and conse-
quently are rectilinear, and take place either in, or perpendicularly
to the plane of polarization. From these hypotheses, combined
with the principle of the superposition of vibrations, the nature of
circularly and elliptically polarized light follows. As to the two
suppositions above mentioned respecting the direction of the vibra-
tions in plane polarized light, there appears to be nothing to choose
between them, so far as the geometrical part of the theory is con-
cerned: they represent observed facts equally well The question
of the direction of the vibrations, it seems, can only be decided, if
decided at all, by a dynamical theory of light. The evidence ac-
cumulated in favour of a particular dynamical theory may be con-
ceived to become so strong as to allow us to regard as decided the
question of the direction of the vibrations of plane polarized light.
It appears, however, that Fresnel's expressions for the intensities,
and the law which gives the velocities of plane waves in different
directions within a crystal, have been deduced, if not exactly, at
least as approximations to the exact result, from different dyna-

* PhUaophical Magazine {New Series), vol. zxn. (1S43) p. 92.
t IHd. p. 262.

THB INTENSITIES OF REFLECTED AND BEFEACTED RAYS. 101

mical theories, in some of which the vibrations are supposed to be
in, and in others perpendicular to the plane of polarization.

It is worthy of remark that, whichever supposition we adopt,
the direction of revolution of an ethereal particle in circularly
polarized light formed in a given way is the same. Similarly, in
elliptically polarized light the direction of revolution is the same
on the two suppositions, but the plane which on one supposition
contains the major axis of the ellipse described, on the other sup-
position contains the minor axis. Thus the direction of revolution
may be looked on as established, even though it be considered
doubtful whether the vibrations of plane polarized light are in, or
perpendicular to the plane of polarization.

The verification of Fresnel's formulse for the three particular
angles of incidence above mentioned is, as we have seen, not suffi-
cient: the formulaB however admit of a very searching comparison
with experiment in an indirect way, which does not require any
photometrical processes. When light, polarized in a plane making
a given angle with the plane of incidence, is incident on the sur-
hce of a transparent medium, it follows from Fresnel's formulae
that both the reflected and the refracted light are plane polarized,
and the azimuths of the planes of polarization are known functions
of the angles of incidence and refraction, and of the azimuth of
the plane of polarization of the incident light, the same formulas
being obtained whether the vibrations of plane polarized light are
supposed to be in, or perpendicular to the plane of polarization.
It is found by experiment that the reflected or refiracted light is
plane polarized, at least if substances of a very high refiractive
power be excepted, and that the rotation of the plane of polariza-
tion produced by reflection or refraction agrees with the rotation
determined by theory. This proves that the two formulae, that is
to say the formula for light polarized in, and for light polarized
•perpendicularly to the plane of incidence, are either both right,
within the limits of error of very precise observations, or both
wrong in the same ratio, where the ratio in question may be any
function of the angles of incidence and refraction. There does not
appear to be any reason for suspecting that the two formulae for
reflection are both wroug in the same ratio. As to the formulae
for refraction, the absolute value of the displacement will depend
on the particular theory of refraction adopted. Perhaps it would
be best, in order to be independent of any particular theoxy, to

[From the Cambridge atui Dublin Mathematical Journal^ VoL iv. p. 194

{Mcuf and November^ 1849).]

On ATTRAcnoNS, and on Clairaut's Theorem.

Clairaut's Theorem is usually deduced as a consequence
of the hypothesis of the original fluidity of the earth, and the
near agreement between the numerical values of the earth's ellip-
ticity, deduced independently from measures of arcs of the meridian
and from pendulum experiments, is generally considered as a
strong confirmation of the hypothesis. Although this theorem is
usually studied in connection with the hypothesis just mentioned,
it ought to be observed that Laplace, without making any assump-
tion respecting the constitution of the earth, except that it consists
of nearly spherical strata of equal density, and that its surface
may be regarded as covered by a fluid, has established a connexion
between the form of the surface and the variation of gravity, which
in the particular case of an oblate spheroid gives directly Clairaut's
Theorem*. If, however, we merely assume, as a matter of obser-
vation, that the earth's surface is a surface of equilibrium, (the
trifling irregularities of the surface being neglected), that is to say
that it is perpendicular to the direction of gravity, then, indepen«
dently of any particular hypothesis respecting the state of the
interior, or any theory but that of universal gravitation, there
exists a necessary connexion between the form of the surface and
the variation of gravity along it, so that the one being given the
other follows. In the particular case in which the surface is an

* See the Micanique Celeste, I4V. ul, or Uie ref«renoe to U in Fratt*8 Mechanics^
Chap. Fiffure qf the Earth,

ON ATTRACTIONS, AND ON CLAIRAUT*S THEOREM. 105

oblate spheroid of small eccentricity, which the measures of arcs
shew to be at least very approximately the form of the earth's
surface, the variation of gravity is expressed by the equation which
is arrived at on the hypothesis of original fluidity. I am at present
engaged in preparing a paper on this subject for the Cambridge
Philosophical Society : the object of the following pages is to give
a demonstration of Clairaut's Theorem, different from the one
there employed, which will not require a knowledge of the pro-
perties of the functions usually known by the name of Laplace's
Functions. It will be convenient to commence with the demon-
stration of a few known theorems relating to attractions, the law
of attraction being that of the inverse square of the distance*.

Preliminary Propositiom respecting AUracHons.

Prop. i. To express the components of the attraction of any
mass in three rectangular directions by means of a single function.

Let m' be the mass of an attracting particle situated at the
point P\ the unit of mass being taken as is usual in central
forces, m the mass of the attracted partiple situated at the point
P, x\ j/, z the rectangular co-ordinates of P' referred to any origin,
a?, y, z those of P ; X, Y^ Z the components of the attraction of
ni on m, measured as accelerating forces, and considered positive
when they tend to increase x,y,z\ then, if PP' = r\

ft!

* My object in giTing these demonstrations is simply to enable a reader who
may not have attended particularly to the theory of attractions to follow with
facility the demonstration here given of Cloiraut's Theorem. In speaking of the
theorems as "known" I have, I hope, sufficiently disclaimed any pretence at
originality. In fact, not one of the ** propositions respecting attractions " is new,
although now and then the demonstrations may differ from what have hitherto
been given. With one or two exceptions, these propositions will aU be found in
a paper by Gauss, of which a translation is published in the third volume of
Taylor's Scientific Memoirs, p. 163. The demonstration here given of Prop. rv. is
(he same as Gauss's; that of Prop, v., though less elegant than Gauss's, appears to
me more natural. The ideas on which it depends render it closely allied to a paper
by Professor Thomson, in the third volume of this Journal (Old Series), p. 71.
Prop. iz. is given merely for the soke of exemplifying the application of the same
mode of proof to a theorem of Gauss's.

106 ON ATTKACnONS, AND

Since r'« = (x'-^)*+(y'-y)* + (^'-^)',

, ,dr / / \ 1 V m df^ d m'

wehaver^^ (^-^)' ^'^^"^ -^==- i^ rfi^dii^^

with similar equations for Y and Z

If instead of a single particle m' we have any number of
attracting particles *»', m"... situated at the points {x, y', «'),
(a?", y", «")...> and if we put

^ + ^'+... = t^=K (1).

weget

If instead of a set of distinct particles we have a continuous
attracting mass M', and if we denote by dm' a differential element
of M\ and replace (1) by

y-lll'-r w.

equations (2) will still remain true, provided at least P be external
to M' ; for it is only in that case that we are at liberty to consider
the continuous mass as the limit of a set of particles which are all
situated at finite distances from P. It must be observed that
should M' occupy a closed shell, within the inner surface of which
P is situated, P must be considered as external to the mass IT.
Nevertheless, even when P lies within M\ or at its surface, the

expressions for Fand dV/dx, namely III —r- and ///(^' — ^) —r >

admit of real integration, defined as a limiting summation, as may
be seen at once on referring if to polar co-ordinates originating
at P ; so that the equations (2) still remain true.

F£OP. n. To express the attraction resolved along any line
by means of the function V,

Let 8 be the length of the given line measured from a fixed
point up to the point P; \ fi, v, the direction-cosines of the
tangent to this line at P, F the attraction resolved along this
tangent; then

ON clairaut's theorem. 107

Now if we restrict oureelves to points lying in the line s, V will
be a function of s alone ; or we may regard it as a function of x, y,
and «, each of which is a function of 8 ; and we shall have, by
Differential Calculus,

dV^dVdx dVdy dVdz^
ds dx ds dt/ ds dz ds '

and since dw/Js = X, dy/ds = fi, dz/ds = v, we get

^-'i (*)•

Prop. hi. To examine the meaning of the function V.

This function is of so much importance that it will be well to
dwell a little on its meaning.

In the first place it may be observed that the equation (1)
or (3) contains a physical definition of V, which has nothing to do
with the system of co ordinates, rectangular, polar, or any other,
which may be used to define algebraically the positions of P and
of the attracting particles. Thus F is to be contemplated as a
function of the position of P in space, if such an expression may
be allowed, rather than as a function of the co-ordinates of P;
although, in consequence of its depending upon the position of P,
V will be a function of the co-ordinates of P, of whatever kind
they may be.

Secondly, it is to be remarked that although an attracted
particle has hitherto been conceived as situated at P, yet V has
a definite meaning, depending upon the position of the point P,
whether any attracted matter exist there or not. Thus V is to be
contemplated as having a definite value at each point of space,
irrespective of the attracted matter which may exist in some
places.

The function V admits of another physical definition which
ought to be noticed. Conceive a particle whose mass is m to move
along any curve from the point P^ to P. If jP be the attraction
of M' resolved along a tangent to m*s path, reckoned as an accele-
rating force, the moving force of the attraction resolved in the
same direction will be niF, and therefore the work done by the
attraction while m describes the elementary arc da will be ulti-
mately mFdSy or by (4) vi . dV/ds . ds. Hence the whole work done
as m moves from P^ to P is equal to m(F— F^, V^ being the

108 ON ATTRACTIONS, AND

value of V at P^. If P^ be situated at an infinite distance, V^
vanishes, and the expression for the work done becomes simply
mV, Hence K might be called the work of the attraction, referred
to a unit of mass of the attracted particle; but besides that such
a name would be inconveniently long, a recognized name already
exists. The function V is called the potential of the attracting
mass*

The first physical definition of V is peculiar to attraction ac-
cording to the inverse square of the distance. According to the
second, V is regarded as a particular case of the more general
function whose partial difiTerential coefficients with respect to x,y,z
are equal to the components of the accelerating force; a function
which exists whenever Xdx + Ydy + Zdz is an exact differential.

Prop. iv. If /S be any closed surface to which all the attract-
ing mass is external, dS an element of 8, cf n an element of the
normal drawn outwards at dS, then

'tu^'-^ (^)'

the integral being taken throughout the whole surface S,

Let m' be the mass of any attracting particle which is situated
at the point P', P being by hypothesis external to S. Through
P draw any right line L cutting S, and produce it indefinitely in
one direction from P. The line L will in general cut S in two
points; but if the surface 8 be re-entrant, it may be cut in four,
six, or any even number of points. Denote the points of section,
taken in order, by P^^ P^, /,, &c., P^ being that which lies nearest
to P. With P for vertex, describe about the line L a conical
surface containing an infinitely small solid angle a, and denote by
J.p J.,... the areas which it cuts out from 8 about the points
Pj, P,.... Let 5j, 5,... be the angles which the normals drawn
outwards at P^, P,... make with the line i, taken in the direction
from Pj to P'; N^, N^... the attractions of m' at P^, P,... resolved
along the normals; Tj, r,... the distances of P^, P,... from P'. It

* [The term ** potential/' as used in the theory of Electricity, may be defined in
the foUowing manner : " The potential at any point P, in the neighbonrhood of
electrified matter, is the amount of work that would be necessary to remove a small
body charged with a unit of negative electricity from that position to an infinite
distance."— w. t.]

!l':

ON clairaut's theorem. 109

is evident that the angles 0^, 0^... will be alternately acute and
obtuse. Then we have

i\rj = ^cos 0^, -BT, = - ^ cos (tt - tf^ &c.

We have also in the limit

A^ =! ar* sec 0^, A^ = or,^ sec (tt — 0^, &c. ;
and therefore If^A^ = am\ N^^ = - am', -^7,^4, = am', &c.;
and therefore, since the number of points P^, P,... is even,
N^A^ + N^^ + N^A^ + N^A^.., = am' — am' + am — am'... = 0.

Now the whole solid angle contained within a conical surface
described with P' for vertex so as to circumscribe fl^ may be divided
into an infinite number of elementary solid angles, to each of which
the preceding reasoning will apply; and it is evident that the
whole surface 8 will thus be exhausted. We have therefore

limit of 2iV^ = 0;

or, by the definition of an integral,

jjNdS^ 0.

The same will be true of each attracting particle m'\ and there-
fore if N refer to the attraction of the whole attracting mass, we
shall still have JJNdS^O. But by (4) N^dVjdn^ which proves
the proposition.

Prop. v. If F be equal to zero at all points of a closed surface
8, which does not contain any portion of the attracting mass, it
must be equal to zero at all points of the space T contained with-
in £r.

For if not, V must be either positive or negative in at least a
certain portion of the space T, and therefore must admit of at least
one positive or negative maximum value F^. Call the point, or
the assemblage of connected points, at which V has its maximum
value Fj, 2\. It is to be observed, first, that T^ may denote either
a space, a surface, a line, or a single point; secondly, that should
F happen to have the same value V^ at other points within T,
such points must not be included in 2\. Then, all round T^ F is
decreasing, positively or negatively according as F^ is positive or
negative. Circumscribe a closed surface 8^ around 7^, lying

108 ON ATTBACTIONS, AND

value of V at P^. If P^, be situated at an infinite distance, V^
vanishes, and the expression for the work done becomes simply
mV. Hence K might be called the work of the attraction, referred
to a unit of mass of the attracted particle; but besides that such
a name would be inconveniently long, a recognized name already
exists. The function V is called the potential of the attracting
mass*.

The first physical definition of V is peculiar to attraction ac-
cording to the inverse square of the distance. According to the
second, V is regarded as a particular case of the more general
function whose partial differential coefficients with respect tox,y,z
are equal to the components of the accelerating force; a function
which exists whenever Xdx + Ydy + Zdz is an exact differential.

Prop. iv. If /8^ be any closed surface to which all the attract-
ing mass is external, dS an element of S, dn im element of the
normal drawn outwards at dS, then

'S'^^-o (5)'

the integral being taken throughout the whole surface S.

Let mf be the mass of any attracting particle which is situated
at the point P', P' being by hypothesis external to 8. Through
P' draw any right line L cutting S, and produce it indefinitely in
one direction from P'. The line L will in general cut S in two
points; but if the surface S be re-entrant, it may be cut in four,
six, or any even number of points. Denote the points of section,
taken in order, by P^, P,, P,, &c., Pj being that which lies nearest
to P'. With P for vertex, describe about the line L a conical
surface containing an infinitely small solid angle a, and denote by
A^, A^,., the areas which it cuts out from 8 about the points
Pj, P,.... Let ^,, 0^... be the angles which the normals drawn
outwards at P^, P,... make with the line Z, taken in the direction
from Pj to P'; N^, N^,,. the attractions of m' at P^, P,... resolved
along the normals; r^, r,... the distances of P^, P,.., from P\ It

* [The term ** potential/' as used in the theory of Electricity, may be defined in
the following manner : ** The potential at any point P, in the neighbourhood of
electrified matter, is the amoant of work that would be necessaiy to remove a small
body charged with a unit of negative electricity from that position to an infinite
distance."— w. t.]

!l'.

ON clairaut's theorem. 109

is evident that the angles 0^, 0^... will be alternately acute and
obtuse. Then we have

-BTj = — 5 cos 0^, J\r, =5 — ^ cos (tt — 0^) &c.

We have also in the limit

-4j =s ar* sec tfj, -4, = or/ sec (tt — tf^, &c.;
and therefore N'^A^ = am', -W^, = — am', -^,^4, = am', &c.;
and therefore, since the number of points Pj, P,... is even,
N^A^-{- N^^ + N^A^ 4- N^A^.,. = am' — am' + am' — am',.. = 0.

Now the whole solid angle contained within a conical surface
described with P' for vertex so as to circumscribe 8 may be divided
into an infinite number of elementary solid angles, to each of which
the preceding reasoning will apply; and it is evident that the
whole surface 8 will thus be exhausted. We have therefore

limit of XNA = 0;

or, by the definition of an integral,

fJNd8=0.

The same will be true of each attracting particle tn; and there-
fore if N refer to the attraction of the whole attracting mass, we
shall still have JJNd8^0. But by (4) N=zdV/dn, which proves
the proposition.

Prop. v. If V be equal to zero at all points of a closed surface
8, which does not contain any portion of the attracting mass, it
must be equal to zero at all points of the space T contained with-
in £r.

For if not, V must be either positive or negative in at least a
certain portion of the space T, and therefore must admit of at least
one positive or negative maximum value V^. Call the point, or
the assemblage of connected points, at which V has its maximum
value Fj, T^ It is to be observed, first, that T^ may denote either
a space, a surface, a line, or a single point; secondly, that should
V happen to have the same value V^ at other points within T,
such points must not be included in 2\. Then, all round 2^, F is
decreasing, positively or negatively according as Fj is positive or
negative. Circumscribe a closed surface 8^ around T^, lying

110 ON ATTRACTIONS, AND

wholly within 5, which is evidently possible. Then if ^^ be drawn
sufficiently close round T^, V will be increasing in passing out-
wards across S^*; and therefore, if n^ denote a normal drawn out-
wards at the element dS^ of /S^, dV/dn^ will be negative or positive

according as V^ is positive or negative, and therefore ll^^^i»

taken throughout the whole surface fif^, will be negative or positive,
which is contrary to Prop. IV. Hence V must be equal to zero
throughout the space T,

Cor. 1. If V be equal to a constant A at all points of the
surface S, it must be equal to ^ at all points within S. For it
may be proved just as before that V cannot be either greater or
less than A within 8.

Cor. 2. If V be not constant throughout the surface 8, and if
A be its greatest, and B its least value in that surface, V cannot
anywhere within 8 be greater than A nor less than B.

Cor. 3. All these theorems will be equally true if the space T
extend to infinity, provided that instead of the value of V at the
bounding surface of T we speak of the value of V at the surface by
which T is partially bounded, and its limiting value at an infinite
distance in T. This limiting value might be conceived to vary
from one direction to another. Thus T might be the infinite space
lying within one sheet of a cone, or hyperboloid of one sheet, or
the infinite space which lies outside a given closed surface S, which
contains within it all the attracting mass. On the latter suppo-
sition, if F be equal to zero throughout /S, and vanish at an infinite
distance, V must be equal to zero everywhere outside S. If V
vanish at an infinite distance, and range between the limits A and
B at the surface 8, V cannot anywhere outside S lie beyond the
limits determined by the two extremes of the three quantities -4,
jB, and 0.

♦ It might, of coarse, be possible to prevent this by drawing Si snfBcicntly
packorod, but S^ is supposed not to be so drawn. Since V is decreasing from T^
outwards, if we consider the loci of the points where V has the values V^ T,, F^...
decreasing by infinitely smaU steps from Fi, it is evident that in the immediate
neighbourhood of 7\ these loci will be closed surfaces, each lying outside the
preceding, the first of which ultimately coincides with T, if T^ be a point, a line, or
a surface, or with the surface of T^ if T^ be a space. If now we take for S^ one of
these *• surfaces of equilibrium," or any surface cutting them at acute angles, what
was asserted in the text respecting S{ will bo true.

ON CLAIRAUT'S tHfiOREM. Ill

Prop. vi. At any point {x, y, z) external to the attracting
mass, the potential V satisfies the partial differential equation

For if V denote the potential of a single particle m\ we have,
employing the notation of Prop, i.,

with similar expressions for d^V'/dt^ and d^V'/dz*] and therefore
V satisfies (6). This equation will be also satisfied by the poten-
tials F^, F'"... of particles m", m'"... situated at finite distances
from the point (a?, y, z)y and therefore by the potential F of all the
particles, since F= V +V" + F"'+ ... Now, by supposing the
number of particles indefinitely increased, and their masses, as
well as the distances between adjacent particles, indefinitely
diminished, we pass in the limit to a continuous mass, of which all
the points are situated at finite distances from the point (x, y, z).
Hence the potential F of a continuous mass satisfies equation (6)
at all points of space to which the mass does not reach.

Scholium to Prop. v. Although the equations (5) and (6)
have been proved independently of each other from the definition
of a potential, either of these equations is a simple analytical con-
sequence of the other* Now the only property of a potential

♦ The equation (6) wiU be proved by means of (5) further on (Prop, vni.), or
rather an equation of which (6) is a particular case, by means of an equation of
which (6) is a particular case. Equation (5) may be proved from (6) by a known
transformation of the equation ///^V dx dy dx=0, where vF denotes the first
member of (6), and the integration is supposed to extend over the space T, For,
taking the first term in v^i we get

///S-*-//(S)*-//(S)*-.

where (^~) , (^\ denote the values of -p at the points where S is cut by

a line drawn parallel to the axis of x through the point whose co-ordinates are
0, y, z. Now if \ be the angle between the normal drawn outwards at the element
of surface dS and the axis of x,

//(g)_.,...//g™,x^. //(|Z)«,..//g».,.-x,^,

where the first integration is to be extended over the portion of S which lies to the

112 ON ATTRACTIONS, AND

assumed in Prop, v, is, that it is a quantity which varies continu-
ously mthin the space T, and satisfies the equation (5) for any
closed surface drawn within T. Hence Prop, v, which was enun-
ciated with respect to the potential of a mass lying outside T, is
equally true with respect to any continuously varying quantity
which within the space T satisfies the equation (6). It should be
observed that a quantity like r"* is not to be regarded as such, if r
denote the distance of the point (a?, y, £) from a point P, which lies
within r, because r~* becomes infinite at P,.

Clairav£8 Theorem.

1. Although the earth is really revolving about its axis, so
that all problems relating to the relative equilibrium of the earth
itself and the bodies on its surface are really dynamical problems,
we know that they may be treated statically by introducing, in
addition to the attraction, that fictitious force which we call the
centrifugal force. The force of gravity is the resultant of the
attraction and the centrifugal force ; and we know that this force
is perpendicular to the general surface of the earth. In fact, by
far the larger portion of the earth's surface is covered by water,
the equilibrium of which requires, according to the principles of
hydrostatics, that its surface be perpendicular to the direction of
gravity ; and the elevation of the land above the level of the sea,
or at least the elevation of large tracts of land, is but trifling com-
pared with the dimensions of the earth. We may therefore regard
the earth's surface as a surface of equilibrium.

positive side of the curve of contact of S and an enveloping cylinder with its gene-
rating lines parallel to the axis of a?, and the second integration over the remainder
of 8, If then we extend the integration over the whole of the surface 5, we get

///S'^''V«««=//^coBX.dS.

Making a similar transformation with respect to the two remaining terms of v^t
and observing that if f^ i^ be for y, « what X is for a?,

cos X 3— + cos ;i J- + eoBp-^=3—-,
dx dy dz dn

we obtain equation (5).

If r be any continuously varying quantity which within the space 7 satisfies
the equation ^V=0, it may be proved that it is always possible to distribute
attracting matter outside T in such a manner as to produce within 7 a potential
equal to V.

ON clairaut's theorem. 113

2. Let the earth be referred to rectangular axes, the axis of z
coinciding with the axis of rotation. Let V be the potential of
the mass, ck> the angular velocity, Z, F, Z the components of the
whole force at the point {Xy y, z) ; then

Now the general equation to surfaces of equilibrium is

j{Xdx + Ydy + Zdz) = const.,
and therefore we must have at the earth's surface

F+K(^+y')=c (7),

where c is an unknown constant. Moreover V satisfies the equa-
tion (6) at all points external to the earth, and vanishes at an
infinite distance. But these conditions are sufficient to determine
V at all points of space external to the earth. For if possible
let V admit of two different values Fj, F, outside the earth, and
let V^—V^^ V\ Since F^ and F, have the same value

c - Jtt>* (a;* + 2/*)

at the surface, F' vanishes at the surface ; and it vanishes likewise
at an infinite distance, and therefore by Prop. v. F'=0 at all points
outside the earth. Hence if the form of the surface be given, F
is determinate at all points of external space, except so far as
relates to the single arbitrary constant c which is involved in its
complete expression.

3. Now it appears from measures of arcs of the meridian,
that the earth's surface is represented, at least very approximately,
by an oblate spheroid of small ellipticity, having its axis of figure
coinciding with the axis of rotation. It will accordingly be more
convenient to refer the earth to polar, than to rectangular co-
ordinates. Let the centre of the surface be taken for origin ; let r
be the radius vector, the angle between this radius and the axis
of Zy the angle between the plane passing through these lines
and the plane xz. Then if the square of the ellipticity be neg-
lected, the equation to the surface may be put under the form

r = a (l-€COs'^) (8);

and from (7) we must have at the surface

F+ Jft)V sia* ^ = c (9).

s. II. 8

114 ON ATTRACTIONS, AND

If we denote for shortness the equation (6) by v^=0, we have
by transformation to polar co-ordinates*

^_(fF, 2d7. 1 d ( . dY\ 1 cfF

= (10).

4. The form of the equations (8) and (9) suggests the occur-
rence of terms of the form V^ (r) +% (r) cos'^ in the value of F.
Assume then

r= -^ (r) + ;)^ (r) cos*^ -f w (11).

We are evidently at liberty to make this assumption, on account
of the indeterminate function w. Now if we observe that

1 d / . >, d.cos'^N o c «zi
we get from (10) and (11)

+ vw = (12).

If now we determine the functions '^, ;^ from the equations

^"(r)+?^'(^)+2^(r) = o (13),

x"W+^x'W--^rxW = (1*).

we shall have yw ■= 0.

By means of (14), equation (13) may be put under the form

V^"(r) + |v^'(r) = -i{x"W + ^xW};

and therefore -^(r)^ — \'^ (r) is a particular integral of (13). The
equations (14), and (13) when deprived of its last term, are easily
integrated, and we get

t(r) = ^+£-JxW. XW=,S+^^ (15).

Now F vanishes at an infinite distance; and the same will be the

• Cambridge Mathematical Journal^ Vol. I. (Old ScrieR), p. 122, or O'Brien's
Tract on the Figure of the Earth, p. 12,

ON clairaut's theorem. 115

case with to provided we take jB = 0, Z> = 0, when we get from
(11) and (15)

^= 7 + ^(co8'^ -*)+'" (16).

5. It remains to satisfy (9). Now this equation may be satis-
fied, so far as the large terms are concerned, by means of the
constant A, since appears only in the small terms. We have
a right then to assume (7 to be a small quantity of the first order.
Substituting in (16) the value of r given by (8), putting the re-
sulting value of V in (9), and retaining the first order only of small
quantities, we get

-(l + €C08'5)+^(cos'5-i) + u?^ + |*a'(l-cos*5)=c...(17),

w^ being the value of w at the surface of the earth. Now the
constants A and C allow us to satisfy this equation without the
aid of w^. Wo get by equating to zero the sum of the constant
terms, and the coefficient of cos'd,

t^s

A G , (o*d
a So'"*" 2 ""

^6 C 6>V ^

(18).

These equations combined with (17) give ii?j = 0. Now we
have seen that w satisfies the equation yw = at all points ex-
terior to the earth, and that it vanishes at an infinite distance ;
and since it also vanishes at the surface, it follows from Prop. V.
that it is equal to zero every where without the earth.

It is true that w^ is not strictly equal to zero, but only to a
small quantity of the second order, since quantities of that order
are omitted in (17). But it follows from Prop. v. Cor. 3, that if
w', V)" be respectively the greatest and least values of w^^ to cannot
anywhere outside the earth lie beyond the limits determined by
the two extremes of the three quantities w, w'\ and 0, and there-
fore must be a small quantity of the second order ; and since we
are only considering the potential at external points, we may omit
w altogether.

If £^ be the mass of the earth, the potential at a very great

8—2

116 ON ATTRACTIONS, AND

distance r is ultimately equal to Ejr. Comparing this with the
equation obtained from (16) by leaving out w, we get

A^E.

The first of equations (18) serves only to determine c in terms
of E^ and c is not wanted. The second gives

whence, we get from (16)

6. If g be the force of gravity at any point of the surface, v
the angle between the veiiiical and the radius vector drawn from
the centre, g cos v will be the resolved part of gravity along the
radius vector ; and we shall have

^cosi. = -J;(r+|Vsin'5) (20),

where after differentiation r is to be put equal to the radius vector
of the surface. Now j^ is a small quantity of the first order, and
therefore cos v may be replaced by 1, whence we get from (8), (19)>
and (20),

5r=^(l+2€Cos'^)-3fT-ia>*a)(cos«^-i)-a>«a(l-cos'tf),

or 5r=(H.e)^-fa»'o+(f©»o-^)co8*^ (21).

At the equator ^ = Jtt ; and if we put O for gravity at the equator,
m for the ratio of the centrifugal force to gravity at the equator,
we get a)*a == m(?, and

(l+fm)(? = (l+e)|,

whence ^=(l+fm-€) ffa* (22);

and (21) becomes ^r = (? {1 + (fm - e) cos*(9} (23).

7. Equation (22) gives the mass of the earth by means of the
value of determined by the pendulum. In the preceding investi-
gation, is the complement of the corrected latitude ; but since 6
occurs only in the small terms, and the squares of small quantities

ON clairaut's theorem. 117

have been omitted throughout, we may regard 6 as the comple-
ment of the true latitude, and therefore replace cos by the sine
of the latitude. In the case of the earth, m is about -^ and e
about ^, and therefore |m — € is positive. Hence it appears
from (23) that the increase of gravity from the equator to the pole
varies as the square of the sine of the latitude, and the ratio which
the excess of polar over equatorial gravity bears to the latter, added
to the ellipticity, is equal to | x the ratio of the centrifugal force
to gravity at the equator.

8. If instead of the equatorial radius a, and equatorial gravity
G, we choose to employ the mean radius a^, and mean gravity O^^
we have only to remark that the mean value of cos* 0, or

^j J cos* 9 sin d0 cUf),

is ^, which gives

a, = a(l-i€), flt^=flt(l+Jm-H

which reduces equations (8), (22), and (23) to

r = a, {l-€(cos«^-i)},
^=(l + fm)(?,a,«,
flr=(?,{l + (|m-€)(cos"d-i)}.

9. We get from (19), for the potential at an external point,

F=|-(e-im)^*(cos*^-i) (24).

Now the attraction of the moon on any particle of the earth,
and consequently the attraction of the whole earth on the moon,
will be very nearly the same as if the moon's mass were collected
at her centre of gravity. Let r be the distance between the
centres of the earth and moon, the moon's north polar distance,
P the attraction of the earth on the moon, resolved along the
radius vector drawn from the earth's centre, Q the attraction per-
pendicular to the radius vector, a force which will evidently lie in
a plane passing through the earth's axis and the centre of the
mOon. Then, supposing Q measured positive towards the equator,
we have from (4),

^~ dr' ^~rd0 '

114 ON ATTRACTIONS, AND

If we denote for shortness the equation (6) by v^=0, we have
by transformation to polar co-ordinates*

^_d*F 2dV 1 d ( . dV\ 1 d'F
^ df^'^r dr'^r'sinedev'^ d0)^r'%m^e d^?

= (10).

4. The form of the equations (8) and (9) suggests the occur-
rence of terms of the form V^ (r) +% (r) cos' 5 in the value of V,
Assume then

r= -^ (r) + ;)^ (r) cos'^ + w (11).

We are evidently at liberty to make this assumption, on account
of the indeterminate function w. Now if we observe that

1 d f . ^ d.cos'^N o fi i/i
we get from (10) and (11)

•f Vw = (12).

If now we determine the functions '^, x ^^^ *^® equations

>^"W + ;^'W + JxW = (13),

x"W + ^x'(»')--^x«=o (1*).

we shall have yw ■= 0.

By means of (14), equation (13) may be put under the form

r(r) + lf'ir)=-i{x" (r) + lx'(r)}i

and therefore y^(r) = '' ^x (^) ^^ a particular integral of (13). The
equations (14), and (13) when deprived of its last term, are easily
integrated, and we get

t(r) = ^+i?-JxW. XW=^+^ (15).

Now F vanishes at an infinite distance; and the same will be the

* Cambridge Mathematical Journal^ Vol. i. (Old Series), p. 122, or O'Brien's
Tract on the Figure of the Earthy p. 12,

ON claibaut's theorem. 115

case with w provided we take 5 = 0, Z> = 0, when we got from
(11) and (15)

^= 7 + J(<=08'^ -*)+'" (16).

5. It remains to satisfy (9). Now this equation may be satis-
fied, so far as the large terms are concerned, by means of the
constant A^ since B appears only in the small terms. We have
a right then to assume (7 to be a small quantity of the first order.
Substituting in (16) the value of r given by (8), putting the re-
sulting value of y in (9), and retaining the first order only of small
quantities, we get

^(l + €Cos'^)-h~(cos*^-i)-hi^, + |'a'(l-cos*^)=c...(17),

■

v)^ being the value of w at the surface of the earth. Now the
constants A and C allow us to satisfy this equation without the
aid of w^. Wo get by equating to zero the sum of the constant
terms, and the coefficient of cos* 5,

t^«

a So'"'' 2 ""
Ae C a)V ^

(18).

These equations combined with (17) give ii?j = 0. Now we
have seen that to satisfies the equation yw = at all points ex-
terior to the earth, and that it vanishes at an infinite distance ;
and since it also vanishes at the surface, it follows from Prop. V.
that it is equal to zero every where without the earth.

It is true that w^ is not strictly equal to zero, but only to a
small quantity of the second order, since quantities of that order
are omitted in (17). But it follows from Prop. v. Cor. 3, that if
w\ V)" be respectively the greatest «and least values of w^, w cannot
anywhere outside the earth lie beyond the limits determined by
the two extremes of the three quantities w?', w\ and 0, and there-
fore must be a small quantity of the second order ; and since we
are only considering the potential at external points, we may omit
w altogether.

If E be the mass of the earth, the potential at a very great

8—2

116 ON ATTRACTIONS, AND

distance r is ultimately equal to Ejr, Comparing this with the
equation obtained from (16) by leaving out w, we get

A^E.

The first of equations (18) serves only to determine c in terms
of E^ and c is not wanted. The second gives

whence, we get from (16)

F=^-(§-ia,v)^(co8*(?-i) (19).

6. If ^r be the force of gravity at any point of the surface, v
the angle between the vertical and the radius vector drawn from
the centre, gcosv will be the resolved part of gravity along the
radius vector ; and we shall have

flrcosi. = -|;(F+^V8m»d) (20),

where after differentiation r is to be put equal to the radius vector
of the surface. Now i; is a small quantity of the first order, and
therefore cos v may be replaced by 1, whence we get from (8), (19)>
and (20),

fir = ^(1 + 2€C0S«^ -3 f^ -i(o*a) (cos'tf-i) -©•a (1 -cos* tf),

or

5r = (l + 6)|-fa,«a+(fo,»a-§*)co8»^ (21).

At the equator tf = Jtt ; and if we put G for gravity at the equator,
m for the ratio of the centrifugal force to gravity at the equator,
we get tt>*a = mG, and

(l+im)G' = (l + e)|,

whence ^=(l+fm-€) Ga^ (22);

and (21) becomes 5r= (? {1 + (|m- e) cos'^} (23).

7. Equation (22) gives the mass of the earth by means of the
value of determined by the pendulum. In the preceding investi-
gation, is the complement of the corrected latitude ; but since
occurs only in the small terms, and the squares of small quantities

ON clairaut's theorem. 117

have been omitted throughout, we may regard 6 as the comple-
ment of the true latitude, and therefore replace cos by the sine
of the latitude. In the case of the earth, m is about -^ and e
about ^^, and therefore |m — e is positive. Hence it appears
from (23) that the increase of gravity from the equator to the pole
varies as the square of the sine of the latitude, and the ratio which
the excess of polar over equatorial gravity bears to the latter, added
to the ellipticity, is equal to | x the ratio of the centrifugal force
to gravity at the equator.

8. If instead of the equatorial radius a, and equatorial gravity
G, we choose to employ the mean radius a^, and mean gravity O^^
we have only to remark that the mean value of cos* 0, or

j^j I QO^^0Aii0d0d4>,

is J, which gives

a, = a(l-i€), flt^=Gf(l+Jm-i6),
which reduces equations (8), (22), and (23) to

r = a,{l-€(cos»^-i)},

^=(l + fm)(?,a/,

5r=(?,{l+(|m-€)(cos«d-i)}.

9. We get from (19), for the potential at an external point,

F=|-(e-Jm)^*(co8»^-i) (24).

Now the attraction of the moon on any particle of the earth,
and consequently the attraction of the whole earth on the moon,
will be very nearly the same as if the moon's mass were collected
at her centre of gravity. Let r be the distance between the
centres of the earth and moon, the moon's north polar distance,
P the attraction of the earth on the moon, resolved along the
radius vector drawn from the earth's centre, Q the attraction per-
pendicular to the radius vector, a force which will evidently lie in
a plane passing through the earth's axis and the centre of the
mbon. Then, supposing Q measured positive towards the equator,
we have from (4),

dr' ^ rdB '

(25).

118 ON ATTRACTIONS, AND

whence, from (24),

p=^_3(e-im)^*(co8'^-i)

Q = 2 (e — im) —j sin cos

The moving force arising from the attraction of the earth on
the moon is a force passing through the centre of the moon, and
having for components MP along the radius vector, and MQ per-
pendicular to the radius vector, M being the mass of the moon ;
and on account of the equality of action and reaction, the moving
force arising from the attraction of the moon on the earth is equal
and opposite to the former. Hence the latter force is equivalent
to a moving force MP passing through the earth's centre in the
direction of the radius vector of the moon, a force MQ passing
through the earth's centre in a direction perpendicular to the
radius vector, and a couple whose moment is MQr tending to turn
the earth about an equatorial axis. Since we only want to deter-
mine the motion of the moon relatively to the earth, the effect of
the moving forces MP, MQ acting on the earth will be fully taken
into account by replacing E in equations (25) hy E-^ M, If /x be
the moment of the couple, we have

/A = 2(e-im)^^^^sin^cos^ (26).

This formula will of course apply, mutatis mutandis, to the moment
of the moving force arising from the attraction of the sun.

10. The force expressed by the second term in the value of P,
in equations (25), and the force Q, or rather the forces thence
obtained by replacing E hy E + M, are those which produce the
only two sensible inequalities in the moon's motion which depend
on the oblateness of the earth. We see that they enable us to
determine the ellipticity of the earth independently of any hypo-
thesis respecting the distribution of matter in its interior.

The moment /x, and the corresponding moment for the sun, are
the forces which produce the phenomena of precession and nuta-
tion. In the observed results, the moments of the forces are
divided by the moment of inertia of the earth about an equatorial
axis. Call this Ea^K ; let M = E/n ; let b be the annual precession,

ON clairaut's theorem. 119

and /the coefficient in the lunar nutation in obliquity; then we
shall have

\ n+lj ^ ' K "^ n+1 K

where A, B^ C denote certain known quantities. Hence the
observed values of h and / will serve to determine the two unknown
quantities w, and the ratio of e — \m to k. If therefore we suppose
e to be known otherwise, we shall get the numerical value of k.

11. In determining the mutual attraction of the moon and
earth, the attraction of the moon has been supposed the same as if
her mass were collected at her centre, which we know would be
strictly true if the moon were composed of concentric spherical
strata of equal density, and is very nearly true of any mass, how-
ever irregular, provided the distance of the attracted body be very
great compared with the dimensions of the attracting mass, and
the centre be imderstood to mean the centre of gravity. It will
be desirable to estimate the magnitude of the error which is likely
to result from this supposition. For this purpose suppose the
moon's surface, or at least a surface of equilibrium drawn imme-
diately outside the moon, to be an oblate spheroid of small ellip-
ticity, having its axis of figure coincident with the axis of rotation.
Then the equation (24) will apply to the attraction of the moon on
the earth, provided we replace JF, a, by M, a', where a' is the
moon's radius, take to denote the angular distance of the radius
vector of the earth from the moon's axis, and suppose e and m to
have the values which belong to the moon. Now E is about 80
times as great as M^ and a about 4 times as great as a', and there-
fore Ed^ is about 1200 times as great as Ma'^. But m is extremely
small in the case of the moon; and there is no reason to think
that the value of e for the moon is large in comparison with its
value for the earth, but rather the contrary; and therefore the
efifect of the moon's oblateness on the relative motions of the
centres of the earth and moon must be altogether insignificant,
especially when we remember that the coefficients of the two
sensible inequalities in the moon's motion depending on the earth's

* l/(n + l) will appear in these equations rather than 1/n, because, if 5 be the
mass, and r, the distance of the sun, the ratio of Mlr^ to Sjrf is equal to l/(ii+ 1)
multiplied by that of (E + M)lr^ to 5/r,>, and the latter ratio is known by the mean
motions of the sun and moon. •

120 ON ATTRACTIONS, AND

oblateness are only about 8". It is to be observed that the suppo-
sition of a spheroidal figure has only been made for the sake of
rendering applicable the equation (24), which had been already
obtained, and has nothing to do with the order of magnitude of the
terms we are considering*.

Although however the effect of the moon's oblateness, or rather
of the possible deviation of her mass from a mass composed of con-
centric spherical strata, may be neglected in considering the motion
of the moon's centre, it does not therefore follow that it ought to
be neglected in considering the moon's motion about her own axis.
For in the first place, in comparing the efifects produced on the
moon and on the earth, the moment of the mutual moving force of
attraction of the moon and earth is divided by the moment of
inertia of the moon, instead of the moment of inertia of the earth,
which is much larger; and in the second place, the efifect now con-
sidered is not mixed up with any other. In fact, it is well known
that the circumstance that the moon always presents the same face
to us has been accounted for in this manner.

12. In concluding this subject, it may be well to consider the
degree of evidence afforded by the figure of the earth in favour
of the hypothesis of the earth's original fluidity.

In the first place, it is remarkable that the surface of the earth
is so nearly a surface of equilibrium. The elevation of the land
above the level of the sea is extremely trifling compared with the
breadth of the continents. The surface of the sea must of course
necessarily be a surface of equilibrium, but still it is remarkable
that the sea is spread so uniformly over the surface of the earth.
There is reason to think that the depth of the sea does not exceed
a very few miles on the average. Were a roundish solid taken at
random, and a quantity of water poured on it, and allowed to
settle under the action of the gravitation of the solid, the proba-
bility is that the depth of the water would present no sort of

* If the expreBsion for V be formed directly, and be expanded according to
inverse powers of r, the first term wiU be MJr, The terms iuvolving r"^ wiU
disappear if the centre of gravity of the moon be taken for origin, those involving
r~> are the terms we are here considering. If the moon*s centre of gravity, or
rather its projection on the apparent disk, did not coincide with the centre of the
disk, it is easy to see the nature of the apparent inequality in the moon's motion
which would thence result.

ON clairaut's theorem. 121

uniformity, and would be in some places very great. Nevertheless
the circumstance that the surface of the earth is so nearly a surface
of equilibrium might be attributed to the constant degradation
of the original elevations during the lapse of ages.

In the second place, it is found that the surface is very nearly
an oblate spheroid, having for its axis the axis of rotation. That
the surface should on the whole be protuberant about the equator
is nothing remarkable, because even were the matter of which the
earth is composed arranged symmetrically about the centre, a
surface of equilibrium would still be protuberant in consequence
of the centrifugal force ; and were matter to accumulate at the
equator by degradation, the ellipticity of the surface of equi-
librium would be increased by the attraction of this matter.
Nevertheless the ellipticity of the earth is much greater than
the ellipticity (Jm) due to the centrifugal force alone, and even
greater than the ellipticity which would exist were the earth
composed of a sphere touching the surface at the poles, and con-
sisting of concentric spherical strata of equal density and of a
spherico-spheroidal shell having the density of the rocks and clay
at the surface*. This being the case, the regularity of the surface
is no doubt remarkable ; and this regularity is accounted for on
the hypothesis of original fluidity.

The near coincidence between the numerical values of the
ellipticity of the terrestrial spheroid obtained independently from
the motion of the moon, from the pendulum, by the aid of
Clairaut's theorem, and from direct measures of arcs, afibrds no
additional evidence whatsoever in favour of the hypothesis of
original fluidity, being a direct consequence of the law of universal
gravitation"!".

* It may be proved without difficulty that the value of e corresponding to this
supposition is -^ nearly, if we suppose the density of the shell to be to the mean
density as 5 to 11.

t With respect to the argument derived from the motion of the moon, this
remark has already been made by Professor O'Brien, who has shewn that if the
form of the surface and the law of the variation of gravity be given independently,
and if we suppose the earth to consist approximately of spherical strata of equal
density, without which it seems impossible to account for the observed regularity of
gravity at the surface, then the attraction on the moon follows as a necessary con-
sequence, independently of any theory but that of universal gravitation. {Tract on
the Figure of the Earth.) If the surface be not assumed to be one of equilibrium,
nor even nearly spherical, and if the component of gravity in a direction perpen-

122 ON AITRACTIONS, AND

If the expression for V given by (24) be compared with the
expression which would be obtained by direct integration, it may
easily be shewn that the axis of rotation is a principal axis, and
that the moments of inertia about the other two principal axes are
equal to each other, so that every equatorial axis is a principal
axis. These results would follow as a consequence of the hypo-
thesis of original fluidity. Still it should be remembered that
we can only affirm them to be accurate to the degree of accuracy
to which we are authorized by measures of arcs and by pendulum
experiments to affirm the surface to be an oblate spheroid.

The phenomena of precession and nutation introduce a new
element to our consideration, namely the moment of inertia of
the earth about an equatorial axis. The observation of these
phenomena enables us to determine the numerical value of the
quantity k, if we suppose e known otherwise. Now, indepen-
dently of any hypothesis as to original fluidity, it is probable that
the earth consists approximately of spherical strata of equal
density. Any material deviation from this arrangement could
hardly fail to produce an irregularity in the variation of gravity,
and consequently in the form of the surface, since we know that
the surface is one of equilibrium. Hence we may assume, when
not directly considering the ellipticity, that the density p is a
function of the distance r from the centre. Now the mean density
of the earth as compared with that of water is known from the
result of Cavendish's experiment, and the superficial density

dicolar to the surface, as well as the form of the surface, be given independently, it
may be shewn that the attraction on an external particle follows, independently of
any hypothesis respecting the distribution of matter in the interior of the earth.
It may be remarked that if the surface bo supposed to differ from a surface of
equilibrium by a quantity of the order of the ellipticity, the component of gravity
in a direction perpendicular to the surface may be considered equal to the wholo
force of gravity. Since however, as a matter of fact, the surface is a surface of
equilibrium, if very trifling irregularities be neglected, it seems better to assume it
to be such, and then the law of the variation of gravity, as well as the attraction on
the moon, foUow from the form of the surface.

It must not here be supposed that these irregularities are actually neglected.
8uch an omission would iU accord with the accuracy of modem measures. In
geodetic operations and pendulum experiments, the direct observations are in fact
reduced to the level of the sea, and so rendered comparable with a theory in which
it is supposed that the earth's surface is accurately a surface of equilibrium. I have
considered this subject in detail in the paper referred to at the beginning of this
article, which has since been read before the Cambridge Philosophical Society.

ON clairaut's theorem. 123

may be considered equal to that of ordinary rocks, or about 2^
times that of water ; and therefore the i*atio of the mean to the
superficial density may be considered known. Take for simplicity
the earth's radius for the unit of length, and let p = Pi when r = 1.
From the mean density and the value of k we know the ratios

of the integrals I pr^dr and I pr^dr to p^. Now it is probable

Jo Jo

that p increases, at least on the whole, from the surface to the
centre. If we assume this to be the case, and restrict p to satisfy
the conditions of becoming equal to p^ when r = l, and of giving
to the two integials just written their proper numerical values,
it is evident that the law of density cannot range within any very
wide limits; and speaking very roughly we may say that the
density is determined.

Now the preceding results will not be sensibly aflFected by
giving to the nearly spherical strata of equal density one form or
another, but the form of the surface will be materially aflFected.
The surface in fact might not be spheroidal at all, or if spheroidal,
the ellipticity might range between tolerably wide limits. But
according to the hypothesis of original fluidity the surface ought
to be spheroidal, and the ellipticity ought to have a certain
numerical value depending upon the law of density.

If then there exist a law of density, not in itself improbable
d pri(yri^ which satisfies the required conditions respecting the
mean and superficial densities, and which gives to the ellipticity
and to the annual precession numerical values nearly agreeing
with their observed values, we may regard this law not only as
in all probability representing approximately the distribution of
matter within the earth, but also as furnishing, by its accordance
with observation, a certain degree of evidence in favour of the
hypothesis of original fluidity. The law of density usually con-
sidered in the theory of the figure of the earth is a law of this
kind.

It ought to be observed that the results obtained relative to
the attraction of the earth remain just the same whether we sup-
pose the earth to be solid throughout or not ; but in founding any
argument on the numerical value of k we are obliged to consider
the state of the interior. Thus if the central portions of the earth
be, as some suppose, in a state of fusion, the quantity £aV must

124 ON ATTRACTIONS, AND

be taken to mean the moment of inertia of that solid, whatever
it may be, which is equivalent to .the solid crust together with
its fluid or viscous contents. On this supposition it is even con-
ceivable that K should depend on the period of the disturbing
force, so that difierent numerical values of k might have to be used
in the precession and in the lunar nutation, in which case the
mass of the moon deduced from precession and nutation would not
be quite correct.

Additional Propositions respecting Attractions,

Although the propositions at the commencement of this paper
were given merely for the sake of the applications made of them
to the figure of the earth, there are a few additional propositions
which are so closely allied to them that they may conveniently be
added here.

!!■

Prop, vii*. If F be the potential of any mass M^, and if if^
be the portion of M^ contained within a closed surface S,

'^d8=-4nrM, (27).

n and dS having the same meaning as in Prop. IV., and the inte-
gration being extended to the whole surfece S.

* This and Prop. iv. are expressed respectively by equations (7) and (8) in the
article by Professor Thomson already referred to (Vol. in. p. 203), where a demon-
stration of a theorem comprehending both founded on the equation

ePv cPv tPv . , ^

s^ + ^»+ ^= -'*''' (»)

is given. In the present paper a different order of investigation is followed ; direct
geometrical demonstrations of the equations

11^ <IS=0 in one case, and // ^ dS= -4irJJfo in another,

are given in Props, rv. and vii. ; and a> new proof of the equation (a) is deduced
from them in Prop. vni.

These equations may be obtained as very particular cases of a general theorem
originally given by Green {Euay on Electricity, p. 12). It will be sufficient to
suppose U=l in Oreen*s equation, and to observe that dw=-dn, and dV=0
or= -4irp, if F be taken to denote the potential of the mass whose attraction is
considered.

ON clairaut's theorem. 125

Let m' be the mass of an attracting particle situated at the
point P' inside S. Through P' draw a right line L, and produce it
indefinitely in one direction. This line will in general cut ;8^ in
one point ; but if 5 be a re-entrant* surface it may be cut by Zr in
three, five, or any odd number of points. About L- describe a
conical surface containing an infinitely small solid angle a, and let
the rest of the notation be as in Prop. IV. In this case the angles
0^f 0^y will be alternately obtuse and acute, and we shall have

^1 = - ^ cos (tt - ^ J = ^ cos 0^ ,

A^ = ar^ sec (tt — ^J = — ar* sec 0^^

and therefore -^i-^i = ~ ^^''

Should there be more than one point of section, the terms N^A^,
N^A^y &c. will destroy each other two and two, as in Prop. rv.
Now all angulai* space around P' may be divided into an infinite
number of solid angles such as a, and it is evident that the whole
surface S will thus be exhausted. We get therefore

limit of XNA = — Sam' = — m'2a ;

or, since 2a = 47r, JJNd8== — 47rm'.

The same formula will apply to any other internal particle, and it
has been shewn in Prop. IV. that for an external particle fJNd8=0.
Hence, adding together all the results, and taking N now to refer
to the attraction of all the particles, both internal and external, we
get JJNdS = — iirM^. But N= d V/dn, which proves the proposi-
tion.

Prop. VIII. At an internal point (x, y, z) about which the
density is p, the potential V satisfies the equation

d^v^d^r^d^v . ,^^.

-d^'^W'^d?'^^ ^^ ^^^^'

Consider the elementary parallelepiped dx dy dz, and apply to
it the equation (27). For the face dy dz whose abscissa is x, the

value of I l-p d/Sis ultimately ^dV/dx . dy dz^ and for the opposite
face it is ultimately + {-j — h -r-^ dxjdydz; and therefore for this

* This term is here used, and has been already used in the demonstration of
Prop. IT., to denote a closed surface which can be cut by a tangent plane.

128 ON ATTRACTIONS, AND

increase continuously in passing from T^ into R. Draw a closed
surface <r lying partly within T^ and partly within ii, and call the
portions lying in T^ and ii, o-^, <r, respectively. Then if i^ be a
normal to o-, drawn outwards, d V/dv will be positive throughout <r,
if o-j be drawn sufficiently close to the space T^ (see Prop. V. and
note), and dV/dv is equal to zero throughout the surface <r,, since

V is constant throughout the space T^] and therefore l\-j-<i<^»

taken throughout the whole surface <r, will be positive, which is
contrary to Prop. IV. Hence V cannot be greater than A in any
portion of 7^ adjacent to T^, and similarly it cannot be less, and
therefore V must have the constant value A throughout T, and
therefore, on account of the continuity of V, at the surface of T.

Combining this with Prop. V. Cor. 1, we see that if V bo
constant throughout the whole surface of a space T which contains
no attracting matter, it will have the same constant value through-
out T; but if Fbe not constant throughout the whole surface, it
cannot be constant throughout any finite portion of T, but only
throughout a surface. Such a surface cannot be closed, but must
abut upon the surface of T, since otherwise V would be constant
within it.

Prop. X. The potential V cannot admit of a maximum or
minimum value in the space T.

It appears from the demonstration of Prop. V. that V cannot
have a maximum or minimum value at a point, or throughout a
line, surface, or space, which is isolated in T. But not even can V
have the maximum or minimum value F, throughout T^ if T^
reach up to the surface S of T; though the term maximum or
minimum is not strictly applicable to this case. By Prop. IX. V
cannot have the value F^ throughout a space, and therefore T^ can
only be a surface or a line.

K possible, let F have the maximum value Fj throughout a
line L which reaches up to 8. Consider the loci of the points
where F has the successive values F,, F,..., decreasing by infi-
nitely small steps from 1''^. In the immediate neighbourhood of
L, these loci will evidently be tube-shaped surfaces, each lying
outside the preceding, the first of which will ultimately coincide
with L, Let « be an element of L not adjacent to S, nor reaching

ON CLAIRAUT*S THEOREM. 120

up to the extremity of L, in case L terminate abniptly. At eaoli
extremity of 8 draw an infinite number of lines of force, that is,
lines traced from point to point in the direction of the force, and
therefore perpendicular to the surfaces of equilibrium. The assem-
blage of these lines will evidently constitute two surfaces cutting
the tubes, and perpendicular to s at its extremities. Call the
space contained within the two surfaces and one of the tubes 7^,,
and apply equation (5) to this space. Since Fis a maximum at Z,
dV/dn is negative for the tube surface of jP,, and it vanishes for
the other surfaces, as readily follows from equation (4). Hence

dV

-J- dS, taken throughout the whole surface T^, is negative,

If.

dn

which is contrary to equation (5). Hence V cannot have a maxi-
mum value at the line L ; and similarly it cannot have a minimum
value.

It may be proved in a similar manner that V cannot have a
maximum or minimum value V^ throughout a surface S^ whicli
reaches up to 8. For this purpose it will be suflScient to draw a
line of force through a point in S^, and make it travel round an
elementary area a which forms p6jrt of S^, and to apply equation
(5) to the space contained between the surface generated by this
line, and the two portions, one on each side of /Sj, of a surface of
equilibrium corresponding to a value of V very little diflferent
from F^.

It should be observed that the space T considered in this
proposition and in the preceding need not be closed : all that is
requisite is that it contain none of the attracting mass. Thus, for
instance, T may be the infinite space surrounding an attracting
mass or set of masses.

It is to be observed also, that although attractive forces have
been spoken of throughout, all that has been proved is equally
true of repulsive forces, or of forces partly attractive and partly
repulsive. In fact, nothing in the reasoning depends upon the
sign of w; and by making m negative we pass to the case of
repulsive forces.

Prop. XI. If an isolated particle be in equilibrium under the
action of forces varying inversely as the square of the distance, the
equilibrium cannot be stable with reference to every possible
s. II. 9

130 ON ATTRACTIONS, AND ON CLAIRAUT'S THEOREM.

displacement, nor unstable, but must be stable with reference to
some displacements and unstable with reference to others; and
therefore the equilibrium of a free isolated particle in such circum-
stances must be unstable*.

For we have seen that F cannot be a maximum or minimum,
and therefore either Fmust be absolutely constant, (as for instance
within a uniform spherical shell), in which case the particle may
be in equilibrium at any point of the space in which it is situated,
or else, if the particle be displaced along any straight line or curve,
for some directions of the line or curve V will be increasing and
for some decreasing. In the former case the force resolved along
a tangent to the particle's path will be directed from the position
of equilibrium, and will tend to remove the particle still farther
from it, while in the latter case the reverse will take place.

* Tliis theorem waa first given bj Mr Eamshaw in his memoir on Molecular
Forces read at the Cambridge Philosophical Society, March 18, 1839 {Tram,
Vol. TEL,), See also a paper by Professor Thomson in the first series of this Journal,
Vol nr. p. 228.

[From the Transactions of the Cambridge PkUosophtcal Society, VoL vui. p. 672.]

On the Variation of Gravity at the Surface of the

Earth.

[Read AjyrU 23, 1849.]

On adoptiDg the hypothesis of the earth's original fluidity,
it has been shewn that the surface ought to be perpendicular to
the direction of gravity, that it ought to be of the form of an oblate
spheroid of small ellipticity, having its axis of figure coincident
with the axis of rotation, and that gravity ought to vary along the
surface according to a simple law, leading to the numerical relation
between the ellipticity and the ratio between polar and equatorial
gravity which is known by the name of Clairaut's Theorem,
Without assuming the earth's original fluidity, but merely sup-
posing that it consists of nearly spherical strata of equal density,
and observing that its surface may be regarded as covered by a
fluid, inasmuch as all observations relating to the earth's figure
are reduced to the level of the sea, Laplace has established a
connexion between the form of the surface and the variation of
gravity, which in the particular case of an oblate spheroid agrees
with the connexion which is found on the hypothesis of original
fluidity. The object of the first portion of this paper is to establish
this general connexion without making any hypothesis whatsoever
respecting the distribution of matter in the interior of the eartli,
but merely assuming the theory of universal gravitation. It ap-
pears that if the form of the surface be given, gravity is determined
throughout the whole surface, except so far as regards one arbitrary
constant which is contained in its complete expression, and which

132 ON THE VARIATION OF GRAVITY

may be determined by the value of gravity at one place. Moreover
the attraction of the earth at all external points of space is de-
termined at the same time ; so that the earth's attraction on the
moon, including that part of it which is due to the earth's ob-
lateness, and the moments of the forces of the sun and moon
tending to turn the earth about an equatorial axis, are found
quite independently of the distribution of matter within the earth.

The near coincidence between the numerical values of the
earth's ellipticity deduced independently from measures of arcs,
from the lunar inequalities which depend on the earth's oblate-
ness, and, by means of Clairaut's Theorem, from pendulum ex-
periments, is sometimes regarded as a confirmation of the hy-
pothesis of original fluidity. It appears, however, that the form
of the surface (which is supposed to be a surface of equilibrium),
suflSces to determine both the variation of gravity and the attrac-
tion of the earth on an external particle*, and therefore the coinci-
dence in question, being a result of the law of gravitation, is no
confirmation of the hypothesis of original fluidity. The evidence
in favour of this hypothesis which is derived from the figure and
attraction of the earth consists in the perpendicularity of the
surface to the direction of gravity, and in the circumstance that
the surface is so nearly represented by an oblate spheroid having
for its axis the axis of rotation. A certain degree of additional
evidence is afiforded by the near agreement between the observed
ellipticity and that calculated with an assumed law of density
which is likely a priori to be not far from the truth, and which
is confirmed, as to its general correctness, by leading to a value
for the annual precession which does not much diflfer from the
observed value.

♦ It has been remarked by ProfesBor O'Brien {Matfiematical TracU^ p. 56) that
if we have given the form of the earth's surface and the variation of gravity, we
have data for determining the attraction of the earth on an external particle, the
earth being supposed to consist of nearly spherical strata of equal density ; so that
the motion of the moon furnishes no additional confirmation of the hypothesis of
original fluidity.

If we have given the component of the attraction of any mass, however irregular
as to its form and interior constitution, in a direction perpendicular to the surface,
throughout the whole of the surface, we have data for determining the attraction at
every external point, as well as the components of the attraction at the surface in
two directions perpendicular to the normal. The corresponding proposition in
Fluid Motion is self-evident.

AT THE SURFACE OF THE EARTH. 133

Since the earth's actual surface is not strictly a surface of
equilibrium, on account of the elevation of the continents and
islands above the sea level, it is necessary to consider in the first
instance in what manner observations would have to be reduced
in order to render the preceding theory applicable. It is shewn in
Art. 13 that the earth may be regarded as bounded by a surface of
equilibrium, and therefore the expressions previously investigated
may be applied, provided the sea level be regarded as the bounding
surface, and observed gravity be reduced to the level of the sea
by taking account only of the change of distance from the earth's
centre. Gravity reduced in this manner would, however, be liable
to vary irregularly from one place to another, in consequence
of the attraction of the land between the station and the surface
of the sea, supposed to be prolonged underground, since this
attraction would be greater or less according to the height of the
station above the sea level. In order therefore to render the
observations taken at different places comparable with one another,
it seems best to correct for this attraction in reducing to the level
of the sea; but since* this additional correction is introduced in
violation of the theory in which the earth's surface is regarded
as one of equilibrium, it is necessary to consider what effect the
habitual neglect of the small attraction above mentioned produces
on the values of mean gravity and of the ellipticity deduced from
observations taken at a number of stations. These effects are
considered in Arts. 17, 18.

Besides the consideration of the mode of determining the values
of mean gravity, and thereby the mass of the earth, and of the
ellipticity, and thereby the effect of the earth's oblateness on the
motion of the moon, it is an interesting question to consider
whether the observed anomalies in the variation of gravity may
be attributed wholly or mainly to the irregular distribution of
land and sea at the surface of the earth, or whether they must
be referred to more deeply seated causes. In Arts. 19, 20, I have
considered the effect of the excess of matter in islands and conti-
nents, consisting of the matter which is there situated above the
actual sea level, and of the defect of matter in the sea, consisting
of the difference between the mass of the sea, and the mass of an
equal bulk of rock or clay. It appears that besides the attraction
of the land lying immediately underneath a continental station.

134 ON THE VABIATION OF GRAVITY

between it and the level of the sea, the more distant portions of
the continent cause an increase in gravity, since the attraction
which they exert is not wholly horizontal, on account of the cur-
vature of the earth. But besides this direct effect, a continent
produces an indirect effect on the magnitude of apparent gravity.
For the horizontal attraction causes the verticals to point more
inwards, that is, the zeniths to be situated further outwards, than
if the continent did not exist ; and since a level surface is every-
where perpendicular to the vertical, it follows that the sea level
on a continent is higher than it would be at the same place if the
continent did not exist Hence, in reducing an observation taken
at a continental station to the level of the sea, we reduce it to
a point more distant from the centre of the earth than if the
continent were away ; and therefore, on this account alone, gravity
is less on the continent than on an island. It appears that this
latter effect more than counterbalances the former, so that on the
whole, gravity is less on a continent than on an island, especially
if the island be situated in the middle of an ocean. This circum-
stance has already been noticed as the result of observation. In
consequence of the inequality to which gravity is subject, de-
pending on the character of the station, it is probable that the
value of the ellipticity which Mr Airy has deduced from his dis-
cussion of pendulum observations is a little too great, on account
of the decided preponderance of oceanic stations in low latitudes
among the group of stations where the observations were taken.

The alteration of attraction produced by the excess and defect
of matter mentioned in the preceding paragraph does not con-
stitute the whole effect of the irregular distribution of land and
sea, since if the continents were cut off at the actual sea level,
and the sea were replaced by rock and clay, the surface so formed
would no longer be a surface of equilibrium, in consequence of
the change produced in the attraction. In Arts 25 — 27, I have
investigated an expression for the reduction of observed gravity to
what would be observed if the elevated solid portions of the earth
were to become fluid, and to run down, so as to form a level bottom
for the sea, which in that case would cover the whole earth. The
expressions would be very laborious to work out numerically, and
besides, they require data, such as the depth of the sea in a great
many places, &c., which we do not at present possess ; but from a

AT THE SURFACE OF THE EARTH. 135

consideration of the general character of the correction, and from
the estimation given in Art. 21 of the magnitude which such
corrections are likely to attain, it appears probable that the ob-
served anomalies in the variation of gravity are mainly due to the
irregular distribution of land and sea at the surface of the earth.

1. Conceive a mass whose particles attract each other ac-
cording to the law of gravitation, and are besides acted on by a
given force/, which is such that if X, Y,Zhe its components along
three rectangular axes, Xdx + Ydif + Zdz is the exact dififerential
of a function U of the co-ordinates. Call the surface of the mass S,
and let V be the potential of the attraction, that is to say, the
function obtained by dividing the mass of each attracting particle
by its distance from the point of space considered, and taking the
sum of all such quotients. Suppose 5 to be a surface of equi-
librium. The general equation to such surfaces is

r+U=c (1),

where c is an arbitrary constant ; and since S is included among
these surfaces, equation (1) must be satisfied at all points of the
surface /9, when some one particular value is assigned to c. For
any point external to 8, the potential V satisfies, as is well known,
the partial differential equation

and evidently V cannot become infinite at any such point, and
must vanish at an infinite distance from £f. Now these conditions
are suflScient for the complete determination of the value of F for
every point external to S^ the quantities TJ and c being supposed
known. The mathematical problem is exactly the same as that of
determining the permanent temperature in a homogeneous solid,
which extends infinitely around a closed space fif, on the conditions,

(1) that the temperature at the surface 8 shall be equal to c — IT,

(2) that it shall vanish at an infinite distance. This problem is
evidently possible and determinate. The possibility has moreover
been demonstrated mathematically.

If TJ alone be given, and not c, the general value of V will
contain one arbitrary constant, which may be determined if we

136 ON THE VARIATION OF GKAVITY

know the value of F, or of one of its differential coefficients, at
one point situated either in the surface 8 or outside it When V
is known, the components of the force of attraction will be obtained
by mere differentiation.

Nevertheless, although we know that the problem is always
determinate, it is only for a very limited i^umber of forms of the
surface S that the solution has hitherto been effected. The
most important of these forms is the sphere. When 8 has very
nearly one of these forms the problem may be solved by approxi-
mation.

2. Let us pass now to the particular case of the earth. Althougb
the earth is really revolving about its axis, so that the bodies on
its surface are really describing circular orbits about the axis of
rotation, we know that the relative equilibrium of the earth itself,
or at least its crust, and the bodies on its surface, would not be
affected by supposing the crust at rest, provided that we introduce,
in addition to the attraction, that fictitious force which we call the
centrifugal force. The vertical at any place is determined by the
plumb-line, or by the surface of standing fluid, and its determi-
nation is therefore strictly a question of relative equilibrium. The
intensity of gravity is determined by the pendulum; but although
the result is not mathematically the same as if the earth were at
rest and acted on by the centrifugal force, the difference is alto-
gether insensible. It is only in consequence of its influence on
the direction and magnitude of the force of gravity that the earth's
actual motion need be considered at all in this investigation : the
mere question of attraction has nothing to do with motion; and
the results arrived at will be equally true whether the earth be
solid throughout or fluid towards the centre, even though, on the
latter supposition, the fluid portions should be in motion relatively
to the crust.

We know, as a matter of observation, that the earth's surface
is a surface of equilibrium, if the elevation of islands and conti-
nents above the level of the sea be neglected. Consequently the
law of the variation of gravity along the surface is determinate, if
the form of the surface be given, the force/ of Art. 1 being in this
case the centrifugal force. The nearly spherical form of the
surfiice renders the determination of the variation easy.

AT THE SURFACE OF THE EARTH. 137

3. Let the earth be referred to polar co-ordinates, the origin
being situated in the axis of rotation, and coinciding with the
centre of a sphere which nearly represents the external surface.
Let r be the radius vector of any point, 6 the angle between the
radius vector and the northern direction of the axis, ^ the angle
which the plane passing through these two lines makes with a
plane fixed in the earth and passing through the axis. Then the
equation (2) which V has to satisfy at any external point becomes
by a common transformation

dr' sm ddd\ dd J sm^ c^* ^ '

Let 0) be the angular velocity of the earth ; then

Z7=ia)Vsin«5,
and equation (1) becomes

7+Ja>Vsin«fl = c (4),

which has to be satisfied at the surface of the earth.

For a given value of r, greater than the radius of the least
sphere which can be described about the origin as centre so as to
lie wholly without the earth, V can be expanded in a series of
Laplace's functions

^0+^1 + ^. + -;

and therefore in general, provided r be greater than the radius of
the sphere above mentioned, V can be expanded in such a series,
but the general term V^ will be a function of r, as well as of
6 and ^. Substituting the above series in equation (3), and
observing that from the nature of Laplace's functions

1 d

sin e dO

we get S [r ^^- - n (n + 1) 7 1 = 0,

where all integral values of n from to oo are to be taken.

Now the differential coefficients of V^ with respect to r are
Laplace's functions of the n*^ order as well as F^ itself; and since
a series of Laplace's functions cannot be equal to zero unless

K

138 ON THE VARUTION OF GRAVITY

the Laplace's functions of the same order are separately equal
to zero, we must have

r-;^^^'-n(n + l) F. = (6).

The integral of this equation is

Y

where F, and Z^ are arbitrary constants so far as r is concerned,
but contain 6 and ^. Since these functions are multiplied by
different powers of r, V^ cannot be a Laplace's function of the n^
order unless the same be true of Y^ and Z». We have for the
complete value of V

Y Y Y

r r r^ » i

Now V vanishes when r = oo , which requires that Z^ =0, Z^ = 0,
&c. ; and therefore

Y Y Y
V=^ + ^' + ^+ (7).

4. The preceding equation will not give the value of the
potential throughout the surface of a sphere which lies partly
within the earth, because although V, as well as any arbitrary but
finite function of 6 and ^, can be expanded in a series of Laplace's
functions, the second member of equation (3) is not equal to
zero in the case of an internal particle, but to — 4?7r/or^, where
p is the density. Nevertheless we may employ equation (7)
for values of r corresponding to spheres which lie partly within
the earth, provided that in speaking of an internal particle we
slightly change the signification of V, and interpret it to mean,
not the actual potential, but what would be the potential if the
protuberant matter were distributed within the least sphere which
cuts the surface, in such a manner as to leave the potential un-
changed throughout the actual surface. The possibility of such a
distribution will be justified by the result, provided the series to
which we are led prove convergent. Indeed, it might easily be shewn
that the potential at any internal point near the surface differs
from what would be given by (7) by a small quantity of the second
order only ; but its differential coeflScient with respect to r, which

AT THE SURFACE OF THE EARTH. 139

gives the component of the attraction along the radius vector,
differs by a small quantity of the first order. We do not, how-
ever, want the potential at any point of the interior, and in fact
it cannot be found without making some hypothesis as to the dis-
tribution of the matter within the earth.

5. It remains now to satisfy equation (4). Let r = a (1 + w)
be the equation to the earth's surface, where t^ is a small quantity
of the first order, a function of 6 and ^. Let u be expanded in a
series of Laplace's functions w^, + Wj + ... The term u^ will vanish
provided we take for a the mean radius, or the radius of a sphere
of equal volume. We may, therefore, take for the equation to
the surface

r = a(l + w^ + w, + ...) (8).

If the surface were spherical, and the earth had no motion of
rotation, V would be independent of and ^, and the second
member of equation (7) would be reduced to its fii-st term. Hence,
since the centrifugal force is a small quantity of the first order, as
well as u, the succeeding terms must be small quantities of the
first order ; so that in substituting in (7) the value of r given by
(8) it will be suflScient to put r = a in these terms. Since the
second term in equation (4) is a small quantity of the first order,
it will be suflScient in that term likewise to put r^a. We
thus get from (4), (7), and (8), omitting the squares of small
quantities,

^•(l_„^_„,_„,...) + J + J+... + ^%m*5 = c (9).

The most general Laplace's function of the order is a con-
stant ; and we have

sin*5 = § + (i-cos'^),

of which expression the two parts are Laplace's functions of the
orders 0, 2, respectively. We thus get from (9), by equating to
zero Laplace's functions of the same order,

y. = a" J^o^ - J«V (i - cos' fl),

140 ON THE VARIATION OF GRAVITY

The first of these equations merely gives a relation between
the arbitrary constants Yq and c; the others determine F^, F,,
&c. ; and we get by substituting in (7)

F=r.g + ^«. + ^* «. + ... )-$\i-cos«tf) (10).

6. Let gr be the force of gravity at any point of the surface of
the earth, dn an element of the normal drawn outwards at that
point; then ^ = — d(F+ U)/dn. Let y^ be the angle between
the normal and the radius vector; then gcosyp^ is the resolved
part of gravity along the radius vector, and this resolved part is
equal to — d ( F+ U) /dr. Now -^ is a small quantity of the first
order, and therefore we may put cos '^ = 1, which gives

where, after differentiation, r is to be replaced by the radius vector
of the surface, which is given by (8). We thus get

-f a)*a (i- cos' 0) - a>*a ( J + J - cos* 6),
which gives, on putting

5-§«*a=G. ^ = m (11).

and neglecting squares of small quantities,

g= G{l-fm(i-cos*^)+ti, + 2w3 + 3M^ } (12).

In this equation G is the mean value of g taken throughout

the w^hole surface, since we know that I ( u^ sin 0dOd<f> = 0, if n

^0 Jo

be different from zero. The second of equations (11) shews that m
is the ratio of the centrifugal force at a distance from the axis
equal to the mean distance to mean gravity, or, which is the same,
since the squares of small quantities are neglected, the ratio of the
centrifugal force to gravity at the equator. Equation (12) makes
known the variation of gravity when the form of the surface is
given, the surface being supposed to be one of equilibrium ; and,
conversely, equation (8) gives the form of the surface if the varia-
tion of gravity be known. It may be obsen^ed that on the latter

AT THE SURFACE OF THE EARTH. 141

supposition there is nothing to determine u^. The most general
form of Wj is

a sin 5 cos ^ + )8 sin 5 sin ^ + 7 cos 0,

where a, j3, 7 are arbitrary constants; and it is very easy to prove
that the co-ordinates of the centre of gravity of the volume are
equal to aa, a/S, ay respectively, the line from which is measured
being taken for the axis of z, and the plane from which <f> is
measured for the plane of wz. Hence the term u^ in (8) may be
made to disappear by taking for origin the centre of gravity of the
volume. It is allowable to do this even should the centre of
gravity fall a little out of the axis of rotation, because the term
involving the centrifugal force, being alrestdy a small quantity of
the first order, would not be aflFected by supposing the origin to be
situated a little out of the axis.

Since the variation- of gravity from one point of the surface to
another is a small quantity of the first order, its expression will
remain the same whether the earth be referred to one origin or
another nearly coinciding with the centre, and therefore a know-
ledge of the variation will not inform us what point has been
taken for the origin to which the surface has been referred.

7. Since the angle between the vertical at any point and the
radius vector drawn from the origin is a small quantity of the first
order, and the angles 0, <f> occur in the small terms only of equa-
tions (8), (10), and (12), these angles may be taken to refer to the
direction of the vertical, instead of the radius vector.

8. If E be the mass of the earth, the potential of its attraction
at a very great distance r is ultimately equal to ^/r. Comparing
this with (10), we get Y^ — Ey and therefore, from the first of
equations (11),

E^ (?a* + §a)V=(?a'(H-§m) (13),

which determines the mass of the earth from the value of deter-
mined by pendulum experiments.

9. If we suppose that the surface of the eai-th may be repre-
sented with sufficient accuracy by an oblate spheroid of small ellip-
ticity, having its axis of figure coincident with the axis of rotation,
equation (8) becomes

r = all + €(i-cos*^)} (14),

142 ON THE VARUTION OF GRAVITY

where e is a constant which may be considered equal to the ellip-
ticity. We have therefore in this case u^ = 0, t^, = ^ — cos* 0,u^ = O
when n>2; so that (12) becomes

5r = (?{l-(fm-6)a-cos«5)} (15),

which equation contains Clairaut'a Theorem. It appears also from
this equation that the value of O which must be employed in (13)
is equal to gravity at a place the square of the sine of whose
latitude is ^.

10. Retaining the same supposition as to the form of the
surface, we get from (10), on replacing F^ by E, and putting in the
small term at the end o) V = m Oa*^ = mEa*,

F=^+(e-im)^a-co8'd) (16).

Consider now the effect of the earth's attraction on the moon.
The attraction of any particle of the earth on the moon, and there-
fore the resultant attraction of the whole earth, will be very nearly
the same as if the moon were collected at her centre. Let there-
fore r be the distance of the centre of the moon from that of the
earth, the moon's North Polar Distance, P the accelerating force
of the earth on the moon resolved along the radius vector, Q the
force perpendicular to the radius vector, which acts evidently in a
plane passing through the earth's axis ; then

dr' ^~rd0'
whence we get from (16)

P = ^V 3 (e - Jm) ^* (i - cos» ^),

G = 2(€-im)^sin^cos^ (17).

The moving forces arising from the attraction of the earth on
the moon will be obtained by multiplying by M, where M denotes
the mass of the moon ; and these are equal and opposite to the
moving forces arising from the attraction of the moon on the earth.
The component MQ of the whole moving force is equivalent to an
equal and parallel force acting at the centre of the earth and a
couple. The accelerating forces acting on the earth will be

AT THE SURFACE OF THE EARTH. 143

obtained by dividing by E; and since we only want to determine
the relative motions of the moon and earth, we may conceive equal
and opposite accelerating forces applied both to the earth and to
the moon, which comes to the same thing as replacing Ehj E+M
in (17). If K be the moment of the couple arising from the
attraction of the moon, which tends to turn the earth about an
equatorial axis, K=^MQr, whence

K=^2 (e-im)^^ Bind cos (18).

The same formula will of course apply, mutatis mutandis, to the
attraction of the sun.

11. The spheroidal form of the earth's surface, and the cir-
cumstance of its being a surface of equilibrium, will afford us some
information respecting the distribution of matter in the interior.
Denoting by x\ y, z' the co-ordinates of an internal particle whose
density is p\ and by x, y, z those of the external point of space to
which V refers, we have

v= [[[ P ^' ^^ ^'

the integrals extending throughout the interior of the earth.
Writing dm' for p dx dy' dz\ putting X, /li, v for the direction-
cosines of the radius vector drawn to the point (a?, y, «), so that
a? = Xr, y = /Ltr, z — Wy and expanding the radical according to
inverse powers of r, we get

V^\jj!dm' + 2 ^///x'rfm' + ^t {5\*-l)ffJx''dm'

+ ^^\,iJJJx'ydm' + (19).

% denoting the sum of the three expressions necessary to form a
symmetrical function. Comparing this expression for Fwith that
given by (10), which in the present case reduces itself to (16), we
get Y^=fffdm'^E, as before remarked, and

JJfx'dm'^0, ////dm' = 0, jjjz'dm'^0 (20),

i S (3V - 1) jjja!' drd + 32X/ii ///x' y' dm'

= (e-Jm)^'(J-C08«^) (21);

144 ON THE VARIATION OF GRAVITY

together with other equations, not written down, obtained by
equating to zero the coeflScients of 1/r*, 1/r^ &c. in (19).

Equations (20) shew that the centre of gravity of the mass
coincides with the centre of gravity of the volume. In treating
equation (21), it is to be remarked that \ fiyV are not independent,
but connected by the equation X' 4-/Lfr*+ 1/^ = 1. If now we insert
X*4-A*'' + »'' as a coefficient in each term of (21) which does not
contain \, /li, or v, the equation will become homogeneous with
respect to X, /i, v, and will therefore only involve the two inde-
pendent ratios which exist between these three quantities, and
consequently we shall have to equate to zero the coefficients of
corresponding powers of X, /li, v. By the transformation just men-
tioned, equation (21) becomes, since cos ^ = v,

S (X' - i/i« - Ji/') Jjjx'^ dm' + SlXtijjjx'y'dm

= (6-im)£'a«(iX« + iM'-§0;
and we get

JSJx'y'dm'^Q, JJJy'zdm'^O, JJJz'x'dm'^O (22),

- Illy'dm' - i Ill^'^dm' - i Ill^'^dm: I ,23)

^''UIIz''dm' + lJJfx''dm' + \JJJy''dm: ^

= i(e-im)JFa*

Equations (22) shew that the co-ordinate axes are principal
axes. Equations (23) give in the first place

which shews that the moments of inertia about the axes of x and
y are equal to each other, as might have been seen at once from
(22), since the principal axes of x and y are any two rectangular
axes in the plane of the equator. The two remaining equations of
the system (23) reduce themselves to one, which is

JUx'^dm: - //^^rf m' = J (e - i m) Ea\

If we denote the principal moments of inertia by A, A, C, this

equation becomes

(7-^=§(e-im)^« (24),

which reconciles the expression for the couple K given by (18)
with the expression usually given, which involves moments of
inertia, and which, like (18), is independent of any hypothesis as
to the distribution of the matter within the earth.

AT THE SURFACE OF THE EARTH. 145

It should be observed that in case the earth be not solid to the
centre, the quantities A, C must be taken to mean what would be
the moments of inertia if the several particles of which the earth
is composed were rigidly connected.

12. In the preceding article the surface has been supposed
spheroidal. In the general case of an arbitrary form we should
have to compare the expressions for F given by (10) and (19). In
the first place it may be observed that the term u^ can always
be got rid of by taking for origin the centre of gravity of the
volume. Equations (20) shew that in the general case, as well
as in the particular case considered in the last article, the
centre of gravity of the mass coincides with the centre of gravity
of the volume.

Now suppress the term u^ in w, and let w = m' + u\ where
w" = |m (^ — cos' ^. Then u' may be expanded in a series of
Laplace's functions u\-{'u\ + .,, ; and since T^ = E\ equation (10)
will be reduced to

^ = ^(J + ^«', + 7«'.-) (25).

If the mass were coUect^jd at the centre of gravity, the second
member of this equation would be reduced to its first terra, which
requires that u\ = 0, u\ = 0, &c. Hence (8) would be reduced to
r = a (1 + w"), and therefore au' is the alteration of the surface
due to the centrifugal force, and au' the alteration due to the
difference between the actual attraction and the attraction of
a sphere composed of spherical strata. Consider at present only
the term u\ of u. From the general form of Laplace's functions
it follows that au\ is the excess of the radius vector of an ellipsoid
not much differing from a sphere over that of a sphere having
a radius equal to the mean radius of the ellipsoid. If we take
the principal axes of this ellipsoid for the axes of co-ordinates,
we shall have

u\ = €'(i - sin' cos* <t>) + e" (^ - sin* 6 sin' <f>) + e" (^ - cos' 0),

e'y e", e'" being three arbitrary constants, and 0, if> denoting angles
related to the new axes of x, y, z in the same way that the
angles before denoted by 5, <f> were related to the old axes.
Substituting the preceding expression for u\ in (25), and com-
paring the result with (19), we shall again obtain equations (22).

s. II. 10

146 ON THE VARIATION OF ORAVITY

Consequently the principal axes of the mass passing through
the centre of gravity coincide with the principal axes of the ellip-
soid. It will be found that the three equations which replace (23)
are equivalent to but two, which are

A - ^Ea' = 5 - y'Ea' = C- K'^">
where A, B, C denote the principal moments.

The permanence of the earth's axis of rotation shews however
that one of the principal axes of the ellipsoid coincides, at least
very nearly, with the axis of rotation ; although, strictly speaking,
this conclusion cannot be drawn without further consideration
except on the supposition that the earth is solid to the centre. If
we assume this coincidence, the term c'" (J — cos' d) will unite
with the term w" due to the centrifugal force. Thus the most
general value of w is that which belongs to an ellipsoid having
one of its principal axes coincident with the axis of rotation, added
to a quantity which, if expanded in a series of Laplace's functions,
would furnish no terms of the order 0, 1, or 2.

It appears from this and the preceding article that the coin-
dence of the centres of gravity of the mass and volume, and that of
the axis of rotation and one of the principal axes of the ellipsoid
whose equation is r = a (1 + u^, which was established by Laplace
on the supposition that the earth consists of nearly spherical strata
of equal density, holds good whatever bo the distribution of matter
in the interior.

13. Hitherto the surface of the earth has been regarded as a
surface of equilibrium. This we know is not strictly true, on ac-
count of the elevation of the land above the level of the sea. The
question now arises. By what imaginary alteration shall we reduce
the surface to one of equilibrium ?

Now with respect to the greater portion of the earth's surface,
which is covered with water, we have a surface of equilibrium
ready formed. The expression level of the sea has a perfectly de-
finite meaning as applied to a place in the middle of a continent,
if it be defined to mean the level at which the sea- water would
stand if introduced by a canal. The surface of the sea, supposed
to be prolonged in the manner just considered, forms indeed a
surface of equilibrium, but the preceding investigation does not
apply directly to this surface, inasmuch as a portion of the at-

AT THE SURFACE OF THE EARTH. 147

tracting matter lies outside it Conceive however the land which
lies above the level of the sea to be depressed till it gets below it,
or, which is the same, conceive the land cut off at the level of the
sea produced, and suppose the density of the earth or rock which
lies immediately below the sea-level to be increased, till the
increase of mass immediately below each superficial element is
equal to the mass which has been removed from above it. The
whole of the attracting matter will thus be brought inside the
original sea-level ; and it is easy to see that the attraction at
a point of space external to the earth, even though it be close
to the surface, will not be sensibly affected. Neither will the
sea-level be sensibly changed, even in the middle of a continent.
For, suppose the sea-water introduced by a pipe, and conceive the
land lying above the sea-level condensed into an infinitely thin
layer coinciding with the sea-level. The attraction of an infinite
plane on an external particle does not depend on the distance of
the particle from the plane ; and if a line be drawn through the
particle inclined at an angle a to the perpendicular let fall on the
plane, and be then made to revolve around the perpendicular, the
resultant attraction of the portion of the plane contained within
the cone thus formed will be to that of the whole plane as versin a
to 1. Hence the attraction of a piece of table-land on a particle
close to it will be sensibly the same as that of a solid of equal
thickness and density comprised between two parallel infinite
planes, and that, even though the lateral extent of the table-land
be inconsiderable, only equal, suppose, to a small multiple of the
length of a perpendicular let fall from the attracted particle on the
further bounding plane. Hence the attraction of the land on the
water in the tube will not be sensibly altered by the condensation
we have supposed, and therefore we are fully justified in regarding
the level of the sea as unchanged.

The surface of equilibrium which by the imaginary displace-
ment of matter just considered has also become the bounding
surface, is that surface which at the same time coincides with
the surface of the actual sea, where the earth is covered by water,
and belongs to the system of surfaces of equilibrium which lie
wholly outside the earth. To reduce observed gravity to what
would have been observed just above this imaginary surface, we
must evidently increase it in the inverse ratio of the square of
the distance from the centre of the earth, without taking ac-

10—2

148 ON THE VARIATION OF GRAVITY

count of the attraction of the table-land which lies between the
level of the station and the level of the sea. The question now
arises, How shall we best determine the numerical value of the
earth's ellipticity, and how best compare the form which results
from observation with the spheroid which results from theory on
the hypothesis of original fluidity ?

14. Before we consider how the numerical value of the earth's
ellipticity is to be determined, it is absolutely necessary that we
define what we mean by ellipticity ; for, when the irregularities of
the surface arc taken into account, the term must be to a certain
extent conventional.

Now the attraction of the earth on an external body, such as
the moon, is determined by the function V, which is given by (10).
In this equation, the term containing r"" will disappear if r be
measured from the centre of gravity ; the term containing r"*, and
the succeeding terms, will be insensible in the case of the moon, or
a more distant body. The only terms, therefore, after the first,
which need be considered, are those which contain r"*. Now the
most general value of w, contains five terms, multiplied by as many
arbitrary constants, and of these terms one is ^ — cos* 0, and the
others contaiu as a factor the sine or cosine of ^ or of 2^. The
terms containing sin ^ or cos ^ will disappear for the reason men-
tioned in Art. 12; but even if they did not disappear their eflFect
would be wholly insensible, inasmuch as the corresponding forces
go through their period in a day, a lunar day if the moon be the
body considered. These terms therefore, even if they existed, need
not be considered ; and for the same reason the terms containing
sin 2<f> or cos 2<f> may be neglected ; so that nothing remains but a
term which unites with the last term in equation (10). Let € be
the coefficient of the term J — cos' in the expansion of u : then e
is the constant which determines the efiect of the earth's oblate-
ness on the motion of the moon, and which enters into the expres-
sion for the moment of the attractions of the sun and moon on the
earth ; and in the particular case in which the earth's surface is an
oblate spheroid, having its axis coincident with the axis of rotation,
6 is the ellipticity. Hence the constant € seems of sufficient
dignity to deserve a name, and it may be called in any case the
ellipticitij.

Let r be the radius vector of the earth's surface, regarded as

AT THE SUBFACE OF THE EARTH. 149

coincident with the level of the sea; and take for shortness
tfi{f{0,<t>)] to denote the mean value of the function / (^, ^)
throughout all angular space, or

Then it follows from the theory of Laplace's functions that

e = ^m{{i-s,n'l)r} (26).

I being the latitude, or the complement of 0. To obtain this
equation it is sufficient to multiply both sides of (8) by l/47r x
(^ — cos* 0) sin 0d0d<t>, and to integrate from 5 = to = ir, and
from ^=0 to ^ = 27r. Since ^ — cos* 5 is a Laplace's function of
the second order, none of the terms at the second side of (8) will
furnish any result except ti,, and even in the case of w, the terms
involving the sine or cosine of ^ or of 2<f> will disappear.

15. Let ^rbe gravity reduced to the level of the sea by taking
account only of the height of the station. Then this is the
quantity to which equation (12) is applicable; and putting for w,
its value we get by means of the properties of Laplace's functions

G^ = m(<7), Oiim'-€) = -^m{{i-sinU)g} (27).

If we were possessed of the values of ^r at an immense number
of stations scattered over the surface of the whole earth, we might
by combining the results of observation in the manner indicated
by equations (27) obtain the numerical values of Q and e. We
cannot, however, obtain by observation the values of g at the
surface of the sea, and the stations on land where the observations
have been made from which the results are to be obtained are not
very numerous. We must consider therefore in what way the
variations of gravity due to merely local causes are to be got rid of,
when we know the causes of disturbance ; for otherwise a local
irregularity, which would be lost in the mean of an immense
number of observations, would acquire undue importance in the
result.

16. Now the most obvious cause of irregularity consists in the
attraction of the land lying between the level of the station and the
level of the sea. This attraction would render the values of g

150 ON THE VARIATION OF GRAVITY

sensibly different, which would be obtained at two stations only a
mile or two apart, but situated at different elevations. To render
our observations comparable with one another, it seems best to
correct for the attraction of the land which lies underneath the
pendulum; but then we must consider whether the habitual
neglect of this attraction may not affect the mean values from
which O and e are to be found.

Let g—g^-{-g\ where g' is the attraction just mentioned, so
that g^ is the result obtained by reducing the observed value of
gravity to the level of the sea by means of Dr Young's formula*.
Let h be the height of the station above the level of the sea, a the
superficial density of the earth where not covered by water ; then
by the formula for the attraction of an infinite plane we have
g = 27rorA. To make an observation, conceived to be taken at the
surface of the sea, comparable with one taken on land, the correc-
tion for local attraction would be additive, instead of subtractive ;
we should have in fact to add the excess of the attraction of a
layer of earth or rock, of a thickness equal to the depth of the sea
at that place, over the attraction of so much water. The formula
g' = 27rcrA will evidently apply to the surface of the sea, provided
we regard A as a negative quantity, equal to the depth of the sea,
and replace cr by cr — 1, the density of water being taken for the
unit of density ; or we may retain a as the coefficient, and diminish
the depth in the ratio of o- to o- — 1.

Let p be the mean density of the earth, then

i7' = 27r<rA = (?|^^ = (??^.

"^ ^pa zpa

If we suppose o-=2J, p = 3J, a = 4000 miles, and suppose h
expressed in miles, with the understanding that in the case of the
sea A is a negative quantity equal to f ths of the actual depth, we
have g = -00017 Gh nearly.

* Phil. Trans, for 1819. Dr Toong's formula is based on the principle of taking
into account the attraction of the table-land existing between the station and the
level of the sea, in reducing the observation to the sea level. On account of this
attraction, the multiplier 2/i/a which gives the correction for elevation alone must
be reduced in the ratio of 1 to 1 -3<y/4/), or 1 to '66 nearly, if <r = 2J, /i)=54. Mr
Airy, observing that the value <r=2^ is a little too small, and ^ = 5} a little too
great, has employed the factor '6, instead of '6G.

AT THE SURFACE OF THE EARTH. 151

17. Consider first the value of 0. We have by the preceding
formula, and the first of equations (27),

Gf = in(5rj)+fl^x00017in(A).

According to Professor Kigaud*s determination, the quantity of
land on the surface of the earth is to that of water as 100 to 276*.
If we suppose the mean elevation of the land ^th of a mile, and
the mean depth of the sea 3^ miles, we shall have

/ix 4x3^x276-4x100 - ,^ ,
m (^) = - ^ ^ 376 = - 1'49 iiearly ;

so that the value of determined by g^ would be too great by
about '000253 of the whole. Hence the mass of the earth deter-
mined by the pendulum would be too great by about the one four-
thousandth of the whole ; and therefore the mass of the moon,
obtained by subtracting from the sum of the masses of the earth
and moon, as determined by means of the coeflBcient of lunar
parallax, the mass of the earth alone, as determined by means of
the pendulum, would be too small by about the one four-thousandth
of the mass of the earth, or about the one fiftieth of the whole.

18. Consider next the value of €. Let c^ be the value which
would be determined by substituting g^ for g in (27), and let

V m {(sin'i - ^)g'] = Gq,

In considering the value of q we may attend only to the land,
provided we transfer the defect of density of the sea with an
opposite sign to the land, because if g' were constant, q would
vanish. This of course proceeds on the supposition that the depth
of the sea is constant. Since € = f^ — g, if q were positive, the
ellipticity determined by the pendulum would appear too great in
consequence of the omission of the force g\ I have made a sort of
rough integration by means of a map of the world, by counting the
quadrilaterals of land bounded each by two meridians distant 10®,
and by two parallels of latitude distant 10®, estimating the fraction
of a broken quadrilateral which was partly occupied by sea. The
number of quadrilaterals of land between two consecutive parallels,
as for example 50" and 60®, was multiplied by 12 (^•— sin'i) cosZ, or
3 cos 3i + cos I, where for I was taken the mean latitude, (55® in the
example,) the sum of the results was taken for the whole surface,

• Cambridge Philotophical Traruactioiu, Vol. vi. p. 297.

152 ON THE VARIATION OF GRAVITY

and multiplied by the proper coefficient. The north pole was
supposed to be surrounded by water, and the south pole by land,
as far as latitude 80®. It appeared that the land lying beyond the
parallels for which sin'i = J, that is, beyond the parallels 35® N. and
35° S. nearly, was almost exactly neutralized by that which lay
within those parallels. On the whole, q appeared to have a very
small positive value, which on the same suppositions as before
respecting the height of the land and the depth of the sea, was
•0000012. It appears, therefore, that the omission of the force g'
will produce no sensible increase in the value of 6, unless the land
be on the whole higher, or the sea shallower, in high latitudes
than in low. If the land had been collected in a great circular
continent around one pole, the value of q would have been '000268 ;
if it had been collected in a belt about the equator, we should
have had j = — '000362. The difference between these values of
q is about one fifth of the whole ellipticity.

19. The attraction ^ is not the only irregularity in the mag-
nitude of the force of gravity which arises from the irregularity in
the distribution of land and sea, and in tlie height of the land and
depth of the sea, although it is the only irregularity, arising from
that cause, which is liable to vary suddenly from one point at the
surface to another not far offi The irregular coating of the earth
will produce an irregular attraction besides that produced by the
part of this coating which lies under and in the immediate neigh-
bourhood of the station considered, and it will moreover cause an
irregular elevation or depression in the level of the sea, and
thereby cause a diminution or increase in the value oi g^.

Consider the attraction arising from the land which lies above
the level of the sea, and from the defect of attracting matter in the
sea. Call this excess or defect of matter the coating of the earth :
conceive the coating condensed into a surface coinciding with the
level of the sea, and let J.S be the mass contained in a small
element A of this surface. Then B^ah in the case of the land,
and 8 = — (<r — 1) A in the case of the sea, h being in that case the
depth of the sea. Let V^ be the potential of the coating, V\ V"
the values of V^ outside and inside the surface respectively. Con-
ceive B expanded in a series of Laplace's functions S^, + S^ + . . ., then
it is easily proved that

AT THE SURFACE OF THE EARTH. 153

^'=*"''{\^-->-£^'*-)

r being the distance of the point considered from the centre.
These equations give

dV , ^ i + 1 fa\t+* ^

dr
dV

7/

= 47r2

I + 1 \a/ j

(29).

dr ^2i

Consider two points, one external, and the other internal,
situated along the same radius vector very close to the surface.
Let E be an element of this surface lying around the radius vector,
an element which for clear ideas we may suppose to be a small
circle of radius 8, and let 8 be at the same time infinitely small
compared with a, and infinitely great compared with the distance
between the points. Then the limiting values of dV'/dr and
dV'jdr will differ by the attraction of the element J?, an attraction
which, as follows from what was observed in Art. 13, will be ulti-
mately the same as that of an infinite plane of the same density,
or 27rS*. The mean of the values of dV jdr and dV'jdr will
express the attraction of the general coating in the direction of
the radius vector, the general coating being understood to mean
the whole coating, with the exception of a superficial element
l3dng adjacent to the points where the attraction is considered.
Denoting this mean by dVJdr, we get, on putting r^a.

dr ^2»+l*

This equation becomes by virtue of either of the equations (28)

dV. V.

dr 2a

(30).

* This result readily follows from equationq (29), which give, on putting r=at

dV" dV . ^. . .
dr dr

This dilterence of attraction at points infinitely close can evidently only arise from
the attraction of the interposed element of surface, which, being ultimately plane,
will act equally at both points ; and, therefore, the attraction will bo in each case
2t5, and will act outwards iu the first case, and inwards in the second.

154 ON THE VARIATION OF GRAVITY

which is a known equation. Let either member of this equation
be denoted by — ^r". Then gravity will be increased by g\ in
consequence of the attraction of the general coating.

20. But besides its direct eflfect, the attraction of the coating
will produce an indirect eflfect by altering the sea-level. Since the
potential at any place is increased by V^ in consequence of the
coating, in passing from what would be a surface of equilibrium if
the coating were removed, to the actual surface of equilibrium
corresponding to the same parameter, {that is, the same value of
the constant c in equation (1),} we must ascend till the labouring
force expended in raising a unit of mass is equal to F^, that is, we
must ascend through a space VJg^ or VJQ nearly. In consequence
of this ascent, gravity will be diminished by the quantity corre-
sponding to the height VJQ^ or A' suppose. If we take account
only of the alteration of the distance from the centre of the earth,
this diminution will be equal to . 2A7a, or 2 VJa^ or 4^", and
therefore the combined direct and indirect eflfects of the general
coating will be to diminish gravity by Sjr".

But the attraction of that portion of the stratum whose thick-
ness is Ky which lies immediately about the station considered,
will be a quantity which involves A' as a factor, and to include this
attraction we must correct for the change of distance' K by Dr
Young's rule, instead of correcting merely according to the square
of the distance. In this way we shall get for the diminution of
gravity due to the general coating, not %g\ but only 4 (1 — 3cr/4p)
g* — g" y or leg' suppose. If <r : p :: 5 : 11, we have A =16*4
nearly.

If we cared to leave the mean value of gravity unaltered, we
should have to use, instead of S, its excess over its mean value \,
In considering however, only the variation of gravity from one place
to another, this is a point of no consequence.

21. In order to estimate the magnitude which the quantity
3n7" is likely to attain, conceive two stations, of which the first is
surrounded by land, and the second by sea, to the distance of 1000
miles, the distribution of land and sea beyond that distance being
on the average the same at the two stations. Then, by hypothesis,
the potential due to the land and sea at a distance greater than

AT THE SURFACE OF THE EARTH. 155

1000 miles is the same at the two stations ; and as we only care
for the diflference between the values of the potential of the earth's
coating at the two stations, we may transfer the potential due to
the defect of density at the second station with an opposite sign
to the first station. We shall thus have around the first station,
taking h! for the depth of the sea around the second station,
S = o-A + (<r — 1) A'. In finding the diflference V of the potentials
of the coating, it will be amply suflScient to regard the attracting
matter as spread over a plane disk, with a radius s equal to 1000
miles. On this supposition we get

=/■•■' •

*/o

^irhsds = iirZs,

Now (? = I TTpa, and therefore

^„ 3F 9&? ^ 9 (rA+((r-l)A' 5^
^ 2a 4pa ^ pa a

Making the same suppositions as before with regard to the
numerical values of cr, p, A, A', and a, we get 3y= '000147 O. This
corresponds to a difference of 6 35 vibrations a day in a seconds'
pendulum. Now a circle with a radius of 1000 miles looks but
small on a map of the world, so that we may readily conceive that
the difference depending on this cause between the number of
vibrations observed at two stations might amount to 15 or 20, that
is 7'5 or 10 on each side of the mean, or even more if the height
of the land or the depth of the sea be under-estimated. This
difference will however be much reduced by using kg' in place of

22. The value of V^ at any station is expressed by a double
integral, which is known if 8 be known, and which may be cal-
culated numerically with suflScient accuracy by dividing the
surface into small portions and performing a summation. Theo-
retically speaking, V^ could be expressed for the whole surface
at once by means of a series of Laplace's functions; the constants
in this series could be determined by integration, or at least the
approximate integration obtained by summation, and then the
value of V^ could be obtained by substituting in the series the

* The effect of the irregularity of the earth's snrface is greater than what is
represented by hg'\ for a reason which will be explained further on (Art. 25).

166 ON THE VARIATION OF GUAVITY

latitude and longitude of the given station for the general latitude
and longitude. But the number of terms which would have to be
retained in order to represent with tolerable accuracy the actual
state of the earth's surface would be so great that the method, I
apprehend, would be practically useless; although the leading
terms of the series would represent the eflFect of the actual
distribution of land and sea in its broad features. It seems
better to form directly the expression for V^ at any station. This
expression may be calculated numerically for each station by
using the value of B most likely to be correct, if the result be
thought worth the trouble; but even if it be not calculated
numerically, it will enable us to form a good estimate of the
variation of the quantity S^r" or kg" from one place to another.

Let the surface be referred to polar co-ordinates originating at
the centre, and let the angles -^j % be with reference to the station
considered what 0, <f> were with reference to the north pole. The
mass of a superficial element is equal to Sa' sin yfrdylrdXi c^i^d its
dbtance from the station is 2a sin ^yft. Hence we have

Fc=a//Scosii^ dyfrdx (31)

Let fi„ be the mean value of B throughout a circle with an
angular radius y^, then the part of V^ which is due to an annulus
having a given infinitely small angular breadth d^jr is proportional
to B^ cos ^yjr, or to S« nearly when yjr is not large. If we regard
the depth of the sea as uniform, we may suppose S = for the
sea, and transfer the defect of density of the sea with an opposite
sign to the land. We have seen that if we set a circle of land
^ mile high of 1000 miles radius surrounding one station against
a circle of sea 3 J miles deep, and of the same radius, surround-
ing another, we get a difference of about J x 1*64! x 6'35, or 3^^
nearly, in the number of vibrations performed in one day by a
seconds' pendulum. It is hardly necessary to remark that high
table-land will produce considerably more effect than land only
just raised above the level of the sea, but it should be observed
that the principal part of the correction is due to the depth of the
sea. Thus it would require a uniform elevation of about 2*1
miles, in order that the land elevated above the level of the sea
should produce as much effect as is produced by the difference
between a stratum of land 3^ miles thick and an equal stratum of
water.

AT THE SURFACE OF THE EARTH. 167

23. These considerations seem sufficient to account, at least in
a great measure, for the apparent anomalies which Mr Airy has
noticed in his discussion of pendulum experiments*. The first
table at p. 230 contains a comparison between the observations
which Mr Airy considers first-rate and theory. The column
headed "Error in Vibrations" gives the number of vibrations
per diem in a seconds' pendulum corresponding to the excess of
observed gravity over calculated gravity. With respect to the
errors Mr Airy expressly remarks " upon sci*utinizing the errors of
the first-rate observations, it would seem that, cceteris paribus,
gravity is greater on islands than on continents." This circum-
stance appears to be fully accounted for by the preceding theory.
The greatest positive errors appear to belong to oceanic stations,
which is just what might be expected. Thus the only errors with
the sign + which amount to 5 are. Isle of France + 7*0 ; Marian
Islands + 6*8 ; Sandwich Islands + 5*2 ; Pulo Gaunsah Lout (a
small island near New Guinea and almost on the equator), + 50.
The largest negative errors are, California — 60; Maranham
— 56 ; Trinidad — 52. These stations are to be regarded as
continental, because generally speaking the stations which are
the most continental in character are but on the coasts of conti-
nents, and Trinidad may be regarded as a coast station. That
the negative errors just quoted are larger than those that stand
opposite to more truly continental stations such as Clermont,
Milan, &c. is no objection, because the errors in such different
latitudes cannot be compared except on the supposition that the
value of the ellipticity used in the comparison is correct.

Now if we divide the 49 stations compared into two groups,
an equatorial group containing the stations lying between latitudes
35® N. and 35* S., and a polar group containing the rest, it will
be found that most if not all of the oceanic stations are contained
in the former group, while the stations belonging to the latter
are of a more continental character. Hence the observations will
make gravity appear too great about the equator and too small
towards the poles, that is, they will on the whole make gravity
vary too little from the equator to the poles; and since the
variation depends upon ^m — 6, the observations will be best
satisfied by a value of € which is too great. This is in fact pre-

* Encyclopadia Metropolitana. Art. Figure of the Earth.

158 ON THE VARIATION OF GRAVITY

cisely the result of the discussion^ the value of € which Mr Airy
has obtained from the pendulum experiments (003635) being
greater than that which resulted from the discussion of geodetic
measures (-003352), or than any of the values (-003370, -003360,
and -003407), obtained from the two lunar inequalities which
depend upon the earth's oblateness.

Mr Airy has remarked that in the high north latitudes the
greater number of errors have the sign + , and that those about
the latitude 45® have the sign — ; those about the equator being
nearly balanced. To destroy the errors in high and mean latitudes
without altering the others, he has proposed to add a term
— A sin*X cos*\, where X is the latitude. But a consideration of the
character of the stations seems sufficient, with the aid of the
previous theory, to account for the apparent anomaly. About
latitude 45" the stations are all continental; in fact, ten con-
secutive stations including this latitude are Paris, Clermont, Milan,
Padua, Fiume, Bordeaux, Figeac, Toulon, Barcelona, New York.
These stations ought, as a group, to appear with considerable nega-
tive errors. Mr Airy remarks " If we increased the multiplier of
sin*\," and consequently diminished the ellipticity, "we might
make the errors at high latitudes as nearly balanced as those at
the equator: but then those about latitude 45" would be still
greater than at present."

The largeness of the ellipticity used in the comparison accounts
for the circumstance that the stations California, Maranham,
Trinidad, appear with larger negative errors than any of the
stations about latitude 45", although some of the latter appear
more truly continental than the former. On the whole it would
seem that the best value of the ellipticity is one which, supposing
it left the errors in high latitudes nearly balanced, would give a
decided preponderance to the negative errors about latitude 45" N.
and a certain preponderance to the positive errors about the
equator, on account of the number of oceanic stations which occur
in low latitudes.

If we follow a chain of stations from the sea inland, or from the
interior to the coast, it is remarkable how the errors decrease
algebraically from the sea inwards. The chain should not extend
over too large a portion of the earth's surface, as otherwise a small
error in the assumed ellipticity might eflfect the result. Thus for

AT THE SURFACE OF THE EARTH. 159

example, Spitzbergen + 4*3, Hammerfest — 0*4, Drontheim — 27.
In comparing Hammerfest with Drontheim, we may regard the
former as situated at the vertex of a slightly obtuse angle, and
the latter as situated at the edge of a straight coast Again,
Dunkirk -01, Paris -1-9, Clermont - 3 9, Figeac - 3*8, Toulon

— O'l, Barcelona 0*0, Fomentera + 02. Again, Padua + 0*7, Milan

— 2*8. Again, Jamaica — 0*8, Trinidad — 5*2.

24. Conceive the correction kg'' calculated, and suppose it
applied, as well as the correction ^g\ to observed gravity reduced
to the level of the sea, or to g^ and let the result be g^; Let e^^ be
the ellipticity which would be determined by means of g,,, €,,+ Ae,,
the true ellipticity. Since g ^^^ g -^ g -\- lcg'\ and therefore
fl^ = fl^y,+^'"*S^"' we get by (27)

A^. = 4^m{a-8in*Z) (/-%")} (32).

Now g' = 27rcrA = 27rS = 27r2S< ; and we get from (30) and (28)

^^ " ^ dr ~2a -'^^''^2i + r

All the terms 8< will disappear from the second side of (32) except
8,, and we therefore get

Hence the correction Ae^^ is less than that considered in Art. 18, in
the ratio of 5 — A: to 5, and is therefore probably insensible on ac-
count of the actual distribution of land and water at the surface of
the earth.

25. Conceive the islands and continents cut oflf at the level of
the sea, and the water of the sea replaced by matter having the
same density as the land. Suppose gravity to be observed at the
surface which would be thus formed, and to be reduced by Dr
Young's rule to the level of what would in the altered state of the
earth be a surface of equilibrium. It is evident that g^^ expresses
the gravity which would be thus obtained.

The irregularities of the earth's coating would still not be
wholly allowed for, because the surface which would be formed
in the manner just explained would no longer be a surface of equi-

16Q ON THE VARIATION OF GRAVITY

librium, in consequence of the fresh distribution of attracting
matter. The surface would thus preserve traces of its original
irregularity. A repetition of the same process would give a surface
still more regular, and so on indefinitely. It is easy to see the
general nature of the correction which still remains. Where a
small island was cut oflF, there was previously no material elevation
of the sea-level, and therefore the surface obtained by cutting off
the island will be very nearly a surface of equilibrium, except in so
far as that may be prevented by alterations which take place on a
large scale. But where a continent is cut off there was a consider-
able elevation in the sea-level, and therefore the surface which is
left will be materially raised above the surface of equilibrium which
most nearly represents the earth's surface in its altered state.
Hence the general effect of the additional correction will be to in-
crease that part of ^" which is due to causes which act on a larger
scale, and to leave nearly unaffected that part which is due to
causes which are more local.

The form of the surface of equilibrium which would be finally
obtained depends on the new distribution of matter, and conversely,
the necessary distribution of matter depends on the form of the
final surface. The determination of this surface is however easy
by means of Laplace's analysis.

26. Conceive the sea replaced by solid matter, of density cr,
having a height from the bottom upwards which is to the depth
of the sea as 1 to cr. Let h be the height of the land above
the actual sea-level, h being negative in the case of the sea,
and equal to the depth of the sea multiplied by 1 — l/o-. Let
X be the unknown thickness of the stratum which must be re-
moved in order to leave the surface a surface of equilibrium,
and suppose the mean value of a; to be zero, so that on the whole
matter is neither added nor taken away. The surface of equili-
brium which would be thus obtained is evidently the same a«
that which would be formed if the elevated portions of the irre-
gular surface were to become fluid and to run down.

Let V be the potential of the whole mass in its first state,
V^ the potential of the stratum removed. The removal of this
stratum will depress the surface of equilibrium by the space
Cr^Vg\ and the condition to be satisfied is, that this new

AT THE SURFACE OF THE EARTH. IGl

surface of equilibrium, or else a surface of equilibrium belong-
ing to the same system, aud therefore derived from the former
by further diminishing the radius vector by the small quantity
c', shall coincide with the actual surface. We must therefore

have

O-'V^-^rc'^x-h (33).

Let h and x be expanded in series of Laplace's functions
A^ + Aj+... and x^-^-x^^ ... Then the value of F. at the sur-
face will be obtained from either of equations (28) by replacing S
by ax and putting r=^a. We have therefore

F, = 47r(ra(jro+K + K + -) (34).

After substituting in (33) the preceding expressions for F„ A,

and x^ we must equate to zero Laplace's functions of the same

order. The condition that a?^ = may be satisfied by means of the

constant c', and we shall have

ft*

(?~\ iiraa ^. * ^Xt^ht.

2t -h 1

which gives, on replacing CT^. 4nraa by its equivalent Sa/p,
_ (2i + l)p _( 3(7 ]

We see that for terms of a high order Xi is very nearly equal
to A<, but for terms of a low order, whereby the distribution of land
and sea would be expressed as to its broad features, Xi is sensibly
greater than hi.

27. Let it be required to reduce gravity g to the gravity
which would be observed, in the altered state of the surface,
along what would then be a surface of equilibrium. Let the cor-
rection be denoted by g' — Sg'\ where g' is the same as before. The
correction due to the alteration of the coating in the manner con-
sidered in Art. 20 has been shewn to be equal to

2^8 - 67r2 ^^^ ,

and the required correction will evidently be obtained by replacing
B by ax. Putting for a?, its value got from (35) we have

g'-Zff"^2^aX ,.^i\":^^^ A* = 27rcrS(l- ,.,.i^7^^., U„
^ -^ (2%+l)p-3a. \ (2« + l)p-3<rJ *'

a IL 11

1C2 ON THE VARIATION OF GRAVITY

which gives, since 27r<rthi = 27rah== g and O-^irpUy

3ff"'= gf^S ,,.^^-^'^., *• (36).

2p {zi+l) p — Sa a ^

If we put cr = 2 J, p = 5 J, a = 4000, and suppose h expressed
in miles, we get

(- 4.5A. + A, + .45A, + .290A, + .2Uh^ + ...) (37).

Had we treated the approximate correction 3g" in the same
manner we should have had

Sg" = G^t^= Ox.00017 X
zpa 2i + 1

(3A^ + h^ + .6A, + .429 A, + .333A, + . . .)
whereas, since A: = 3 (1 — cr/p), we get

^ 2pa (2t+l)p

(l,636K + Moh^ + .327A, + .234A, + .182A, + ...) (38).

The general expressions for 3g"\ Sg^', and kg" shew that the
approximate correction kg* agrees with the true correction 9g'"
so far as regards terms of a high order, whereas the leading terms,
beginning with the first variable term, are decidedly too small ;
so that, as far as regards these terms, Sg"* is better represented
by 3y than by kg". This agrees with what has been already
remarked in Art. 25.

If we put g — g + ^9*'=ff,n* ^^^ suppose G and e determined
by means of g^^^, small corrections similar to those already investi-
gated will have to be applied in consequence of the omission of the
quantity g' — Sg" in the value of g. The correction to e would
probably be insensible for the reason mentioned in Art. 18. If
we are considering only the variation of gravity, we may of
course leave out the term h^.

The series (37) would probably be too slowly convergent to be
of much use. A more convergent series may be obtained by sub-
tracting kg" from 3g"\ since the terms of a high order in 3g" are
ultimately equal to those in kg". We thus get

3(7'"= A-^"+G^x. 00017 X

(- C.136A,+ .455Aj + .123A, + :05GA,+ .032A, + ...) .... (39),

AT THE SURFACE OF THE EARTH. 163

which gives g" if g" be known by quadratures for the station
considered.

Although for facility of calculation it has been supposed that
the sea was first replaced by a stratum of rock or earth of less
thickness, and then that the elevated portions of the earth's
surface became fluid and ran down, it may be readily seen that it
would come to the same thing if we supposed the water to remain
as it is, and the land to become fluid and run down, so as to form
for the bottom of the sea a surface of equilibrium. The gravity
g^,^ would apply to the earth so altered.

28. Let us return to the quantity V^ of Art. 19, and consider
how the attraction of the earth's irregular coating affects the
direction of the vertical Let I bo the latitude of the station,
which for the sake of clear ideas may be supposed to be situated
in the northern hemisphere, «r its longitude west of a given p!ace,
f the displacement of the zenith towards the south produced by
the attraction of the coating, i\ its displacement towards the east.
Then

^ Ga dl ' '^ " Ga d^ '

because oT^dVJdl and secl.o'^dVJdvT are the horizontal compo-
nents of the attraction towards the north and towards the west
respectively, and O may be put for g on account of the smallness
of the displacements.

Suppose the angle ;^ of Art. 22 measured from the meridian,
so as to represent the noiih azimuth of the elementary mass
Sa' sin •^'^d^* On passing to a place on the same meridian
whose latitude is l-\-dl, the angular distance of the elementary
mass is shortened by cos x • ^^> ^^^ therefore its linear distance,
which was a chord '^, or 2a sin \-^, becomes

2a sin J^ — a cos J^ cos ;^. dl.

Hence the reciprocal of the linear distance is increased by

l/4a . cos \-^ cosec' ^-^ cos x • ^^

and therefore the part of V^ due to this element is increased by

\ia cos' i*^ cosec J^ cos x • dy^dxdl.

Hence we have

dV^ a rfcOS* Jilr COSY.,, , , .,^.

■di--2n sLV^ ^^^ ^*^^-

w— ^

1C4 ON THE VARIATION OF GRAVITY

Although the quantity under the integral sign in this expres-
sion becomes infinite when '^ vanishes, the integral itself has a
finite value, at least if we suppose B to vary continuously in the
immediate neighbourhood of the station. For if S becomes S'
when X becomes x + tt, we may replace S under the integral sign
by S — S*, and integrate from ;^«0 to x=7r, instead of inte-
grating from X = to ^ = ^Tr, and the limiting value of
(S — S')/sin i*^ when -^ vanishes is 4dhld-^, which is finite.

To get the easterly displacement of the zenith, we have only to
measure ^ froRi the west instead of from the north, or, which
comes to the same, to write x + i^ ^^^ X» ^^^ continue to measure
X from the north. We get

sec i - * = — ^ //cos' i*^ cosec J-^ sin ^ • Bd^dx . • .(41).

29. The expressions (40) and (41) are not to be applied to
points very near the station if S vary abruptly, or even very
rapidly, about such points. Recourse must in such a case be had
to direct triple integration, because it is not allowable to consider
the attracting matter as condensed into a surface. If however S
vary gradually in the neighbourhood of the station, the expression
(40) or (41) may bo used without further change. For if we
modify (40) in the way explained in the preceding article, or else
by putting the integral under the form

/o'/o^ cos* J^ cosec i^ cos X (S - 8^) d^dx,

where Sj denotes the value of Z at the station, we see that the
part of the integral due to a very small area surrounding the
station is very small. If 8 vary abruptly, in consequence suppose
of the occurrence of a cliflf, we may employ the expressions (40),
(41), provided the distance of the cliff from the station be as much
as three or four times its height

These expressions shew that the vertical is liable to very
irregular deviations depending on attractions which are quite
local. For it is only in consequence of the opposition of attractions
in opposite quarters that the value of the integral is not con-
siderable, and it is of course larger in proportion as that opposition
is less complete. Since sin ^-^ is but small even at the distance
of two or three hundred miles, a distant coast, or on the other
hand a distant tract of high land of considerable extent, may

AT THE SURFACE OF THE EARTH. 165

produce a sensible effect ; although of course in measuring an arc
of the meridian those attractions may be neglected which arise
from masses which are so distant as to affect both extremities
of the arc in nearly the same way.

If we compare (40) or (41) with the expression for g** or g**\
we shall see that the direction of the vertical is liable to far more
irregular fluctuations on account of the inequalities in the earth's
coating than the force of gravity, except that part of the force
which has been denoted by g\ and which is easily allowed for.
It has been supposed by some that the force of gravity alters
irregularly along the earth's surface ; and so it does, if we compare
only distant stations. But it has been already remarked with
what apparent regularity gravity when corrected for the inequality
^ appears to alter, in the direction in which we should expect, in
passing from one station to another in a chain of neighbouring
stations.

30. There is one case in which the deviation of the vertical
may become unusually large, which seems worthy of special con-
sideration.

For simplicity, suppose S to be constant for the land, and equal
to zero for the sea, which comes to regarding the land as of
constant height, the sea as of uniform depth, and transferring
the defect of density of the sea with an opposite sign to the land.
Apply the integral (40) to those parts only of the earth's surface
which are at no great distance from the station considered, so that
we may put cos ^-^ = 1, sin \'^ = J*^ = «/2a, if 8 be the distance
of the element, measured along a great circle. In going from the
station in the direction determined by the angle ;^, suppose that
we pass from land to sea at distances ^^ s^, «g,... and from sea
to land at the intermediate distances «,, «^... On going in the
opposite direction suppose that we pass from land to sea at the
distances *_j, 8_^, 8_^, ... and from sea to land at the distances
*-»> *-4"«. Then we get from (40),

dV

-21 = ^^ /{'^S ^i - log «-i - (log *, - log O + log ^. - log «.3

— ...} cos;^ . d^.

If the station be near the coast, one of the terms log«,, log jj.,
will be large, and the zenith will be sensibly displaced towards the

166 ox THE VARUTION OF GRAVITY

sea by the irregular attraction. On account of tbe shelving of the
coast, the preceding expression, which has been formed on the
supposition that S vanished suddenly, would give too great a
displacement ; but the object of this article is not to perform any
precise calculation, but merely to shew how the analysis indicates
a case in which there would be unusual disturbance. A cliff
bounding a tract of table-land would have the same sort of effect
as a coast, and indeed the effect might be greater, on account of
the more sudden variation of S. The effect would be nearly the
same at equal horizontal distances from the edge above and
below, that distance being supposed as great as a small multiple
of the height of the cliff, in order to render the expression (40)
applicable without modification.

31. Let us return now to the force of gravity, and leaving the
consideration of the connexion between the irregularities of gravity
and the irregularities of the earth's coating, and of the possibility
of destroying the former by making allowance for the latter, let us
take the earth such as we find it, and consider further the con-
nexion bet>\een the variations of gravity and the irregularities of
the surface of equilibrium which constitutes the sea-level.

Equation (12) gives the variation of gravity if the form of the
surface be known, and conversely, (8) gives the form of the surface
if the variation of gravity be known. Suppose the variation of
gravity known by means of pendulum-experiments performed at a
great many stations scattered over the surface of the earth ; and
let it be required from the result of the observations to deduce
the form of the surface. According to what has been already
remarked, a scries of Laplace's functions would most likely be
practically useless for this purpose, unless we are content with
merely the leading terms in the expression for the radius vector ;
and the leading character of those terras depends, not necessarily
upon their magnitude, but only on the wide extent of the ine-
qualities which they represent. We must endeavour therefore
to reduce the determination of the radius vector to quadratures.

For the sake of having to deal with small terms, let g be
represented, as well as may be, by the formula which applies to an
oblate spheroid, and let the variable term in tbe radius vector be
calculated by Clairaut's Theorem. Let g^ be calculated gravity.

AT THE SURFACE OF THE EARTH. 167

r, the calculated radius vector, and -put g=g^ + Agy r = r^-\-a^u.
Suppose A^ and Au expanded in series of Laplace's functions.
It follows from (12) that A^ will have no term of the order 1 ;
indeed, if this were not the case, it might be shewn that the
mutual forces of attraction of the earth's particles would have a
resultant Moreover the constant term in Ag may be got rid of by
using a different value of 0. No constant term need be taken in
the expansion of Au, because such a term might be got rid of by
using a different value of a, and a of course cannot be determined
by pendulum-experiments. The term of the first order will dis-
appear if r be measured from the common centre of gravity of
the mass and volume. The remaining terms in the expansion
of Aw will be determined from those in the expansion of Ag by
means of equations (8) and (12).

Let A(7 = (?(v, + t;, + r,+ ...) (42),

and we shall have

Aw = t;, + iv3 + iv, + (43).

Suppose Ag = GF{0, <f>). Let -^ be the angle between the
directions determined by the angular co-ordinates 0, <f> and ^, 4>\

Let (1 — 2fcos'^+ f)^ be denoted by B, and let Q^ be the coef-
ficient of f* in the expansion of BT* in a series according to ascend-
ing powers of f. Then

V, = ?^ j' f'F{ff, <!>') Q, sin ffd0'd<l>\

and therefore if f be supposed to be less than 1, and to become 1
in the limit, we shall have 47rAM = limit of

j' ^F{ff, f )(5?Q,+irQs- + ^ r' (?,+ ...)sin^d5'df ...(44).
Now assume

and we shall have

^=5Q.+7?<?....+(2t+i)r<2.+ ...;

wheuce we get, putting Z for if' - Q, - ?Q., 7 = 2/f ~* d . ^Z.

168 ON THE VARIATION OF GRAVITY

Integrating by parts, and observing that 7 vanishes with ^, we get

The last integral may be obtained by rationalization. If we
assume JS = w — f, and observe that Qo = 1, Q^ = cos '^, and that
ti; = 1 when ^ vanishes, we shall find

iJr-«^?=cos^.log i3^ - (1 +cos^) ^^^ - 2 cos^ . log -2~ .

Whenf=lwehaveZ=(2-2cos^)-*-(l+cos^),t(;=l+2sini^,
and

ji^^Zd^^ - 2 sin i^ (1 - sin ^yfr) - cos ^ log {sin J^ (1 + sin if)}.

Putting/ {f) for the value of 7 when 5"= 1, we have
f(y^) = cosec i'f + 1 — 6 sin J^

— 5 cos '^ — 3 cos ^ log {sini'^(l +sin J^)} (45).

In the expression for Au, we may suppose the line from which
ff is measured to be the radius vector of the station considered.
We thus get, on replacing F{d', <f>) by O'^Ag, and employing the
notation of Art. 22,

^" = i^(?^-^'^5'-/W sinfdirdx (46).

32. Jjet Ag=sg' + A'g, Then A'g is the excess of observed
gravity reduced to the level of the sea by Dr Young's rule over
calculated gravity; and of the two parts g* and A'g of which Ag
consists, the former is liable to vary irregularly and abruptly from
one place to another, the latter varies gradually. Hence, for the
sake of interpolating between the observations taken at diflFerent
stations, it will be proper to separate A^ into these two parts, or,
which comes to the same, to separate the whole integral into two
parts, involving g' and A'g respectively, so as to get the part of Au
which is due to g' by our knowledge of the height of the land and
the depth of the sea, and the part which depends on A'g by the
result of pendulum-experiments. It may be observed that a con-
stant error, or a slowly varying error, in the height of the land
would be of no consequence, because it would enter with opposite
signs into g' and A'g,

It appears, then, that the results of pendulum-experiments
furnish sufficient data for the determination of the variable part of

AT THE SURFACE OF THE EABTH. 169

the radius vector of the earth's surface, and consequently for the
determination of the particular value which is to be employed at
any observatory in correcting for the lunar parallax, subject how-
ever to a constant error depending on an error in the assumed
value of a.

33. The expression for g'" in Art. 27 might be reduced to
quadratures by the method of Art, 31, but in this case the inte-
gration with respect to f could not be performed in finite terms, and
it would be necessary in the first instance to tabulate, once for all,
an integral of the form /o*/(?' ^^^ ^) cif for values of '^, which need
not be numerous, from to tt. This table being made, the tabu-
lated function would take the place oi f(yfr) in (46), and the rest
of the process would be of the same degree of diflBculty as the
quadratures expressed by the equations (31) and (46).

34. Suppose Au known approximately, either as to its general
features, by means of the leading terms of the series (43), or in
more detail from the formula (46), applied in succession to a great
many points on the earth's surface. By interpolating between
neighbouring places for which Au has been calculated, find a
number of points where Aw has one of the constant values — 2)8,
— ^, 0, yS, 2)8..., mark these points on a map of the world, and join
by a curve those which belong to the same value of Aw. We shall
thus have a series of contour lines representing the elevation or
depression of the actual sea-level above or below the surface of
the oblate spheroid, which has been employed as most nearly
representing it. If we suppose these lines traced on a globe, the
reciprocal of the perpendicular distance between two consecutive
contour lines will represent in magnitude, and the perpendicular
itself in direction, the deviation of the vertical from the normal to
the surface of the spheroid, or rather that part of the deviation
which takes place on an extended scale : for sensible deviations
may be produced by attractions which are merely local, and which
would not produce a sensible elevation or depression of the sea-
level ; although of course, as to the merely mathematical question,
if the contour lines could be drawn suflSciently close and exact,
even local deviations of the vertical would be represented.

Similarly, by joining points at which the quantity denoted in
Art. 19 by Vc has a constant value, contour lines would be formed

170 ON THE VARIATION OF GRAVITY

representiDg the elevation of the actual sea-level above what
would be a surface of equilibrium if the earth's irregular coating
were removed. By treating 7, in the same way, contour lines
would be formed corresponding to the elevation of the actual
sea-level above what would be the sea-level if the solid portions of
the earth's crust which are elevated were to become fluid and to
run down, so as to form a level bottom for the sea, which would in
that case cover the whole earth.

These points of the theory are noticed more for the sake of
the ideas than on account of any application which is likely to be
made of them; for the calculations indicated, though possible with
a sufficient collection of data, would be very laborious, at least if
we wished to get the results with any detail.

35. The squares of the ellipticity, and of quantities of the
same order, have been neglected in the investigation. Mr Airy,
in the Treatise already quoted, has examined the consequence, on
the hypothesis of fluidity, of retaining the square of the ellipticity,
in the two extreme cases of a uniform density, and of a density
infinitely great at the centre and evanescent elsewhere, and has
found the correction to the form of the surface and the variation of
gravity to be insensible, or all but insensible. As the connexion
between the form of the surface and the variation of gravity fol-
lows independently of the hypothesis of fluidity, we may infer that
the terms depending on the square of the ellipticity which would
appear in the equations which express that connexion would be
insensible. It may be worth while, however, just to indicate the
mode of proceeding when the square of the ellipticity is retained.

By the result of the first approximation, equation (1) is satis-
fied at the surface of the earth, as far as regards quantities of the
first order, but not necessarily further, so that the value of V + U
at the surface is not strictly constant, but only of the form c + H,
where -ff is a small variable quantity of the second order. It is
to be observed that V satisfies equation (3) exactly, not approxi-
mately only. Hence we have merely to add to F a potential V
which satisfies equation (3) outside the earth, vanishes at an
infinite distance, and is equal to B" at the surface. Now if we
suppose F' to have the value H at the surface of a sphere whose
radius is a, instead of the actual surface of the earth, we shall only

AT THE SURFACE OF THE EARTH. 171

commit an error which is a small quantity of the first order com-
pared with H, and II is itself of the second order, and therefore
the error will be only of the third order. But by this modifica-
tion of one of the conditions which F ' is to satisfy, we are enabled
to find F' just as F was found, and we shall thus have a solution
which is correct to the second order of approximation. A repeti-
tion of the same process would give a solution which would be
correct to the third order, and so on. It need hardly be remarked
that in going beyond the first order of approximation, we must
distinguish in the small terms between the direction of the vertical,
and that of the radius vector.

[From the Beport of the British Association for 1849. Part li. p. 10.]

On a Mode of Measuring the Astigmatism of a Defective

Eye.

Besides the common defects of long sight and short sight,
there exists a defect, not very uncommon, which consists in the
eye's refracting the rays of light with diflFerent power in diflFerent
planes, so that the eye, regarded as an optical instrument, is not
symmetrical about its axis. This defect was first noticed by the
present Astronomer Royal in a paper published about 20 years
ago in the Transactions of the Cambridge Philosophical Society.
It may be detected by making a small pin-hole in a card, which is
to be moved from close to the eye to arm's length, the eye mean-
while being directed to the sky, or any bright object of suflBcient
size. With ordinary eyes the indistinct image of the hole remains
circular at all distances ; but to an eye having this peculiar defect
it becomes elongated, and, when the card is at a certain distance,
passes into a straight line. On further removing the card, the
image becomes elongated in a perpendicular direction, and finally,
if the eye be not too long-sighted, passes into a straight line
perpendicular to the former. Mr Airy has corrected the defect in
his own case by means of a spherico-cylindrical lens, in which the
required curvature of the cylindrical surface was calculated by
means of the distances of the card from the eye when the two focal
lines were formed. Others however have found a difficulty in
preventing the eye from altering its state of adaptation during the
measurement of the distances. The author has constructed an
instrument for determining the nature of the required lens, which
is based on the following proposition : —

Conceive a lens ground with two cylindrical surfaces of equal
radius, one concave and the other convex, with their axes crossed

Oy A MODE OP MEASCniNG, &C. 173

at right angles ; call such a lens an astigmatic lens ; let the reci-
procal of its focal length in one of the principal planes be called its
power, and a line parallel to the axis of the convex surface its
astigmatic axis. Then if two thin astigmatic lenses be combined
with their astigmatic axes inclined at any angle, they will be
equivalent to a third astigmatic lens, determined by the following
construction : — In a plane perpendicular to the common axis of
the lenses, or axis of vision, draw through any point two straight
lines, representing in magnitude the powers of the respective
lenses, and inclined to a fixed line drawn arbitrarily in a direc-
tion perpendicular to the axis of vision at angles equal to twice
the inclinations of their astigmatic axes, and complete the
parallelogram. Then the two lenses will be equivalent to a single
astigmatic lens, represented by the diagonal of the parallelogram
in the same way in which the single lenses are represented by the
sides. A piano-cylindrical or spherico-cylindrical lens is equi-
valent to a common lens, the power of which is equal to the semi-
sum of the reciprocals of the focal lengths in the two principal
planes, combined with an astigmatic lens, the power of which is
equal to their semi-diflFerence,

If two piano cylindrical lenses of equal radius, one concave and
the other convex, be fixed, one in the lid and the other in the
body of a small round wooden box, with a hole in the top and
bottom, so as to be as nearly as possible in contact, the lenses
will neutralize eacli other when the axes of the surfaces are
parallel ; and, by merely turning the lid round, an astigmatic lens
may be formed of a power varying continuously from zero to twice
the astigmatic power of either lens. When a person who has the
defect in question has turned the lid till the power suits his eye,
an extremely simple numerical calculation, the data for which are
furnished by the chord of double the angle through which the lid
has been turned, enables him to calculate the curvature of the
cylindrical surface of a lens for a pair of spectacles which will
correct the defect of his eye.

[The proposition here employed is easily demonstrated by a
method founded on the notions of the theory of undulations,
though of course, depending as it does simply on the laws of
reflection and refraction, it does not involve the adoption of any
particular theory of light.

174 ON A MODE OF MEASURING

Consider a thin lens bounded by cylindrical surfaces, the axes
of the cylinders being crossed at right angles. Refer points in the
neighbourhood of the lens to the rectangular axes of x, y, z^ the
axis of z being the axis of the lens, and those of x and y parallel
to the axes of the two cylindrical surfaces respectively, the origin
being in or near the lens, suppose in its middle point. Let r, s,
measured positive when the surfaces are convex, be the radii of
curvature in the planes of xz^ yz respectively. Then if T be the
central thickness of the lens, the thickness near the point {x, y)
will be

very nearly. As T is constant, and is supposed very small, we may
neglect it, and regard the thickness as negative, and expressed by
the second term in the above formula. The incident pencil being
supposed to be direct, or only slightly oblique, and likewise slender,
the retardation of the ray which passes through the point {x, y)
may be calculated as if it were incident perpendicularly on a
parallel plate of thickness

so that if £ be the retardation, measured by equivalent space in
air, and /x be the index of refraction

-ie=0*-i).j(^4-^)

=0*-i).i(J+j)(^+i^ + 0*-i).i(J-J)(a^-3^.

The efifect therefore of our lens, to the lowest order of approxi •
mation, which gives the geometrical foci in the principal planes, is
the same as that of two thin lenses placed in contact, one an
ordinary lens, and the other an astigmatic lens. If / be the radius
of curvature of the piano-spherical lens equivalent to the ordinary
lens, and r" that of the astigmatic lens, we have

;=j(l+l), 4,=i(---).

r \r 8 J r \r a J

as above enunciated. If p be the power of the astigmatic lens^

P = Cf*-l)-i(r--j)'

THE ASTIGMATISM OF A DEFECTIVE EYE. 175

and for the retardation produced by this lens alone

where p, 6 are polar co-ordinates in the plane of xy.

If two thin astigmatic lenses of powers p, p* and with their
astigmatic axes inclined at azimuths a, a to the axis of y be com-
bined, we shall have for the combination

-R = ipp* cos 2 (^ - a) + ipp* cos 2(0- a%

which is the same as would be given by a single astigmatic lens of
power Pj at an azimuth a^, provided

pp^ cos 2 (^ - a) + p'p* cos 2 (5 - a') = p,p' cos 2 (^ - a,),

which will be satisfied for all values of provided

p cos 2a -\-p cos 2x' =Pj cos 22^,
^ sin 22 + p' sin 2a' =Pj sin 22^.

These two equations geometrically interpreted give the propo-
sition enunciated above for the combination of astigmatic lenses.]

[From the Beport of the BriiUh Association for 1849. Part ii. p. 11.]

On the Determination of the Wave Length coiuiESPONDiNa

WITH ANY POINT OF THE SPECTRUM.

Mr Stokes said it was well known to all engaged in optical
researches that Fraunhofer had most accurately measured the wave
lengths of seven of the principal fixed lines of the spectrum. Now
he found that by a very simple species of interpolation, which he
described, he could find the wave length for any point intermediate
between the two of them. He then exemplified the accuracy to
be obtained by his method by applying it to the actually known
points, and shewed that in these far larger intervals than he ever
required to apply the method to the error was only in the eighth,
and in one case in the seventh, place of decimals. By introducing
a term depending on the square into the interpolation still greater
accuracy was attainable. The mode of interpolation depended on
the known fact that, if substances of extremely high refractive
power be excepted, the increment A/x of the refractive index in
passing from one point of the spectrum to another is nearly propor-
tional to the increment A\"* of the squared reciprocal of the wave
length. Even in the case of flint glass, the substance usually
employed in the prismatic analysis of light, this law is nearly true
for the whole spectrum, and will be all but exact if restricted to
the interval between two consecutive fixed lines. Hence we have
only to consider /x as a function, not of X, but of X"*, and then take
proportional parts.

On examining in this way Fraunhofer*s indices for flint glass,
it appeared that the wave length B\ of the fixed line B was too
great by about 4 in the last, or eighth, place of decimals. It is

ON THE DETERMINATION OF THE WAVE LENGTH, &C. 177

remarkable that the line B was not included in Fraunhofer's
second and more accurate determination of the wave lengths, and
that the proposed correction to BX is about the same, both as to
sign and magnitude, as one would have guessed from Fraunhofer's
own corrections of the other wave lengths, obtained from his
second series of observations,

[A map of the spectrum laid down according to the values of
X"* instead of X refers equally to a natural standard, that is, one
independent of the material of any prism, and is much more con-
venient for comparison with spectra obtained by dispersion, not
diflFraction.]

s. n. 12

[From the Transactions of the Cambridge Philosophical Society, Vol. viii.

p. 707.]

Discussion of a Differential Equation relating to the

breaking of railway bridges.

[Read May 21, 1849.]

To explain the object of the following paper, it will be best to
relate the circumstance which gave rise to it. Some time ago
Professor Willis requested my consideration of a certain diflFeren-
tial equation in which he was interested, at the same time explain-
ing its object, and the mode of obtaining it. The equation will be
found in the first article of this paper, which contains the substance
of what he communicated to me. It relates to some experiments
which have been performed by a Royal Commission, of which Pro-
fessor Willis is a member, appointed on the 27th of August, 1847,
" for the purpose of inquiring into the conditions to be observed
by engineers in the application of iron in structures exposed to
violent concussions and vibration." The object of the experiments
was to examine the eflFect of the velocity of a train in increasing or
decreasing the tendency of a girder bridge over which the train
is passing to break under its weight. In order to increase the
observed effect, the bridge was purposely made as slight as possible :
it consisted in fact merely of a pair of cast or wrought iron bars,
nine feet long, over which a carriage, variously loaded in different
sets of experiments, was made to pass with different velocities.
The remarkable result was obtained that the deflection of the
bridge increased with the velocity of the carriage, at least up to a
certain point, and that it amounted in some cases to two or three
times the central statical deflection, or that which would be pro-
duced by the carriage placed at rest on the middle of the bridge.
It seemed highly desirable to investigate the motion mathemati-
cally, more especially as the maximum deflection of the bridge,
considered as depending on the velocity of the carriage, had not

DISCUSSION OF A DIFFERENTIAL EQUATION, &C. 179

been reached in the experiments*, in some cases because it corre-
sponded to a velocity greater than any at command, in others
because the bridge gave way by the fracture of the bars on increas-
ing the velocity of the carriage. The exact calculation of the
motion, or rather a calculation in which none but really insignifi-
cant quantities should be omitted, would however be extremely
difficult, and would require the solution of a partial differential
equation with an ordinary differential equation for one of the
equations of condition by which the arbitrary functions would have
to be determined. In fact, the forces acting on the body and on
any element of the bridge depend upon the positions and motions,
or rather changes of motion, both of the body itself and of every
other element of the bridge, so that the exact solution of the
problem, even when the deflection is supposed to be small, as it is
in fact, appears almost hopeless.

In order to render the problem more manageable. Professor
Willis neglected the inertia of the bridge, and at the same time
regarded the moving body as a heavy particle. Of course the
masses of bridges such as are actually used must be considerable ;
but the mass of the bars in the experiments was small compared
with that of the carriage, and it was reasonable to expect a near
accordance between the theory so simplified and experiment.
This simplification of the problem reduces the calculation to an
ordinary differential equation, which is that which has been already
mentioned ; and it is to the discussion of this equation that the
present paper is mainly devoted.

This equation cannot apparently be integrated in finite terms f,
except for an infinite number of particular values of a certain
constant involved in it ; but I have investigated rapidly convergent
series whereby numerical results may be obtained. By merely
altering the scale of the abscisssB and ordinates, the differential
equation is reduced to one containing a single constant /8, which is
defined by equation (5). The meaning of the letters which appear
in this equation will be seen on referring to the beginning of
Art. 1. For the present it will be sufficient to observe that /8
varies inversely as the square of the horizontal velocity of the

* The details of the experiments will be found in the Report of the Commission,
to which the reader is referred.

t [The integral can be expressed by definite mtcgrols. See Art. 7, and last
paragraph bnt one in the paper.]

12—2

180 DISCUSSION OF A DIFFERENTIAL EQUATION

body, so that a small value of )8 corresponds to a high velocity, and
a large value to a small velocity.

It appears from the solution of the differential equation
that the trajectory of the body is unsymmetrical with respect to
the centre of the bridge, the maximum depression of the body occur-
ring beyond the centre. The character of the motion depends mate-
rially on the numerical value of yS. When /8 is not greater than
i, the tangent to the trajectory becomes more and more inclined
to the horizontal beyond the maximum ordinate, till the body gets
to the second extremity of the bridge, when the tangent becomes
vertical. At the same time the expressions for the central deflec-
tion and for the tendency of the bridge to break become infinita
When is greater than ^, the analytical expression for the ordi-
nate of the body at last becomes negative, and afterwards changes
an infinite number of times from negative to positive, and from
positive to negative. The expression for the reaction becomes
negative at the same time with the ordinate, so that in fact the
body leaps.

The occurrence of these infinite quantities indicates one of two
things : either the deflection really becomes very large, after which
of course we are no longer at liberty to neglect its square ; or else
the effect of the inertia of the bridge is really important Since
the deflection does not really become very great, as appears from
experiment, we are led to conclude that the effect of the inertia is
not insignificant, and in fact I have shewn that the value of the
expression for the vis viva neglected at last becomes infinite.
Hence, however light be the bridge, the mode of approximation
adopted ceases to be legitimate before the body reaches the second
extremity of the bridge, although it may be sufficiently accurate
for the greater part of the body's course.

In consequence of the neglect of the inertia of the bridge, the
differential equation here discussed fails to give the velocity for
which T, the tendency to break, is a maximum. When /8 is a
good deal greater than \, T is a maximum at a point not very
near the second extremity of the bridge, so that we may apply the
result obtained to a light bridge without very material error. Let
jTj be this maximum value. Since it is only the inertia of the
bridge that keeps the tendency to break from becoming extremely
great, it appears that the general eflfect of that inertia is to
preserve the bridge, so that we cannot be far wrong in regarding

RELATING TO THE BREAKING OF RAILWAY BRIDGE.'^. 181

7\ as a superior limit to the actual tendency to break. When fi is
very large, T^ may be calculated to a sufficient degree of accuracy
with very little trouble.

Experiments of the nature of those which have been mentioned
may be made with two distinct objects; the one, to analyse experi-
mentally the laws of some particular phenomenon, the other, to
apply practically on a large scale results obtained from experi-
ments made on a small scale. With the former object in view,
the experiments would naturally be made so as to render as con-
spicuous as possible, and isolate as far as might be, the effect which
it was desired to investigate; with the latter, there are certain
relations to be observed between the variations of the different
quantities which are in any way concerned in the result. These
relations, in the case of the particular problem to which the present
paper refers, are considered at the end of the paper.

1. It is required to determine, in a form adapted to numerical
computation, the value of y in terms of x\ where y is a function
of x' defined by satisfying the differential equation

dy_^ hy'

(1),

dx"" ^ {2cx' ^xy

with the particular conditions

y=0, ^, = 0, whena;' = (2),

the value of y' not being wanted beyond the limits and 2c of x\
It will appear in the course of the solution that the first of the
conditions (2) is satisfied by the complete integral of (1), while the
second serves of itself to determine the two arbitrary constants
which appear in that integral.

The equation (1) relates to the problem which has been ex-
plained in the introduction. It was obtained by Professor Willis
in the following manner. In order to simplify to the very utmost
the mathematical calculation of the motion, regard the carriage as
a heavy particle, neglect the inertia of the bridge, and suppose the
deflection very small. Let x\ y be the co-ordinates of the moving
body, X being measured horizontally from the beginning of the
bridge, and y vertically downwards. Let M be the mass of the
body, V its velocity on entering the bridge, 2c the length of the
bridge, g the force of gravity, 8 the deflection produced by the

182 DISCUSSION OF A DIFFERENTIAL EQUATION

body placed at rest on the centre of the bridge, B the reaction
between the moving body and the bridge. Since the deflection is
very small, this reaction may be supposed to act vertically, so that
the horizontal velocity of the body will remain constant, and there-
fore equal to V. The bridge being regarded as an elastic bar or
plate, propped at the extremities, and supported by its own stiflF-
ness, the depth to which a weight will sink when placed in succes-
sion at diflferent points of the bridge will vary as the weight
multiplied by (2g»' — a? ')', as may be proved by integration, on
assuming that the curvature is proportional to the moment of the
bending force. Now, since the inertia of the bridge is neglected,
the relation between the depth y to which the moving body has
sunk at any instant and the reaction B will be the same as if 22
were a weight resting at a distance x' from the extremity of the
bridge ; and we shall therefore have

y' = CM (2cx - xy,

C being a constant, which may be determined by observing that
we must have y== 8 when R = Mg and a?' = c; whence

Mgc^'
We get therefore for the equation of motion of the body

cTy' g^!/_

/«M »

dx'
which becomes on observing that -r- = V

d?y' _ g g<f y'

'«M >

which is the same as equation (1), a and h being defined by the
equations

^=^ y=2» ^^ yTg (^)-

2. To simplify equation (1) put

X = 2cx, y ' = 1 6c*a6'*y , 6 = 4c"A
whicli gives

dx' ^ (x-a^y ^*'-

RELATING TO THE BREAKli^G OF RAILWAY BRIDGES. 183

It is to be observed that x denotes the ratio of the distance of the
body from the beginning of the bridge to the length of the bridge;
y denotes a quantity from which the depth of the body below the
horizontal plane in which it was at first moving may be obtained
by multiplying by 16c*a6"' or 16/S; and yS, on the value of which
depends the form of the body's path, is a constant defined by the
equation

^ = 4F*5 (^^-

3. In order to lead to the required integral of (4), let us first
suppose that x is very small. Then the equation reduces itself to

da? '^ a? ^^^'

of which the complete integral is

y=^ + ilxi+^i^ + i?xi-^i~^ (7).

and (7) is the approximate integral of (4) for very small values of
X, Now the second of equations (2) requires that 4=0, i? = 0*,
80 that the first term in the second member of equation (7) is the
leading term in the required solution of (4).

4. Assuming in equation (4) y = {x — i>^ z, we get

^.{(a;-a^)'^} + /3z = /3 (8).

Since (4) gives y = (a? — a:*)' when )8 = oo , and (5) gives )8 = oo
when F= 0, it follows that z is the ratio of the depression of the
body to the equilibrium depression. It appeal's also from Art. 3,
that for the particular integral of (8) which we are seeking, z
is ultimately constant when x is very small.

* When /3>J, the last two terms in (7) take the form «* {Cco8(glogx)
+ D8in((7logj:)}; and if ^j denote this qnantity we cannot in Btrictness speak of
the limiting value of dyjdx when x=0. If we give x a small positive valae, which
we then suppose to decrease indefinitely, dyjdx will fluctuate between the constantly
increasing limits =t x-l^/!(iC + <77))«+(iD-5C)-}, or =fca;-iv{/3(C«+ D«)}, since
9 =>/(/3- i)* But the body is supposed to enter the bridge horizontally, that is, in
the direction of a tangent, since the bridge is supposed to be horizontal, so that we
must clearly have C« + I>3=0, and therefore C=0, D=0. When /3=J the last two

terms in (7) take the form x^ (E + F\ogx), and we must evidently have £=0, F=-.0.

184 DISCUSSION OF A DIFFERENTIAL EQUATION

To integrate (8) assume then

z ^ A^ + A^x + A^c^ -{-,., = t A ff (9),

and we get

S(i + 2)(i + l) J,a?*-22(i + 3)(i + 2)^^*
or

+ (f + l)(i + 2)ilja?' = i8 (10)

where it is to be observed that no coefficients -4| with negative
suffixes are to be taken.

Equating to zero the coefficients of the powers 0, 1, 2... of a? in
(10), we get

(2H.i8)A = A
(6+i8)^,-12il,= 0, &c.
and generally

{{% + 1) (»• + 2) + ^} 4 - 2 (» + 1) (t + 2) A^,

+ (t + l)(t + 2)^,^ = (11).

The first of these equations gives for A^ the same value which
would have been got from (7). The general equation (11), which
holds good from i = 1 to i = oo , if we conventionally regard -4,, as
equal to zero, determines the constants -4^, A^, -4,... one after
another by a simple and uniform arithmetical process. It will be
rendered more convenient for numerical computation by putting it
under the form

/3

A=M..-fA^jjl-(-^

(12);

H-2)+/8J-

for it is easy to form a table of differences as we go along ; and
when i becomes considerable, the quantity to be subtracted from
il^j + A -4^ will consist of only a few figures.

6. When { becomes indefinitely great, it follows from (11)
or (12) that the relation between the coefficients A^ is given by
the equation

A-2^^, + J,_ = (13),

of which the integral is

il, = (7 + Cf (14).

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 185

Hence the ratio of consecutive coefficients is ultimately a ratio
of equality, and therefore the ratio of the (i + l)th term of the
series (9) to the ith is ultimately equal to x. Hence the series is
convergent when x lies between the limits — 1 and + 1 ; and it
is only between the limits and 1 of a: that the integral of (8)
is wanted. The degree of convergency of the series will be ulti-
mately the same as in a geometric series whose ratio is x,

6. When x is moderately small, the series (9) converges so
rapidly as to give z with little trouble, the coefficients A^, -4,...
being supposed to have been already calculated, as far as may be
necessary, from the formula (12). For larger values, however, it
would be necessary to keep in a good many terms, and the labour
of calculation might be abridged in the following manner.

When i is very large, we have seen that equation (12) reduces
itself to (13), or to A^A^_^ = 0, or, which is the same, AM, = 0.
When i is large, A^A^ will be small ; in fact, on substituting in the
small term of (12) the value of A^ given by (14), we see that
A*A^ is of the order t~\ Hence AM,, AM, ... will be of the orders
f^', i^..., so that the successive differences of At will rapidly de-
crease. Suppose f terms of the series (9) to have been calculated
directly, and let it be required to find the remainder. We get by
finite integration by parts

X* x*^^ af^

XAfc* = const. + At =- - AAf -.■ ttj + AM,

and taking the sum between the limits i and oo we get
A/c* + A.^^a}^^ + ... to inf.

z will however presently be made to depend on series so rapidly
convergent that it will hardly be worth while to employ the series
(15), except in calculating the scries (9) for the particular value \
of Xy which will be found necessary in order to determine a certain
arbitrary constant*.

* A mode of caloalating the value of z tat x= .J will presently be given, which is
easier tlian that here mentioned, unless /3 be very large. See equation (42) at the
end of tliis paper.

186 DISCUSSION OF A DIFFEBENTIAL EQUATION

7. If the constant term in equation (4) be omitted, the equa-
tion reduces itself to

g + ^y =0 (16).

The form of this equation suggests that there may be an inte-
gral of the form y = a?"* (1 — a?)*. Assuming this expression for trial,
we get

=y{m(m— 1)— 2m(m+n— l)j?+(m+7i)(m+n— l)a?'}.

The second member of this equation will be proportional to y, if

m + w-l = (17),

and will be moreover equal to — )8y, if

m'-m + )8 = (18).

It appears from (17) that m, n are the two roots of the quad-
ratic (18). We have for the complete integral of (16)

y = -4a?"*(l-a:)»4-5^*(l-^)"* (19).

The complete integral of (4) may now be obtained by replacing
the constants -4, B by functions jR, 8 of x, and employing the
method of the variation of parameters. Putting for shortness

a;^ (1 - xY = u, 0^(1- ar)"* = V,

we get to determine It and S the equations

dR , dS ^
u-r- + v-T- = 0,
dx dx

du dR dv dS ^

Since v-j — w-T- = m — n, we get from the above equations

dR ^ ^v dS^ _ I3u
dx 171 — 71* dx m — n*

whence we obtain for a particular integral of (4)

y= -^L~(l-a:)" fa;"(l-a:)"'(ir-a;*(l-a?)'-fa;"'(l-a:)»da;l...(20);
m—n (Jo Jo )

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 187

and the complete integral will be got by adding together the
second members of equations (19), (20). Now the second member
of equation (20) varies ultimately as a;", when x is very small, and
therefore, as shewn in Art. 3, we must have ^ = 0, 5 = 0, so that
(20) is the integral we want.

When the roots of the quadratic (18) are real and commen-
surable, the integrals in (20) satisfy the criterion of integrability,
so that the integral of (4) can be expressed in finite terms without
the aid of definite integrals. The form of the integral will, how-
ever, be complicated, and y may be readily calculated by the
method which applies to general values of yS.

8. Since I F{x)dx=\ F{x)dx—\ F{l'-x)dx, we have
from (20)

y=— ^ {x'il-x)' j ar{l-x)''cLB-af{l-x)'' j z'^l-x^dx]

+ '^{x"(l-a;)"[ ' {l-x)''x''dx-x''{l-x)' f '' {l-xyardx].

If we put f(x) for the second member of equation (20), the
equation just written is equivalent to

f(x)=f{l-x) + <l>{x) (21),

where

^(x) =-^ {x"'(l-a:)" f x'{l-x)''dx-x'{l-x)'' f ar(l-xydx} . . . (22).

1^ ^ J Q Jo

Now since 7n + n = 1,

{af'[l-'xydx^L[x-'--l)'^dx^-{w-\io-iy^w^dw=^

At the limits a? = and a? = 1, we have t^ = oo and w = 1, « = oo and
5 = 0, whence if I denote the definite integral,

/=/>-(i-.)-rf.=/;^

(1 + «)'

We get by integration by parts

j(r+^'" 2(l+«)''*"2J(l+«)''

188 DISCUSSION OF A DIFFERENTIAL EQUATION

and again by a formula of reduction

Now /8 being essentially positive, the roots of the quadratic (18)
are either real, and comprised between and 1, or else imaginary
with a real part equal to |. In either case the expressions which
are free from the integral sign vanish at the limits 5 = and « = oo ,
and we have therefore, on replacing m (1 — m) by its value 13,

2Jo 1 + J? •

■ The function <f> (x) will have diflferent forms according as the
roots of (18) are real or imaginary. First suppose the roots real,
and let m = J + r, n = J — ?•, so that

r = Vi^ (23).

In this case w is a real quantity lying between and 1, and we
have therefore by a known formula

/

"-"-^cZs

£ ^^ '^ ^ '^ (24)

l+« sin mTT cos^'Tr ^

whence we get from (22), observing that the two definite integrals
in this equation are equal to each other,

^(.) = . . ^•-- V.-x'Ir-^V- f^fl (25).

^^ "^ 4rcosr7r l\l-^/ \l—xj) ^ '

This result might have been obtained somewhat more readily
by means of the properties of the first and second Eulerian inte-
grals.

When /8 becomes equal to J, r vanishes, the expression for
^ (d?) takes the form ^, and we easily find

<^(^) = JV^^r^'log^^ (26).

When )8 > J, the roots of (18) become imaginary, and r becomes
p V — 1, where

P = V^~i (27).

The formula (25) becomes

& (.r) = ^ Vl^IPsin ip logr^^) (i?8).

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 189

li f{x) be calculated from x = to x = i, equation (21) will
enable us to calculate it readily from a? = ^ to a? = 1, since it is easy
to calculate ^ (x),

9. A series of a simple form, which is more rapidly convergent
than (9) when x approaches the value ^, may readily be investi-
gated.

Let 0? = ^ (1 + tt?) ; then substituting in equation (8) we get

i-^^{(l-v,yz]+^z = fi (29).

Assume

z^B, + B^w^-{'B^w\..='ZB,w^' (30),

then substituting in (29) we get

l.B,{2i {2i - 1) w*^ - 2 (2i 4 2) (2i + 1) tu^

+ (2i + 4) (2i + 3) ti7«^ + 4^M?''} = 4^,
or,

S {i{2i - 1) 5, - 2 [i(2i - 1) - ^] B^, + i(2i - 1) 5^,} vT" = 2/3.

This equation leaves B^ arbitrary, and gives on dividing by
i(2t — 1), and putting in succession i= 1, i = 2, &c.,

/3

-B. - 2 (l - y^j 5, = 2/3 (31),

5,-2(1- 2^g)5. + 5. = 0,&c.;

and generally when i > 1,

5, = 5,. + A5,.-.^2?fj^5._. (32).

The constants 5,, i?,,... being thus determined, the series (30)
will be an integral of equation (29), containing one arbitrary con-
stant. An integral of the equation derived from (29) by replacing
the second member by zero may be obtained in just the same way
by assuming 2 = CqW + C^w^+ ,,. when C^, C,. . . will be determined
in terms of C^, which remains arbitrary. The series will both be
convergent between the limits w = — 1 and w = 1, that is, between
the limits x = and ir= 1. The sum of the two series will be the
complete integral of (29), and will be equal to (a? — a;")"*/(a?) if the

100 DISCUSSION OF A DIFFERENTIAL EQUATION

constants B^, C^ be properly determined. Denoting the sums of
the two series by F^ (w), F^ (w) respectively, and writing a (x) for
{x — x'y*f{x)y SO that z = <T {x), we get

<T(x) = F.{w)-\~F,{tc\ a(l^x) = F,(w)-F,(w);

and since 2F^ (w) = o- (a:) - o- (1 — re) = (a: — a^)"^ (f> {x) by (21), we get

<T{x)^F,{w) + \{x-xy'4>{xl )

a[i^x)=F^{w)--\{x-x'r4>m ^^ ^'

To determine B^ we have

B,=o(\) (34),

which may be calculated by the series (9).

10. The series (9), (30) will ultimately be geometric series
with ratios x^ w^, or x, {2x — 1)', respectively. Equating these
ratios, and taking the smaller root of the resulting quadratic, we
get a; = J. Hence if we use the series (9) for the calculation of
a [x) from a: = to a; = J, and (30) for the calculation of <t (x) from
x = \ to x = ^, we shall have to calculate series which are ulti-
mately geometric series with ratios ranging from to ^.

Suppose that we wish to calculate a (x) or z for values of x
increasing by '02. The process of calculation will be as follows.
From the equation (2+/8)-4^ = )8 and the general formula (12),
calculate the coefficients A^, A^, A^,,,, as far as may be necessary.
From the series (9), or else from the series (9) combined with the
formula (15), calculate o- (J) or 5„, and then calculate 5^, i?,...
from equations (31), (32). Next calculate a-(x) from the series
(9) for the values -02, '04,... -26 of ar, and F,{w) from (30) for the
values *04, '08..., '44 of w, and lastly {x — x^f^<f>{x) for the values
•52, '54..., '98 of X. Then we have o-(a;) calculated directly from
a? = to ic = '26 ; equations (33) will give o- {x) from x = '28 to
a? = '72, and lastly the equation o- (a*) = o- (1 - a*) + (a: — x^f^ (f) (x)
will give a {x) from a? = '74 to a: = 1.

11. The equation (21) will enable us to express in finite terras
the vertical velocity of the body at the centre of the bridge. For
according to the notation of Art. 2, the horizontal and vertical co-
ordinates of the body are respectively 2ca; and IG/Sy, and we have
also d . 2cx/dt = F, whence, if v be the vertical velocity, we get

_dA6St/dx_SSV .

dx at

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 191

But (21) gives/' (J) = i ^' (i), whence if Ve be the value of v at the
centre, we get from (25) or (28)

t;- = ^— , or = (3o),

* c cos nr c (e^' + e"'*') ^ ^

according as ^ < > J.

In the extreme cases in which Fis infinitely great and infinitely
small respectively, it is evident that Vc must vanish, and therefore
for some intermediate value of V, Vc must be a maximum. Since
Voe:j3~^ when the same body is made to traverse the same bridge
with different velocities, v<, will be a maximum when p or 5 is a
minimum, where

p = 2y3'* cos rTT, g = ^"* (e^' + e"'*').

Putting for cos rir its expression in a continued product, and
replacing r by its expression (23) in terms of 13, we get

whence

,=s.-.(i-4i-^(.-L-«^

dlogp 1 1 1

(36).

The same expression would have been obtained for dlogq/d^.
Call the second member of equation (3G) i^(yS), and let —N, P be
the negative and positive parts respectively of F (fi). When /8 = 0,

-A/'= 00 , and P = = — o +:r^j ... = 1, and therefore F{0) is nega-

tive. When ff becomes infinite, the ratio of P to iV becomes
infinite, and therefore F {jS) is positive when yS is sufficiently large ;
and F (/8) alters continuously with 13. Hence the equation -F (/S) =
must have at least one positive root. But it cannot have more
than one; for the rates of proportionate decrease of the quantities
N, P, or - 1/N, dN/dff, - 1/P . dP/d/3, are respectively

1 (1.24-iSr + (2. 34-/8)"'+...
$' (1.2 + /3)-»4 (2.3+^r + ...'

and the several terms of the denominator of the second of these
expressions are equal to those of the numerator multiplied by
1 . 2 +/8. 2 . 3 + )8,... respectively, and tlierefore the denominator is

192 DISCUSSION OF A DIFFERENTIAL EQUATION

equal to the numerator multiplied by a quantity greater than
2 + A and therefore greater than yS; so that the value of the
expression is less than 1/yS. Hence for a given infinitely small
increment of fi the change — dN in N bears to -W a greater ratio
than — dP bears to P, so that when N is greater than or equal to
P it is decreasing more rapidly than P, and therefore after having
once become equal to P it must remain always less than P. Hence
Vc admits of but one maximum or minimum value, and this must
evidently be a maximum.

When ^ = i, -^=2, and P< t— + 5— « + ... or < 1, and there-

fore F {P) has the same sign as when fi is indefinitely small.
Hence it is 9 and not p which becomes a minimum. Equating
dq/dfi to zero, employing (27), and putting 2*7rp = log, f, we find

^^^^=log.r+9r'(log.C)-.

The real positive root of this equation will be found by trial to
be 36-3 nearly, which gives p = '5717, /3 = i + p» = '5768. If V^ be
the velocity which gives Vc a maximum, v^ the maximum value of
Vc, Uihe velocity due to the height S, we get

^^==\/ 4^5 = 1^'^^^"* = ?^*! ^- ^^^^^"

F, = -4655 ^ U, v^ = -6288 U.

12. Conceive a weight W placed at rest on a point of the
bridge whose distance from the first extremity is to the whole
length as ^ to 1. The reaction at this extremity produced by W
will be equal to (1 — a:) TT, and the moment of this reaction about
a point of the bridge whose abscissa 2cx^ is less than 2cx will be
2c (1 — a?) ajjTT. This moment measures the tendency of the bridge
to break at the point considered, and it is evidently greatest when
x^ = a?, in which case it becomes 2c (1 — x)xW, Now, if the inertia
of the bridge be neglected, the pressure R produced by the moving
body will be proportional to (a? — aj')"*y, and the tendency to break
under the action of a weight equal to R placed at rest on the
bridge will be proportional to (1 — x) a: x (a? — a?)^ y, or to (x — a?") z.

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 193

Call this tendency T, and let T be so measured that it may be
equal to 1 when the moving body is placed at rest on the centre of
the bridge. Then T = C (^ - a:*) z, and 1 = (7 ( J - i), whence

The tendency to break is actually liable to be somewhat greater
than T, in consequence of the state of vibration into which the
bridge is thrown, in consequence of which the curvature is alter-
nately greater and less than the statical curvature due to the same
pressure applied at the same point. In considering the motion of
the body, the vibrations of the bridge were properly neglected, in
conformity with the supposition that the inertia of the bridge is
infinitely small compared with that of the body.

The quantities of which it will be most interesting to calculate
the numerical values are z, which expresses the ratio of the de-
pression of the moving body at any point to the statical depression,
T, the meaning of which has just been explained, and y\ the actual
depression. When z has been calculated in the way explained in
Art. 10, T will be obtained by multiplying by 4 {x —a?'), and then
yjS will be got by multiplying T by 4 (a; — a;*).

13. The following Table gives the values of these three quan-
tities for each of four values of /8, namely ^, ^, ^, and J, to which
correspond r = J, r = 0, p = i, p=l, respectively. In performing
the calculations I have retained five decimal places in calculating
the coeflBcients A^, A^y A^,,. and 5^,, i?j, B^... and four in calcula-
ting the series (9) and (30). In calculating ^ (x) I have used four-
figure logarithms, and I have retained three figures in the result.
The calculations have not been re-examined, except occasionally,
when an irregularity in the numbers indicated an error.

14. Let us first examine the progress of the numbers. For
the first two values of ^, z increases from a small positive quantity
up to 00 as a; increases from to 1. As far as the table goes, z is
decidedly greater for the second of the two values of ^ than for
the first. It is easily proved however that before x attains the
value 1, z becomes greater for the first value of )8 than for the
second. For if we suppose x very little less than 1,/(1 —x) will
be extremely small compared with ff> (x), or, in case <f} (x) contain a
sine, compared with the coefl&cient of the sine. Writing x^ for

s. n. 13

194

DISCUSSION OF A DIFFERENTIAL EQUATION

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RELATING TO THE BREAKING OF RAILWAY BRIDGES. 195

1 —a?, and retaining only the most import^int term inf(x), we get
from (21), (25), (26), and (28)

f(x) = . a: i"*", = ^rr x} log — ,

or

= —-^ -aTisinfolog-") (37)

according as /8 < J, /8 = J, or ^ > J ; and z will be obtained by
dividing /(a?) by x* nearly. Hence if i > ^9, > /8j > 0, « is ultimately
incomparably greater when /8 = )3j than when ^=^0^, and when
fi = 0, than when fi = l Since/(0) = ^, = ^9 (2 + 0)" = {2^' + IT\
/(O) increases with /3, so that /(a?) is at first larger when 0=^fi^
than when /3 =/3j, and afterwards smaller.

When /8 > i, « vanishes for a certain value of x, after which it
becomes negative, then vanishes again and becomes positive, and
80 on an infinite number of times. The same will be true of T.
If p be small, /(a:*) will not greatly differ, except when x is nearly
equal to 1, from what it would be if p were equal to zero, and
therefore f{x) will not vanish till x is nearly equal to 1. On the
other hand, if p be extremely large, which corresponds to a very
slow velocity, « will be sensibly equal to 1 except when x is nearly
equal to 1, so that in this case alsof{x) will not vanish till x is
nearly equal to 1. The table shews that when )3=i, f{x) first
vanishes between x = '98 and x=l, and when /3 = | between x = '94
and X = '96. The first value of x for which f{x) vanishes is pro-
bably never much less than 1, because as /3 increases from | the
denominator p (e^' -f e"**') in the expression for ^ {x) becomes
rapidly large.

15. Since when ^ > J, T vanishes when x = 0, and again for a
value of X less than 1, it must be a maximum for some inter-
mediate value. When /3 = ^ the table appears to indicate a maxi-
mum beyond x = '98. When /3 = f , the maximum value of T is
about 2*61, and occurs when x = '86 nearly. As /3 increases
indefinitely, the first maximum value of T approaches indefinitely
to 1, and the corresponding value of x to ^, Besides the first
maximum, there are an infinite number of alternately negative
and positive maxima ; but these do not correspond to the problem,
for a reason which will be considered presently.

196 DISCnSStOH OF A DIFFERENTIAL EQUATION

16. The following curves represent the trajectory of the body
for the four values of ^ contained in the preceding table. These
curves, it must be remembered, correspond
to the ideal limiting case in which the inertia
of the bridge is infinitely smalL

In this figure the right line AB repre-
sents the bridge in its position of equi- \ ^
librium, and at the same time represents
the trajectoiy of the body in the ideal limit-
ing case in which /9 = or 7" = oo . AeeeB
represents what may be called the equilibrium
trajectory, or the curve the body would de-
scribe if it moved along the bridge with an
infinitely small velocity. The trajectories
corresponding to the four values of ^ con-
tained in the above table are marked by
1,1,1,1; 2,2,2; 3,3,3; 4,4,4.4, respec-
tively. The dotted curve near B is meant
to represent the parabolic arc which the body
really describes after it rises above the hori-
zontal line AB*. G is the centre of the
right line AB: the curve AeeeB is symme-
trical with respect to an ordinate drawn
through C.

17. The inertia of the bridge being neg-
lected, the reaction of the bridge against the
body, as already observed, will be repre-
sented by Cif/(x — a:")', where C depends on
the length and stiffness of the bridge. Since
this expression becomes negative with y, the
preceding solution will not he appUcable
beyond the value of x for which y first
vanishes, unless we suppose the body held
down to the bridge by some contrivance. If
it be not so held, which in fact is the case,

. it will quit the bridge when y becomes nega-

" The dotted curve ought to have been drawn wholly ontaide the full carve.
The two ourrea touch each other at the point where they are cot by the line ACB,
as IB leprenented ia the figure.

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 197

live. More properly speaking, the bridge will follow the body, in
consequence of its inertia, for at least a certain distance above the
horizontal line AB, and will exert a positive pressure against the
body : but this pressure must be neglected for the sake of consist-
ency, in consequence of the simplification adopted in Art. 1, and
therefore the body may be considered to quit the bridge as soon as
it gets above the line AB. The preceding solution shews that
when /3 > J the body will inevitably leap before it gets to the end
of the bridge. The leap need not be high; and in fact it is
evident that it must be very small when is very large. In
consequence of the change of conditions, it is only the first maxi-
mum value of T which corresponds to the problem, as has been
already observed.

18. According to the preceding investigation, when /8 < i the
body does not leap, the tangent to its path at last becomes vertical,
and T becomes infinite. The occurrence of this infinite value
indicates the failure, in some respect, of the system of approxima-
tion adopted. Now the inertia of the bridge has been neglected
throughout; and, consequently, in the system of the bridge and
the moving body, that amount of labouring force which is requisite
to produce the vis viva of the bridge has been neglected. If f , rf
be the co-ordinates of any point of the bridge on the same scale on
which X, y represent those of the body, and f be less than x, it may
be proved on the supposition that the bridge may be regarded at
any instant as in equilibrium, that

f^t+r^J^'^Ti"^ ^^^'

When X becomes very nearly equal to 1, y varies ultimately as
(1 — a?)*^, and therefore t) contains terms involving (1 — x)"^, and
(drj/dxy, and consequently {drj/dty contains terms involving
(1 — a?)"*"^. Hence the expression for the vis viva neglected at
last becomes infinite ; and therefore however light the bridge may
be, the mode of approximation adopted ceases to be legitimate
before the body comes to the end of the bridge. The same result
would have been arrived at if /3 had been supposed equal to or
greater than J.

19. There is one practical result which seems to follow from
the very imperfect solution of the problem which is obtained when

198 DISCUSSION OF A DIFFERENTIAL EQUATION

the inertia of the bridge is neglected. Since this inertia is the
main cause which prevents the tendency to break from becoming
enormously great, it would seem that of two bridges of equal length
and equal strength, but unequal mass, the lighter would be the
more liable to break under the action of a heavy body moving
swiftly over it. The effect of the inertia may possibly be thought
worthy of experimental investigation.

20. The mass of a rail on a railroad must be so small com-
pared with that of an engine, or rather with a quarter of the mass
of an engine, if we suppose the engine to be a four-wheeled one,
and the weight to be equally distributed between the four wheels,
that the preceding investigation must be nearly applicable till the
wheel is very near the end of the rail on which it was moving,
except in so far as relates to regarding the wheel as a heavy point.
Consider the motion of the fore wheels, and for simplicity suppose
the hind wheels moving on a rigid horizontal plane. Then the
fore wheels can only ascend or descend by the turning of the whole
engine round the hind axle, or else the line of contact of the hind
wheels with the rails, which comes to nearly the same thing. Let
M be the mass of the whole engine, I the horizontal distance
between the fore and hind axles, h the horizontal distance of the
centre of gravity from the latter axle, k the radius of gyration
about the hind axle, x, y the coordinates of the centre of one of the
fore wheels, and let the rest of the notation be as in Art. 1. Then
to determine the motion of this wheel we shall have

whereas to determine the motion of a single particle whose mass is
I J/" we should have had

M^^M __ Gy

Now h must be nearly equal to \l, and h? must be a little greater
than JZ*, say equal to Ji', so that the two equatipns are very nearly
the same.

Hence, /8 being the quantity defined by equation (5), where S
denotes the central statical deflection duo to a weight \Mgy it
appears that the rail ought to be made so strong, or else so short,

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 199

as to render /3 a good deal larger than J. In practice, however, a
rail does not rest merely on the chairs, but is supported throughout
its whole length by ballast rammed underneath.

21. In the case of a long bridge, fi would probably be large in
practice. When ^ is so large that the coefficient ^irfp (€'*' + €'"'*'),
or ir^e"^ nearly, in <f> (x) may be neglected, the motion of the
body is sensibly symmetrical with respect to the centre of the
bridge, and consequently T, as well as y, is a maximum when a? = ^.
For this value of x we have 4 (a? — a:*) = 1, and therefore z = T=^y,
Putting (7, for the (i + 1)*** term of the series (9), so that (74 = -4^2"*,
we have for a: = J

y= C, + C, + C, + (39)

wnere ^o-g+yS' ^^ 6 + /3'

and generally,

^_ (t + l)(t4-2) _

^' - li + 1) (; + 2) + /3 ^^'-^ ^ ^'-«*'

whence Tis easily calculated. Thus for /3 = 5 we have ir^ €"'^ =*031
nearly, which is not large, and we get from the series (39) T= 1*27
nearly. For /3 = 10, the approximate value of the coefficient in
^ (x) is '0048, which is very small, and we get r= 1*14. In these
calculations the inerti^ of the bridge has been neglected, but the
effect of the inertia would probably be rather to diminish than to
increase the greatest value of T.

22. The inertia of a bridge such as one of those actually in
use must be considerable : the bridge and a carriage moving over
it form a dynamical system in which the inertia of all the parts
ought to be taken into account. Let it be required to construct
the same dynamical system on a different scale. For this purpose
it will be necessary to attend to the dimensions of the different
constants on which the unknown quantities of the problem depend,
with respect to each of the independent units involved in the
problem. Now if the thickness of the bridge be regarded as very
small compared with its length, and the moving body be regarded
as a heavy particle, the only constants which enter into the prob-
lem are My the mass of the body, M\ the mass of the bridge, 2c,
the length of the bridge, S, tlic central statical deflection, F, the

200 DISCUSSION OF A DIFFERENTIAL EQUAllON

horizontal velocity of the body, and g, the force of gravity. The
independent units employed in dynamics are three, the unit of
length, the unit of time, and the unit of density, or, which is equi-
valent, and which will be somewhat more convenient in the present
case, the unit of length, the unit of time, and the unit of mass.
The dimensions of the several constants if, M\ &c., with respect
to each of these units are given in the following table.

Unit of length. Unit of time. Unit of mass.

M and M\ 1

c and iS. 1

F. 1-10

g, 1-2

Now any result whatsoever concerning the problem will consist
of a relation between certain unknown quantities x\ a?" ... and the
six constants just written, a relation which may be expressed by

f{x\ x\ ...itf, M\ c, S, V,g) = (40).

But by the principle of homogeneity and by the preceding table
this equation must be of the form

{x_ x^ ^' ^ n

^Vxy (O'*" ^' c' eg;

where (x), {x') ..., denote any quantities made up of the six
constants in such a manner as to have with respect to each of the
independent units the same dimensions as x\ x' ..,, respectively.
Thus, if (40) be the equation which gives the maximum value T^
of T in terms of the six constants, we shall have but one unknown
quantity x\ where a?' = r, and we may take for {x),Mcg, or else
JK/'F*. If (40) be the equation to the trajectory of the body, we
shall have two unknown constants, x, x\ where x is the same as
in Art. 1, and x = y\ and we may take {x) = c, (x') = c. The
equation (41) shews that in order to keep to the same dynami
cal system, only on a .diflferent scale, we jnust alter the quantities
J/, 3r, &c. in such a manner that

M' X J/, 5« c, F* oc eg,

and consequently, since g is not a quantity which we can alter at
pleasure in our experiments, F must vary as sic, A small system
constructed with attention to the above variations forms an exact
(li/namical model uf a larger system with respect to which it may

= (41),

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 201

be desired to obtain certain results. It is not even necessary for
the truth of this statement that the thickness of the large bridge
be small in comparison with its length, provided that the same
proportionate thickness be preserved in the model.

To take a numerical example, suppose that we wished, by
means of a model bridge five feet long and weighing 100 ounces,
to investigate the greatest central deflection produced by an
engine weighing 20 tons, which passes with the successive velo-
cities of 80, 40, and 50 miles an hour over a bridge 50 feet long
weighing 100 tons, the central statical deflection produced by
the engine being one inch. We must give to our model carriage
a weight of 20 ounces, and make the small bridge of such a stiff-
ness that a weight of 20 ounces placed on the centre shall cause
a deflection of -j^th of an inch ; and then we must give to the
carriage the successive velocities of 3^10, 4^10, 5\/10, or 9*49,
12-65, 15-81 miles per hour, or 13-91, 1855, 2319 feet per second.
If we suppose the observed central deflections in the model to be
•12, '16, '18 of an inch, we may conclude that the central deflec-
tions in the large bridge corresponding to the velocities of 30, 40,
and 50 miles per hour would be 1*2, 1-6, and 1*8 inch.

Addition to the preceding Paper,

Since the above was written. Professor Willis has informed me
that the values of /3 are much larger in practice than those which
are contained in Table I., on which account it would be interesting
to calculate the numerical values of the functions for a few larger
values of )8. I have accordingly performed the calculations for
the values 3, 5, 8, 12, and 20. The results are contained in
Table II. In calculating z from a? = to a: = '5, I employed the
formula (12), with the assistance occasionally of (15). I worked
with four places of decimals, of which three only are retained.
The values of z for a; = '5, in which case the series are least con-
vergent, have been verified by the formula (42) given below : the
results agreed within two or three units in the fourth place of
decimals. The remaining values of z were calculated from the

202 .

DISCUSSION OF A DIFFERENTIAL EQUATION

>»:&5

N

o

09

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09

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II

II
00.

II

00.

l|
00.

00

II

II

CO

II

II
00.

OJ

00.

00

It

u

CO
II

'^coc<»»oe«iHoo»ooi>QOOOO»0'^co'«^
coc(i^Qt^C9Udcocooeoc&cpc4t«o^C4eo

0»-lC««^»OI>-0000000»aOI>»0"^CQ»-lO

C<»0>OO»CQ0t*l>Q0
Or-I0ie0k0t>009>0

Saot*ciooc<»o«oi>QOJop'^i-4"^*He<i'i|oo
rHcoao»oc^QOC<»e<iaoi>o»OQOo^i>0«p

0^-IO^CO»p^-®0*H»H»7^^HO»^-^pCQ^H1Hp

t-?OOt<»»OeOOOOCO'«^"^iHCOCO«OCOOQ>'>

SOO«?OeCi-400'^OOt-»-IQDI>QO*HOa®^
»H oi ep ip t> op ^ •H w CO CQ fH o ^- CO o O p
^^^^^^•••••, :^

O^opoc<^»«o•^59^cocoo6'«^co1-i^-^r^o^o>
pp7<«'^«5ap»H«"rr«^'Tj<epfHi>cp»Hoi

t»e<i»'5coe<iO'*««t-<opt»o»oooocoxiHO>
1HeO'^0^«qDO>pppppc»op^<-o^eplH

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Soc<»'«^cot»coaoo»iHcoQcocoxeoe«i»ot<»
e00pCQkOC0C0e0QP<HO$Q0t>900»t«-C0C0

iHep^^i>Qpojpp»H»Hpo»apb»4a"ri«cptH

UdiHCOQ'^^t«C9eOQ09>»ibb*C0tOe0C03bkO

iHcOTi<^t-ooo>piHr-ii-ii-(poat-»i3cp^eo

eoi>iH'«^-*ii<o«oc<i»0'^i>rf<o>o»oo«oe<it»
•^Qpcot^iH'.i«i>eJC5r^c<icogot^w3e<it-«»9co

•HW'^»pi>QpC5p»HOIepcpc5*HC>^*He9«

^^^^^^^ ' ' * *l^ +

""""""""" '7

iHOOC»OCCOpO>COQOOiHQO<HC09>COr«eO

_ eokot«Qo«HC)eQ^iAOto>oeo^<Hccr<*^'<^
o»o>9d9>odppppppppppp^cdOdO>

eOUdUdUd^010»<HC»<H

S^-l'^t»OC0*'5 00Ci»H
O)d»0»00000r-I

• •••••••••

§koe^oOrHco'^-^Qoeoc90>^eooiQO(i'««t
«0t*-iHUa0»'Nt*-i-l»500pOt*OQ««0Q00
op op Od o» o» p p 7^ *H fH ^ ^ fH fH p ^ CO a5 a3

"^»QOOeot*i-ioconr>i>"^iHot»cocooo-^<ooo
b*r«-t«QOooc»oorHC9e9cocO'<^coc40>eococsi
-^AAAA^AA^- • -i^

ppiH<Hcicimep^'^kpud^^b*b»a5dppp

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 203

expression for {x — a?)'^<f> {x). The values of T and y/8 were
deduced from those of ^, by merely multiplying twice in succes-
sion by 4a? (1 —a?). Professor Willis has laid down in curves the
numbers contained in the last five columns. In laying down
these curves several errors were detected in the latter half of the
Table, that is, from x = '55 to x '95. These errors were corrected
by re-examining the calculation ; so that I feel pretty confident
that the table as it now stands contains no errors of importance.

The form of the trajectory will be sufficiently perceived by
comparing this table with the curves represented in the figure.
As fi increases, the first point of intersection of the trajectory with
the equilibrium trajectory eee moves towards A. Since £: = 1 at
this point, we get from the part of the table beaded " z/* for the
abscissa of the point of intersection, by taking proportional parts,
•34, '29, '26, "24, and '22, corresponding to the respective values
3, 5, 8, 12, and 20 of /3. Beyond this point of intersection the
trajectory passes below the equilibrium trajectory, and remains
below it during the greater part of the remaining course. As /8
increases, the trajectory becomes more and more nearly sym-
metrical with respect to C : when /3 = 20 the deviation from sym-
metry may be considered insensible, except close to the extremities
A, By where however the depression itself is insensible. The
greatest depression of the body, as appears from the column which
gives y\ takes place a little beyond the centre; the point of
greatest depression approaches indefinitely to the centre as )3
increases. This greatest depression of the body must be carefully
distinguished from the greatest depression of the bridge, which
is decidedly larger, and occurs in a different place, and at a dif-
ferent time. The numbers in the columns headed " T'* shew that
r is a maximum for a value of x greater than that which renders
y' a maximum, as in fact immediately follows from a consideration
of the mode in which y' is derived from T, The first maximum
value of T, which according to what has been already remarked
is the only such value that we need attend to, is about 1*59 for
/8 = 3, 1-33 for /3 = 5, 1-19 for /3 = 8, 1-11 for /3 = 12, and 1*06 for
/3 = 20.

When )3 is equal to or greater than 8, the maximum value
of T occurs so nearly when a: = '5 that it will be sufficient to sup-
pose X = 5. The value of z, T, or y'/S for x—*5 may be readily

204 DISCUSSION OF A DIFFERENTIAL EQUATION

calculated by the method explained in Art. 21. I have also ob-
tained the following expression for this particular value

« = 2/3-4/y

1.2-h)3 2.3 + /3'*"3.4-h/3 •••] ^^^^•

When /3 is small, or only moderately large, the series (42)
appears more convenient for numerical calculation, at least with
the assistance of a table of reciprocals, than the series (39), but
when /3 is very large the latter is more convenient than the
former. In using the series (42), it will be best to sum the series
within brackets directly to a few terms, and then find the re-
mainder from the formula

The formula (42) was obtained from equation (20) by a trans-
formation of the definite integral. In the transformation of Art. 8,
the limits of 8 will be 1 and oo , and the definite integral on which
the result depends will be

r

as.

The formula (42) may be obtained by expanding the denomi-
nator, integrating, and expressing m in terms of 0.

In practice the values of /8 are very large, and it will be con-
venient to expand according to inverse powers of /3. This may be
easily effected by successive substitutions. Putting for shortness
a- — a;* = X, equation (4) becomes by a slight transformation

and we have for a first approximation y = X^, for a second

and so on. The result of the successive substitutions may be ex-
pressed as follows :

y = ^-/J-A-^X' + ^Z'^Z-^X'-&c (43),

where each term, taken positively, is derived from the preceding by
differentiating twice, and then multiplying by jB^^X*.

For such large values of /3, we need attend to nothing but the
value of 2? for ic = J, and this may be obtained from (43) by putting

RELATING TO THE BREAKING QV RAILWAY BRIDGES. 205

« =s J, after differentiation, and multiplying by 16. It will how-
ever be more convenient to replace a? by ^ (1 + w), which gives
cP/da^ = 4 . cP/dv^ ; -X" = ^^ IF, where F = (1 - w^. We thus get
from (43)

where we must put w^O after differentiation, if we wish to get
the value of « for a? = i^. This equation gives, on performing the
differentiations and multiplications, and then putting w=0,

i? = l+/9-^ + f^ + 13/3-»+ (44).

In practical cases this series may be reduced to 1 + )8"\ The
latter term is the same as would be got by taking into account the
centrifugal force, and substituting, in the small term involving that
force, the radius of curvature of the equilibrium trajectory for the
radius of curvature of the actual trajectory. The problem has
already been considered in this manner by others by whom it has
been attacked.

My attention has recently been directed by Professor Willis
to an article by Mr Cox On the Dynamical Deflection and Strain
of Railway Girders, which is printed in The Civil Engineer and
Architect's Journal for September, 1848. In this article the
subject is treated in a very original and striking manner. There
is, however, one conclusion at which Mr Cox has arrived which
is so directly opposed to the conclusions to which I have been led,
that I feel compelled to notice it. By reasoning founded on the
principle of vis viva, Mr Cox has arrived at the result that the
moving body cannot in any case produce a deflection greater than
double the central statical deflection, the elasticity of the bridge
being supposed perfect. But among the sources of labouring force
which can be employed in deflecting the bridge, Mr Cox has omitted
to consider the vis viva arising from the horizontal motion of the
body. It is possible to conceive beforehand that a portion of this
vis viva should be converted into labouring force, which is ex-
pended in deflecting the bridge. And this is, in fact, precisely
what takes place. During the first part of the motion, the hori-
zontal component of the reaction of the bridge against the body
impels the body forwards, and therefore increases the vis viva due
to the horizontal motion ; and the labouring force which produces
this increase being derived from the bridge, the bridge is less

206 DISCUSSION OF A DIFFERENTIAL EQUATION

deflected than it would have been had the horizontal velocity of the
body been unchanged. But during the latter part of the motion
the horizontal component of the reaction acts backwards, and a
portion of the vis viva due to the horizontal motion of the body is
continually converted into labouring force, which is stored up in the
bridge. Now, on account of the asymmetry of the motion, the
direction of the reaction is more inclined to the vertical when the
body is moving over the second half of the bridge than when it is
moving over the first half, and moreover the reaction itself is
greater, and therefore, on both accounts, more vis viva depending
upon the horizontal motion is destroyed in the latter portion of
the body's course than is generated in the former portion; and
therefore, on the whole, the bridge is more deflected than it would
have been had the horizontal velocity of the body remained un-
changed.

It is true that the change of horizontal velocity is small ; but
nevertheless, in this mode of treating the subject, it must be taken
into account. For, in applying to the problem the principle of
vis viva, we are concerned with the square of the vertical velocity,
and we must not omit any quantities which are comparable with
that square. Now the square of the absolute velocity of the body
is equal to the sum of the squares of the horizontal and vertical
velocities ; and the change in the square of the horizontal velocity
depends upon the product of the horizontal velocity and the
change of horizontal velocity; but this product is not small in
comparison with the square of the vertical velocity.

In Art. 22 1 have investigated the changes which we are allowed
by the general principle of homogeneous quantities to make in
the parts of a system consisting of an elastic bridge and a travel-
ling weight, without aflfecting the results, or altering anything but
the scale of the system. These changes are the most general that
we are at liberty to make by virtue merely of that general prin-
ciple, and without examining the particular equations which relate
to the particular problem here considered. But when we set down
these equations, we shall see that there are some further changes
which we may make without affecting our results, or at least
without ceasing to be able to infer the results which would be
obtained on one system from those actually obtained on another.

In an apparatus recently constructed by Professor Willis, which

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 207

will be described in detail in the report of the commission, to which
the reader has already been referred, the travelling weight moves
over a single central trial bar, and is attached to a horizontal arm
which is moveable, with as little friction as possible, about a
fulcrum carried by the carriage. In this form of the experiment,
the carriage serves merely to direct the weight, and moves on rails
quite independent of the trial bar. For the sake of greater gene-
rality I shall suppose the travelling weight, instead of being free,
to be attached in this manner to a carriage.

Let M be the mass of the weight, including the arm, h the
radius of gyration of the whole about the fulcrum, A the horizontal
distance of the centre of gravity from the fulcrum, I the horizontal
distance of the point of contact of the weight with the bridge, ir, y
the co-ordinates of that point at the time U f, ^ those of any
element of the bridge, JB the reaction of the bridge against the
weight, M' the mass of the bridge, B,\ R" the vertical pressures
of the bridge at its two extremities, diminished by the statical
pressures due to the weight of the bridge alone. Suppose, as
before, the deflection to be very small, and neglect its square.

By D'Alembert's principle the efifective moving forces reversed
will be in statical equilibrium with the impressed forces. Since
the weight of the bridge is in equilibrium with the statical pres-
sures at the extremities, these forces may be left out in the equa-
tions of equilibrium, and the only impressed forces we shall have
to consider will be the weight of the travelling body and the
reactions due to the motion. The mass of any element of the
bridge will be M'/2c . df very nearly ; the horizontal efifective force
of this element will be insensible, and the vertical efifective force
will be if 72c . d*rj/dt^ . df , and this force, being reversed, must be
supposed to act vertically upwards.

The curvature of the bridge being proportional to the moment
of the bending forces, let the reciprocal of the radius of curvature
be equal to K multiplied by that moment. Let A, B be the
extremities of the bridge, P the point of contact of the bridge
with the moving weight, Q any point of the bridge between A
and P. Then by considering the portion -4. Q of the bridge we get,
taking moments round Q,

208 DISCUSSION OF A DIFFERENTIAL EQUATION

71 being the same function of I' that 17 is of f. To determine K,
let £1 be the central statical deflection produced by the weight Mg
resting partly on the bridge and partly on the fulcrum, which is
equivalent to a weight hjl, Mg resting on the centre of the bridge.
In this ca^se we should have

d^" 21 ^'

Integrating this equation twice, and observing that df)/d^=:0
when ^ = c, and 77 = when f = 0, and, that S is the value of 7)
when f = c, we get

'-^^ w

Returning now to the bridge in its actual state, we get to de-
termine m, by taking moments about B,

. -K'.2c-i2(2c-a;) + |^J ^(2c-f)(if = (47).

Eliminating R between (45) and (47), putting for -K'its value
given by (46), and eliminating t by the equation dx/dt = F, we get

-i-|£.!(-')f«-^'[<-«/:Sr.r

firs*^"-*'"'*']} <*'>'

This equation applies to any point of the bridge between A
and P. To get the equation which applies to any point between
P and B, we should merely have to write 2c — f for f , 2c — a: for x.

If we suppose the fulcrum to be very nearly in the same hori-
zontal plane with the point of contact, the angle through which
the travelling weight turns will be y/l very nearly ; and we shall
have, to determine the motion of this weight,

M]^V'^=^Mghl^BP (49).

We have also the equations of condition,

1; = when a? = 0, for any value of ^ from to 2c ; \

rf = y when f = a?, for any value of x from to 2c ; >. . . (50).

1; = when f = 0or = 2c; y = and dy/dx = when x = 0)

BELATINO TO THE BREAKING OP RAILWAY BRIDGES. 209

Now the general equations (48), (or the equation answering to
it which applies to the portion PB of the bridge,) and (49), com-
bined with the equations of condition (50), whether we can manage
them or not, are sufficient for the complete determination of the
motion, it being understood that ^ and cfty/df vary continuously in
passing from AP to PB, so that there is no occasion formally to
set down the equations of condition which express this circum-
stance. Now the form of the equations shews that, being once
satisfied, they will continue to be satisfied provided 17 x y,
^ oc a; X c, and

These variations give, on eliminating the variation of B,

y oc 5, -p5§ « y^ , 3^v « p (51).

Although g is of course practically constant, it has been
retained in the variations because it may be conceived to vary,
and it is by no means essential to the success of the method that
it should be constant. The variations (51) shew that if we have
any two systems in which the ratio of Mk^ to MP is the same, and
we conceive the travelling weights to move over the two bridges
respectively, with velocities ranging from to x , the trajectories
described in the one case, and the deflections of the bridge, corre-
spond exactl}' to the trajectories and deflections in the other case,
so that to pass from one to the other, it will be suflScient to alter
all horizontal lines on the same scale as the length of the bridge,
and all vertical lines on the same scale as the central statical
deflection. The velocity in the one system which coiTesponds to a
given velocity in the other is determined by the second of the
variations (51).

We may pass at once to the case of a free weight by putting
A = A: = /, which gives

yocS, V'Socgc^, M oc M (52).

The second of these variations shews that corresponding veloci-
ties in the two systems are those which give the same value to the
constant fi. When £f x c we get V* x gc, which agrees with
Art 22.

S. II. 14

210 DISCUSSION OF A DIFFERENTIAL EQUATION

In consequence of some recent experiments of Professor Willis's,
from which it appeared that the deflection produced by a given
weight travelling over the trial bar with a given velocity was in
some cases increased by connecting a balanced lever with the
centre of the bar, so as to increase its inertia without increasing its
weight, while in other cases the deflection was diminished, I have
been induced to attempt an approximate solution of the problem,
taking into account the inertia of the bridge. I find that when we
replace each force acting on the bridge by a uniformly distributed
force of such an amount as to produce the same mean deflection
as would be produced by the actual force taken alone, which
evidently cannot occasion any very material error, and when we
moreover neglect the difference between the pressure exerted by
the travelling mass on the bridge and its weight, the equation
admits of integration in finite terms.

Let the notation be the same as in the investigation which
immediately precedes; only, for simplicity's sake, take the length
of the bridge for unity, and suppose the travelling weight a heavy
particle. It will be easy in the end to restore the general unit of
length if it should be desirable. It will be requisite in the first
place to investigate the relation between a force acting at a given
point of the bridge and the uniformly distributed force which
would produce the same mean deflection.

Let a force F act vertically downwards at a point of the bridge
whose abscissa is x, and let y be the deflection produced at that
point. Then, f , ^ being the co-ordinates of any point of the bridge,
we get from (38)

/,Vf=|f + 4(i^}.

To obtain faV^^» we have only got to write 1 — a? in place
of X. Adding together the results, and observing that, according
to a formula referred to in Art. 1, y = 16S . F/Mg . iu^ (1 — xY, we
obtain

!:vdS'-^lx(l-x)+a^{l-xr} (53);

and this integral expresses the mean deflection produced by the
force F, aiiice the length of the bridge is unity.

RELATING TO THE BREAKING OF RAILWAY BRIDGES. 211

Now suppose the bridge subject to the action of a uniformly
distributed force F\ In this case we should have

-^=^{ij?"f-/.*(f-r)Fdr}=iZi"(f-r).

iDtegratiog this equation twice, and observing that diy/df =
when f = i, and 17 = when f = 0, and that (46) gives, on putting
Z = A and c = i, K = 4i8S/Mg, we obtain

'7=^(f-2r+r) (54).

This equation gives for the mean deflection

/.Vf=|g^ (55);

and equating the mean deflections produced by the force F acting
at the point whose abscissa is x, and by the uniformly distributed
force F, we get F' = uF, where

u = Dx(l-x) + 5a^{l'-xy (56).

Putting fi for the mean deflection, expressing F in terms of fi,
and slightly modifying the form of the quantity within parentheses
in (54), we get for the equation to the bridge when at rest under
the action of any uniformly distributed force

'7=5/*if(i-f)+r(i-m (57).

If D be the central deflection, rj=D when f = J ; so that
D : /Lt :: 25 : 16.

Now suppose the bridge in motion, with the mass M travelling
over it, and let a?, y be the co-ordinates of M. As before, the
bridge would be in equilibrium under the action of the force
M (g " (Py/df) acting vertically downwards at the point whose
abscissa is x, and the system of forces such as M^d^.d^rj/df acting
vertically upwards at the several elements of the bridge. Accord-
ing to the hypothesis adopted, the former force may be replaced by
a uniformly distributed force the value of which will be obtained
by multiplying by w, and each force of the latter system may be
replaced by a uniformly distributed force obtained by multiplying
by u, where u is what u becomes when f is put for x. Hence if
F^ be the whole uniformly distributed force we have

^>=^(^-S)«-^7!S"''^^ ('«>•

212 DISCUSSION OF A DIFFERENTIAL EQUATION

Now according to our hypothesis the bridge must always have
the form which it would assume under the action of a uniformly
distributed force; and therefore, if /a be the mean deflection at the
time t, (57) will be the equation to the bridge at that instant.
Moreover, since the point (x, y) is a point in the bridge, we must
have 17 = y when f = a?, whence y = ^lu. We have also

We get from (55), F^ = 5MgfjL/28. Making these various sub-
stitutions in (58), and replacing d/dt by V . d/da?, we get for the
differential equation of motion

'^^-jf^-jfr..^»-l|j/T.g m

Since fi is comparable with S, the several terms of this equa-
tion are comparable with

respectively. If then V^8 be small compared with ^r, and likewise
M small compared with M\ we may neglect the third term, while
we retain the others. This term, it is to be observed, expresses
the difference between the pressure on the bridge and the weight
of the travelling mass. Since c = J, we have V*S/g = 1/16)8, which
will be small when /3 is large, or even moderately large. Hence
the conditions under which we are at liberty to neglect the differ-
ence between the pressure on the bridge and the weight of the
travelling mass are, first, that )8 be large, secondly, that the mass
of the travelling body be small compared with the mass of the
bridge. If iS be large, but M be comparable with M\ it is true
that the third term in (59) will be small compared with the lead-
ing terms; but then it will be comparable with the fourth, and the
approximation adopted in neglecting the third term alone would
be faulty, in this way, that of two small terms comparable with
each other, one would be retained while the other was neglected.
Hence, although the absolute error of our results would be but
small, it would be comparable with the difference between the
results actually obtained and those which would be obtained on
the supposition that the travelling mass moved with an infinitely
small velocity.

EELATINO TO THE BREAKING OF RAILWAY BRIDGES. 213

Neglecting the third term in equation (59), and putting for u
its value, we get

^ + q*,i^2q'S(x^2a?'\-x') (60),

where

63% _1008Af)8 ....

^^SlM'irS" SIM' ^^^^•

The linear equation (60) is easily integrated. Integrating, and
determining the arbitrary constants by the conditions that /t = 0,
and dfjk/dx = 0, when a: = 0, we get

94 )

+ -^(l-cosqx)[ (62);

and we have for the equation to the trajectory

y = 5fjL{x -Zaf" + x*) ==5fi{X + X") (63),

where as before Z = a? (1 — a?).

When F = 0, g = x, and we get from (62), (63), for the
approximate equation to the equilibrium trajectory,

y^lOS{X+Xy (64);

whereas the true equation is

y = 165Z" (65).

Since the forms of these equations are very diflferent, it will be
proper to verify the assertion that (64) is in fact an approximation
to (65). Since the curves represented by these equations are both
symmetrical with respect to the centre of the bridge, it will be
sufficient to consider values of x from to ^, to which correspond
values of X ranging from to J. Denoting the error of the
formula (64), that is the excess i)( the y in (64) over the y in (65),
by 8S, we have

8 = - 6Z* + 20Z» + lOZ*,

^^ = 4(_3 + 15X + 10Z«)Z^.

E(|uating dS/dx to zero, we get X= 0, a? = 0, S = 0, a maximum ;
Jf = *1787, a; = '233, 8 = — '067, nearly, a minimum; and x = i,
8 = — '023, nearly, a maximum. Hence tlie greatest error in the

214 DISCUSSION OF A DIFFERENTIAL EQUATION

approximate value of the ordinate of the equilibrium trajectory is
equal to about the one-fifteenth of S.

Putting /A =/ip + /ij, y = yo + yi» where fi^, y^ are the values of
/i, y for 5 = 00 , we have

fl2 /I 12\ 24 1

/Lt, = 2S|^a?(l-ar)-(^- + -jjsingx+-i^(l-cosgx)L

y,= 5^(l-^){l+a:(l-^)}M, (67).

The values of /Lt^ and y^ may be calculated from these formulae
for different values of q, and they are then to be added to the
values of /a^, y^, respectively, which have to be calculated once for
alL If instead of the mean deflection fi, we wish to employ the
central deflection D, we have only got to multiply the second sides
of equations (62), (66) by ^, and those of (63), (67) by if, and to
write D for /i. The following table contains the values of the
ratios of D and y to iST for ten different values of g, as well as for
the limiting value 9=00, which belongs to the equilibrium tra-
jectory.

The numerical results contained in Table III. are represented
graphically in figs. 2 and 3 of the woodcut on p. 216, where how-
ever some of the curves are left out, in order to prevent confusion
in the figures. In these figures the numbers written against the
several curves are the values of 2y/ir to which the curves respect-
ively belong, the symbol 00 being written against the equilibrium
curves. Fig. 2 represents the trajectory of the body for different
values of 9, and will be understood without further explanation.
In the curves of fig. 3, the ordinate represents the deflection of
the centre of the bridge when the moving body has travelled over
a distance represented by the abscissa. Fig. 1, which represents
the trajectories described when the mass of the bridge is neglected,
is here given for the sake of comparison with fig. 2. The num-
bers in fig. 1 refer to the values of fi. The equilibrium curve
represented in this figure is the true equilibrium trajectory ex-
pressed by equation (65), whereas the equilibrium curve repre-
sented in fig. 2 is the approximate equilibrium trajectory ex-
pressed by equation (64). In fig. 1, the body is represented as
flying off near the second extremity of the bridge, which is in fact
the case. The numerous small oscillations which would take
place if the body were held down to the bridge could not be

BBLATING TO THB BREAKIXQ OF BAILWAY BRIDGES. 215

TABLE III.

X

Values of — when — is equal to

O IT

r

1

2

3

4

5

6

8

10

12

16

1
00

•00
•06
•10
•15
•20
•26
•30
•35
•40
•46
•50
•55
•60
•65
•70
•75
•80
•85
•90
•95
100

•000
•004
•009
•017
-025
•041
•056
-070
•100
•134
•169
•213
•256
•306
•359
•419
•475
•533
•586
-646
•699

•000

•004

•018

•028

•052

•093

•144

•214

•300

•399

•516

•640

•776

•913

1-050

1-181

1-296

1-399

1-476

1-525

1-640

•000

•005

•022

•048

•099

•177

•282

•418

•578

•757

•947

1-139

1-321

1-482

1-609

1-691

1-717

1-681

1-588

1-402

1-158

•000

•006

•027

•075

•159

•285

•461

•650

•870

1-097

1-310

1-491

1-619

1-681

1-663

1-560

1-371

1-106

•776

•400

•000

•000
•007

•037

•108

•231

•406

•626

•871

1-115

1-332

1-492

1-574

1-562

1-454

1-257

•990

•677

•350

•037

--234

-•446

•000

•008

•053

•146

•309

•531

•787

1045

1-265

1-412

1-460

1-403

1-250

1027

•769

•517

•303

•149

•064

•025

•019

•000

•014

•081

•234

•469

•746

1^003

1-180

1-238

1-178

1-036

•870

-739

•682

•695

•746

•777

-733

•579

•321

•000

•000
•019
•117
•327
•607
•871
1-031
1052
•967
•859
•812
•860
•969
1-054
1031
•869
•604
•325
•117
•021
•001

•000

•025
•158
•412
•696
•884
•915
•845
•796
•856
1-004
1-127
1-115
•948
•718
•549
•499
•516
•477
•296
-•001

•000
•041
•239
•530
•707
•707
•689
•814
1^017
1-097
•991
•862
•872
•959
•924
•707
•472
•384
•885
•276
•000

•000
•166
•807
•449
•580
•696
•794
•873
•980
•966
•977
•965
•980
•878
•794
•696
-580
•449
•807
•156
•000

X

Values of 7$ when — is equal to

A

1

2

3

4

5

6

8

10

12

16'

00

•00
•05
•10
•16
•20
•25
•80
•35
•40
•45
•50

•60
•65
•70
•75
•80
•85
•90
•95
1-00

•000
•001
•003
•008
-015
•029
•045
•063
•096
-133
-169
-210
•244
-274
•292
•298
•282
•245
-184
-103
•000

000
001

004
013
031
056
•117
•191
•285
394
•516
•632
•739
•816
•854
•842
•770
•644
•463
•243
•000

•000
•001
•007
•022

•059

•126

•230

•374

-550

•748

•947

1-126

1-258

1-325

1-308

1-205

1-020

-774

•498

•224

•000

•000

•001

•008

•034

•095

•203

•366

-581

-828

1085

1-310

1-473

1-542

1-502

1-352

1-111

•814

•509

•244

-064

•000

•000

•001

•012

•050

•137

•290

•509

-778

1-062

1-316

1-492

1-65.)

1-487

1-300

1022

•705

•402

•161

•012

- -037

-000

•000

•001

•017

•067

•184

•378

•640

•934

1-205

1-395

1-460

1-387

1191

•917

•626

•369

•180

•069

-020

-004

•000

•000
•002
•025
•108
•279
•532
•814
1-054
1-178
1-164
1-036
•860
•704
•609
•565
•532
•462
•337
•182
•051
•000

•000
•003
•037
•150
•360
•621
•839
•940
•921
•849
•812
•850
•923
•942
•839
•619
•359
•149
•037
•003
•000

•000
•004
•050
•190
•414
•630
•744
•755
•759
•846
1-004
1-114
1-062
-848
•584
•391
•297
•237
•160
•047
•000

•000
•006
•076
•244
•420
•504
•560
•727
•969
1-084
•991
•852
•830
•857
•752
•488
•280
•178
•121
•044
•000

•000
•025
•096
•207
•844
•496
•646
•780
•886
•964
•977
•954
•886
•780
•646
•496
•844
•207
•096
•026
•000

216

DISCUSSION OF A DIFFEBENTIAL EQUATION

properly represented in the figure without using a much larger
scale. The reader is however requested to bear in mind the
existence of these oscillations, as indicated by the analysis, because,

Fig. I -^Forms of the trajectory when^is very large.

if the ratio of M to M' altered continuously from oo to 0, they
would probably pass continuously into the oscillations which are
80 conspicuous in the case of the larger values of q in fig. 2. Thus
the consideration of these insignificant oscillations which, strictly
speaking, belong to fig. 1, aids us in mentally filling up the gap
which corresponds to the cases in which the ratio of if to if ' is
neither very small nor very large.

As everything depends on the value of q, in the approximate
investigation in which the inertia of the bridge is taken into
account, it will be proper to consider further the meaning of this
constant. In the first place it is to be observed that although
M appears in equation (61), q is really independent of the mass
of the travelling body. For, when if alone varies, ff varies in-
versely as S, and 8 varies directly as M, so that q remains constant.
To get rid of the apparent dependence of q on M, let S^ be the
central statical deflection produced by a mass equal to that of the

RELATING TO THE BEEA&ING OF ItAILWAT BRIDQES. 21*7

bridge, and at the same time restore the general unit of length.
If x continue to denote the ratio of the abscissa of the body to the
length of the bridge, q will be numerical, and therefore, to restore
the general unit of length, it will be sufficient to take the general
expression (5) for fi. Let moreover t be the time the body takes
to travel over the bridge, so that 2c = Vr; then we get

«=3T^ (^^^-

If we suppose r expressed in seconds, and 8^ in inches, we must
put g = 32*2 X 12 = 386, nearly, and we get,

'-^ ■•• w-

Conceive the mass M removed ; suppose the bridge depressed
through a small space, and then left to itself. The equation of
motion will be got from (59) by putting Jlf=0, where M is not
divided by S, and replacing M/S by M'/8^, and V. djdx by didt
We thus get

and therefore, if P be the period of the motion, or twice the time
of oscillation £com rest to rest,

^ = 2.y|^>;5 = 2.J (70).

Hence the numbers 1, 2, 3, &c., written at the head of Table III.
and against the curves of figs. 2 and 3, represent the number of
quarter periods of oscillation of the bridge which elapse during
the passage of the body over it. This consideration will materially
assist us in understanding the nature of the motion. It should be
remarked too that q is increased by diminishing either the velocity
of the body or the inertia of the bridge.

In the trajectory 1, fig. 2, the ordinates are small because the
body passed over before there was time to produce much deflection
in the bridge, at least except towards the end of the body's course,
where even a large deflection of the bridge would produce only a
small deflection of the body. The corresponding deflection curve,
(curve 1, fig. 3,) shews that the bridge was depressed, and that its
deflection was rapidly increasing, when the body left it. When

$1S DISCUSSION OF A DIFFERENTIAL EQUATION

the body is made to move with velocities successively one-half and
one- third of the former velocity, more time is allowed for deflecting
the bridge, and the trajectories marked 2, 3, are described, in
which the ordinates are far larger than in that marked 1. The
deflections too, as appears from fig. 3, are much larger than before,
or at least much larger than any deflection which was produced in
the first case while the body remained on the bridge. It appears
from Table III., or from fig. 3, that the greatest deflection occurs
in the case of the third curve, nearly, and that it exceeds the
central statical deflection by about three-fourths of the whole.
When the velocity is considerably diminished, the bridge has time
to make several oscillations while the body is going over it. These
oscillations may be easily observed in fig. 3, and their effect on
the form of the trajectory, which may indeed be readily under-
stood from fig. 3, will be seen on referring to fig. 2.

When q is large, as is the case in practice, it will be sufficient
in equation (6G) to retain only the term which is divided by the
first power of q. With this simplification we get

D. 25 u, 25 . .^-v

•#=16^ =-8^ '"^«^ (^^)'

80 that the central deflection is liable to be alternately increased
and decreased by the fraction 25/8gr of the central statical deflec-
tion. By means of the expressions (61), (69), we get

25 .« /^' .„.•>'«.

l-"y»?="^^ ™.

It is to be remembered that in the latter of these expressions
the units of space and time are an inch and a second respectively.
Since the diflorence between the pressure on the bridge and weight
of the body is neglected in the investigation in which the inertia
of the bridge is considered, it is evident that the result will be
sensibly the same whether the bridge in its natural position be
straight, or be slightly raised towards the centre, or, as it is tech-
nically termed, cambered. The increase of deflection in the case
first investigated would be diminished by a camber.

In this paper the problem has been worked out, or worked out
approximately, only in the two extreme cases in which the mass of
the travelling body is infinitely great and infinitely small respect-
ively, compared with the mass of the bridge. The causes of the

BBLATINQ TO THE BH£A£INQ OF KAILWAT BRIDGES. 21 d

increase of deflection in these two extreme cases are quite distinct.
In the former case, the increase of deflection depends entirely on
the difference between the pressure on the bridge and the weight
of the body, and may be regarded as depending on the centrifugal
force. In the latter, the effect depends on the manner in which
the force, regarded as a function of the time, is applied to the
bridge. In practical cases the masses of the body and of the
bridge are generally comparable with each other, and the two
effects are mixed up in the actual result. Nevertheless, if we find
that each effect, taken separately, is insensible, or so small as to be
of no practical importance, we may conclude without much fear of
error that the actual effect is insignificant. Now we have seen
that if we take only the most important terms, the increase of
deflection is measured by the fractions l/)9 and 25/8^ of 8. It is
only when these fractions are both sqaall that we are at liberty to
neglect all but the most important terms, but in practical cases
they are actually small. The magnitude of these fractions will
enable us to judge of the amount of the actual effect.

To take a numerical example lying within practical limits, let
the span of a given bridge be 44 feet, and suppose a weight equal
to I of the weight of the bridge to cause a deflection of ^ inch.
These are nearly the circumstances of the Ewell bridge, mentioned
in the report of the commissioners. In this case, ^^ = | x '2 = *15;
and if the velocity be 44 feet in a second, or 30 miles an hour, we
have T = 1, and therefore from the second of the formulae (72),

I? = -0434, q = 721 = 45*9 x y .
Sq ^ 4

The travelling load being supposed to produce a deflection of
•2 inch, we have /3 = 127, l/)8=0079. Hence in this case the
deflection due to the inertia of the bridge is between 5 and 6 times
as great as that obtained by considering the bridge as infinitely
light, but in neither case is the deflection important. With a
velocity of 60 miles an hour the increase of deflection •0434/S would
be doubled.

In the case of one of the long tubes of the Britannia bridge fi
must be extremely large; but on account of the enormous mass of
the tube it might be feared that the effect of the inertia of the tube
itself would be of importance. To make a supposition every way

220 DISCUSSION OP A DIPFEKENTIAL EQUATION, &C.

disadvantageous, regard the tube as unconnected with the rest of
the structure, and suppose the weight of the whole train collected
at one point. The clear span of one of the great tubes is 460 feet,
and the weight of the tube 1400 tons. When the platform on
which the tube had been built was removed, the centre sank 10
inches, which was very nearly what had been calculated, so that
the bottom became very nearly straight, since, in anticipation of
the deflection which would be produced by the weight of the tube
itself, it had been originally built curved upwards. Since a uni-
formly distributed weight produces the same deflection as g ths of
the same weight placed at the centre, we have in this case
iSj = f X 10 = 16; and supposing the train to be going at the rate
of 30 miles an hour, we have t = 460 -5- 44 = lO'o, nearly. Hence
in this case 25/82 ~ '04fS, or J^ nearly, so that the increase of de-
flection due to the inertia of the bridge is unimportant.

In conclusion, it will be proper to state that this "Addition"
has been written on two or three difl*erent occasions, as the reader
will probably have perceived. It was not until a few days after
the reading of the paper itself that I perceived that the equation
(16) was integrable in finite terms, and consequently that the
variables were separable in (4). I was led to try whether this
might not be the case in consequence of a remarkable numerical
coincidence. This circumstance occasioned the complete remodel-
ling of the paper after the first six articles. I had previously
obtained for the calculation of z for values of x approaching 1, in
which case the series (9) becomes inconvenient, series proceeding
according to ascending powers of 1 - a?, and involving two arbitrary
constants. The determination of these constants, which at first
appeared to require the numerical calculation of five series, had
been made to depend on that of three only, which were ultimately
geometric series with a ratio equal to |.

The fact of the integrability of equation (4) in the form given
in Art 7, to which I had myself been led from the circumstance
above mentioned, has since been communicated to me by Mr
Cooper, Fellow of St John's College, through Mr Adams, and by
Professors Maliusten and A. F. Svanberg of IJpsala through Pro-
fessor Thomson; and I take this opportunity of thanking these
mathematicians for the communication.

[From the Cambridge and Dublin Mathematical Journal, Vol. iv. p. 219

{November, 1849)].

Notes on Hydrodynamics.

IV. — On Waves.

The theory of waves has formed the subject of two profound
memoirs by MM. Poisson and Cauchy, in which some of the
highest resources of analysis are employed, and the results deduced
from expressions of great complexity. This circumstance might
naturally lead to the notion that the subject of waves was unap-
proachable by one who was either unable or unwilling to grapple
with mathematical difficulties of a high order. The complexity,
however, of the memoirs alluded to arises from the nature of the
problem which the authors have thought fit to attack, which is the
determination of the motion of a mass of liquid of great depth
when a small portion of the surface has been slightly disturbed in
a given arbitrary manner. But after all it is not such problems
that possess the greatest interest. It is seldom possible to realize
in experiment the conditions assumed in theory respecting the
initial disturbance. Waves are usually produced either by some
sudden disturbing cause, which acts at a particular part of the
fluid in a manner too complicated for calculation, or by the wind
exciting the surface in a manner which cannot be strictly investi-
gated. What chiefly strikes our attention is the propagation of
waves already produced, no matter how : what we feel most desire
to investigate is the mechanism and the laws of such propagation.
But even here it is not every possible motion that may have been
excited that it is either easy or interesting to investigate; there
are two classes of waves which appear to be especially worthy of

attention.

222 XOTES ON HYDRODTyAMCS.

The first consists of those whose length is very great compared
with the depth of the fluid in which they are propagated. To this
class belongs the great tidal wave which, originally derived from
the oceanic oscillations produced by the disturbing forces of the
sun and moon, is propagated along our shores and up our channels.
To this class belongs likewise that sort of wave propagated along a
canal which Mr Russell has called a solitary ware. As an example
of this kind of wave may be mentioned the wave which, when a
canal boat is stopped, travels along the canal with a velocity
depending, not on the previous velocity of the boat, but merely
upon the form and depth of the canaL

The second class consists of those waves which Mr Russell has
called oscillatory. To this class belong the waves produced by the
action of wind on the surface of water, from the ripples on a pool
to the long swell of the Atlantic. By the waves of the sea which
are referred to this class must not be understood the surf which
breaks on shore, but the waves produced in the open sea, and
which, after the breeze that has produced them has subsided,
travel along without breaking or undergoing any material change
of form. The theory of oscillatory waves, or at least of what may
be regarded as the tjrpe of oscillatory waves, is sufficiently simple,
although not quite so simple as the theory of long waves.

Theory of Long Waves.

Conceive a long wave to travel along a uniform canal. For the
sake of clear ideas, suppose the wave to consist entirely of an
elevation. Let k be the greatest height of the surface above the
plane of the surface of the fluid at a distance from the wave, where
the fluid is consequently sensibly at rest ; let X be the length of
the wave, measured suppose from the point where the wave first
becomes sensible to where it ceases to be sensible on the opposite
side of the ridge ; let 6 be the breadth, and h the depth of the
canal if it be rectangular, or quantities comparable with the
breadth and depth respectively if the canal be not rectangular.
Then the volume of fluid elevated will be comparable with hXJc
As the wave passes over a given particle, this volume (not how-
ever consisting of the same particles be it observed) will be trans-

ON WAVES. 223

ferred from the one side to the other of the particle in question.
Consequently if we suppose the horizontal motions of the particles
situated in the same vertical plane perpendicular to the length of
the canal to be the same, a supposition which cannot possibly give
the greatest horizontal motion too great, although previously to
investigation it might be supposed to give it too small, the hori-
zontal displacement of any particle will be comparable with bTJc/bh
or \k/h. Hence if X be very great compared with A, the horizontal
displacements and horizontal velocities will be very great compared
with the vertical displacements and vertical velocities. Hence we
may neglect the vertical effective force, and therefore regard the
fluid as in equilibrium, so far as vertical forces are concerned, so
that the pressure at any depth 8 below the actual surface will be
gpS, g being the force of gravity, and p the density of the fluid, the
atmospheric pressure being omitted. It is this circumstance that
makes the theory of long waves so extremely simple. If the canal
be not rectangular, there will be a slight horizontal motion in a
direction perpendicular to the length of the canal ; but the corre-
sponding effective force may be neglected for the same reason as
the vertical effective force, at least if the breadth of the canal be
not very great compared with its depth, which is supposed to be
the case; and therefore the fluid contained between any two
infinitely close vertical planes drawn perpendicular to the length of
the canal may be considered to be in equilibrium, except in so far
as motion in the direction of the length of the canal is concerned.
It need hardly be remarked that the investigation which applies
to a rectangular canal will apply to an extended sheet of standing
fluid, provided the motion be in two dimensions.

Let X be measured horizontally in the direction of the length
of the canal ; and at the time t draw two planes perpendicular to
the axis of x, and passing through points whose abscissae are a?'
and x' + daf. Then if 17 be the elevation of the surface at
any point of the horizontal line in which it is cut by the first
plane, ri + dr}ldx' . dx will be the elevation of the surface where
it is cut by the second plane. Draw a right line parallel to the
axis of X, and cutting the planes in the points P, P'. Then if
S be the depth of the line PP' below the surface of the fluid
in equilibrium, the pressures at P, P' will be gp (S + ri) and
gp (8 + 17 + drj/dx . dx) respectively ; and therefore the difference

224 NOTES ON HYDRODYNAMICS.

of pressures will be gp dr^lda!. daf. About the line PP* describe
an infinitely thin cylindrical surface, with its generating lines per-
pendicular to the planes, and let k be the area which it cuts from
either plane ; and consider the motion of fluid which is bounded
by the cylindrical surface and the two planes. The difference of
the pressures on the two ends is ultimately gpK dr}/dx\ da! , and the
mass being pK dx\ the accelerating force is g drildx\ Hence the
effective force is the same for all particles situated in the same
vertical plane perpendicular to the axis of x ; and since the parti-
cles are supposed to have no sensible motion before the wave
reaches them, it follows that the particles once in a vertical plane
perpendicular to the length of the canal remain in such a vertical
plane throughout the motion.

Let X be the abscissa of any plane of particles in its position of
equilibrium, a; -f f the common abscissa of the same set of particles
at the time t, so that f and 17 are functions of x and t Then
equating the effective to the impressed accelerating force, we get

de ^ dx' ^^^'

and we have a?' =a? + f (2).

Thus far the canal has been supposed to be not necessarily
rectangular, nor even uniform, provided that its form and dimen-
sions change very slowly, nor has the motion been supposed to be
necessarily very small If we adopt the latter supposition, and
neglect the squares of small quantities, we shall get from (1)
and (2)

de—^di <^^-

It remains to form the equation of continuity. Suppose the
canal to be uniform and rectangular, and let b be its breadth and h
its depth. Consider the portion of fluid contained between two
vertical planes whose abscissae in the position of equilibrium are x
and X -h dx. The volume of this portion is expressed by bh dx. At
the time t the abscissae of the bounding planes of particles are
a? + f and a? + f + (1 + d^/dx) dx ; the depth of the fluid contained
between these planes is A + ^ ; and therefore the expression for
the volume is b (h + 17) (1 + d^/dx) dx. Equating the two expres-

ON WAVES. 225

siona for the volume, dividing by bdx, and neglecting the product
of the two small quantities, we get

''i+''=« w-

Eliminating f between (3) and (4), we get

-dt^=^^dx* <^')-

The complete integral of this equation is

V=f{^-^/(gh)t] + F{x■^^{gh)t} (6),

where/, F denote two arbitrary functions. This integral evidently
represents two waves travelling, one in the positive, and the other
in the negative direction, with a velocity equal to i/(gh), or to
that acquired by a heavy body in falling through a space equal
to half the depth of the fluid. It may be remarked that the
velocity of propagation is independent of the density of the
fluid

It is needless to consider the determination of the arbitrary
functions /, F by means of the initial values of r) and drj/dtt sup-
posed to be given, or the reflection of a wave when the canal is
stopped by a vertical barrier, since these investigations are pre-
cisely the same as in the case of sound, or in that of a vibrating

* This equation is in fact a second integral of the ordinary equation of con-
tinuity, corrocted so as to suit the particular case of motion which is under con*
sideration. For motion in two dimensions the latter equation is

du dv ^ / \

dx + dy=° (") =

and denoting by rf the vertical displacement of any particle, we have

d^ dri'

''-dr ''-It'

Substituting in (a), and integrating with respect to f, we get

S-|'=^(''^) W'

^ (x, y) denoting an arbitrary fimotion of x, y, that is, a quantity which may vary
from one particle to another, but is independent of the time. To determine ^ we
must observe that when any particle is not involved in the wave V=0, and | does
not vary in passing from one particle to another, and therefore ^(x, y)=0, Inte>
grating equation (b) with respect to y from ^=0 to y = /i + i7, observing that { is
independent of y, and that the limits of rf are and 17, and neglecting 17 d^Jdx^
which is a small quantity of the second order, we get the equation in the
text.

S. II. 1^

•

22G NOTES oS HYDK0J)YNAM1CS.

string. The only thing peculiar to the present problem consists in
the determination of the motion of the individual particles.

It is evident that the particles move in vertical planes parallel
to the length of the canal. Consider an elementary column of
fluid contained between two such planes infinitely close to each
other, and two vertical planes, also infinitely close to each other,
perpendicular to the length of the canal. By what has been
already shewn, this column of fluid will remain throughout the
motion a vertical column on a rectangular base ; and since there
can be no vertical motion at the bottom of the canal, it is evident
that the vertical displacements of the several particles in the
column will be proportional to their heights above the base. Hence
it will be sufficient to determine the motion of a particle at the
surface ; when the motion of a particle at a given depth will be
found by diminishing in a given ratio the vertical displacement of
the superficial particle immediately above it, without altering the
horizontal displacement.

The motion of a particle at the surface is defined by the values
of 7) and f . The former is given by (G), where the fimctions /, F
arc now supposed known, and the latter will be obtained from (4)
by integration. Consider the case in which a single wave con-
sisting of an elevation is travelling in the positive direction ; lot
X be the length of the wave, and suppose the origin taken at the
posterior extremity of the wave in the position it occupies when
^ = 0: then we may suppress the second function in (6), and
we shall have /(a?) = from a;= — x toa? = 0, and from ir = X to
a? = + 00 , SLndf(x) will be positive from a? = to a? = X. Let

c = VW (7),

80 that c is the velocity of propagation, and let the position of
equilibrium of a particle be considered to be that which it occu-
pies before the wave reaches it, so that f vanishes for a? = + oo .
Then we have from (4) and (6)

^=-lj\du^ = jJjXx^ct)<Lv (8).

Consider a particle situated in front of the wave when t = 0,
80 that a?>X. Since /(a?)=sO when x>\, we shall have
/(a? — c^) = 0, until d = a? — X. Consequently from (6) and (8)
there will be no motion until t^^x-^X/c, when the motion will
commence. Suppose now that a very small portion only of the

ON WAVES. 227

wave, of length 8, haa passed over the particle considered. Then
x — ct^X — s; and we have from (6) and (8)

V =/(>. - s), ^=1 j'_f{\ -s)ds = l j'fiX -s)ds:

for since f(x) vanishes when x>\, we may replace the limits

— X and 8 hy and «. Since I /(X — «)cJ» is equal to s mul-

tiplied by the mean value of/(X — «) from to 5, and this mean
value is comparable with /(X — s), it follows that f is at first very
small compared with 17. Hence the particle begins to move verti-
cally; and since rf is positive the motion takes place upwards.
As the wave advances, f becomes sensible, and goes on increasing
positively. Hence the particle moves forwai-ds as well as upwards.
When the ridge of the waves reaches the particle, 17 is a maxi-
mum ; the upward motion ceases, but it follows from (8) that f is
then increasing most rapidly, so that the horizontal velocity is
a maximum^ As the wave still proceeds, 17 begins to decrease,
and f to increase less rapidly. Hence the particle begins to
descend, and at the same time its onward velocity is checked.
As the wave leaves the particle, it may be shewn just as before
that the final motion takes place vertically downwards. When the
wave has passed, 17 = 0, so that the particle is at the same height
from the bottom as at first ; but f is a positive constant, equal to

jj f{x) dx, or to ^ j bf{x) clx,

that is, to the volume elevated divided by the area of the section
of the canal. Hence the particle is finally deposited in advance of
its initial position by the space just named.

If the wave consists of a single depression, instead of a single
elevation, everything is the same as before, except that the parti-
cle is depressed and then raised to its original height, in place of
being first raised and then depressed, and that it is moved back-
wards, or in a direction contrary to that of propagation, instead of
being moved forwards.

These results of theory with reference to the motions of the in-
dividual particles may be compared with Mr Russeirs experiments
described at page 342 of his second report on waves*.

* Beport of the 14th meeting of the British Association. Mr Bnssell's fixst
report is contained in the Beport of the 7th meeting.

228 NOTES ON HYDRODYNAMICS.

In the preceding investigation the canal has been supposed
rectangidar. A very trifling modification, however, of the pre-
ceding process will enable us to find the velocity of propagation in
a uniform canal, the section of which is of any arbitrary contour.
In fact, the dynamical equation (3) will remain the same as before ;
the equation of continuity alone will have to be altered. Let A be
the area of a section of the canal, b the breadth at the surface of
the fluid ; and consider the mass of fluid contained between two
vertical planes whose abscissae in the position of equilibrium are
X and x-{- dx, and which therefore has for its volume Adx, At the
time t, the distance between the bounding planes of particles is
(1 + d^.Jx) dx, and the area of a section of the fluid is A-{-brf
nearly, so that the volume is

(^ + ^)(l+2)rf^, or (A + bT, + A^)dx
nearly. Equating the two expressions for the volume, we get

Comparing this ei [nation with (4), we see that it is only
necessary to write A/b for h ; so that if c be the velocity of
propagation,

'f) (»)•

-ye

The formula (9) of course includes (7) as a particular case.
The latter was given long ago by Lagrange* : the more compre-
hensive formula (9) was first given by Prof Kelland"f-, though at
the same time or rather earlier it was discovered independently

* Berlin Memointj 178(>, p. 192. In this memoir Lagrange has obtained the
velocity of ])ropagation by very simple reasoning. Laplace had a little earlier (Mim,
ile VAcadimie for 1776, p. 512) given the exprestuon (see equation (29) of this note)
for the velocity of pro])agation of oscillatory waves, which when h is very small
compared uith X reduces itself to Lagrange's formula, but had made an unwarrant-
able cxteuHion of the application of the formula. In the Micanique Analytique
Lagrange has obtained analytically the expression (7) for the velocity of propagation
when the depth is smaU, whether the motion take place in two or three dimensions,
by assuming the result of an investigation relating to sound.

For a full account of the various theoretical investigations in the theory of
waves, which had been made at the date of publication, as well as for a number of
interesting experiments, the reader is referred to a work by the brothers Weber,
entitled WelUrUehrt auf ExperimenU gegrUndet, Leipzig, 1825.

f Transactiom of the Royal Society of Edinburgh^ Vol. xrr. pp. 524, 580.

ON WAVES. 229

by Green*, in the particular ease of a triangular canal. These
formulae agree very well with experiment when the height of the
waves is small, which has been supposed to be the case in the
previous investigation, as may be seen from Mr Russell's reports. A
table containing a comparison of theory and experiment in the
case of a triangular canal is given in Green's paper. In this table
the mean error is only about l/60th of the whole velocity.

As the object of this note is merely to give the simplest cases
of wave motion, the reader is referred to Mr Airy's treatise on tides
and waves for the effect produced by a slow variation in the dimen-
sions of the canal on the length and height of the wavef, as well
as for the effect of the finite height of the wave on the velocity of
propagation. With respect to the latter subject, however, it must
be observed that in the case of a solitary wave artificially excited
in a canal it does not appear to be suflScient to regard the wave as
infinitely long when we are investigating the correction for the
height; it appears to be necessary to take account of the finite
length, as well as finite height of the wave.

Theory of Oscillatory Waves.

In the preceding investigation, the general equations of hydro-
djmamics have not been employed, but the results have been
obtained by referring directly to first principles. It will now be
convenient to employ the general equations. The problem which
it is here proposed to consider is the following.

The surface of a mass of fluid of great depth is agitated by a
series of waves, which are such that the motion takes place in two
dimensions. The motion is supposed to be small, and the squares
of small quantities are to be neglected. The motion of each
particle being periodic, and expressed, so far as the time is con-
cerned, by a circular function of given period, it is required to
determine all the circumstance of the motion of the fluid. The
case in which the depth is finite and uniform will be considered
afterwards.

* Transactions of Vie Cambridge Philosophical Society , Vol. vii. p. 87.
t Encyclop<edia Metropolitan/i. Art. 260 of the treatise.

230 NOTES ON HYDRODYNAMICS.

It must be observed that the supposition of the periodicity of
the motion is not, like the hypothesis of parallel sections, a mere
arbitrary hypothesis introduced in addition to our general equa-
tions, which, whether we can manage them or not, are sufficient
for the complete determination of the motion in any given case.
On the contrary, it will be justified by the result, by enabling us
to satisfy all the necessary equations ; so that it is used merely to
define, and select from the general class of possible motions, that
particular kind of motion which we please to contemplate.

Let the vertical plane of motion be taken for the plane of xy.
Let X be measured horizontally, and y vertically upwards from the
mean surface of the fluid. If a, 6 be the co-ordinates of any parti-
cle in its mean position, the co-ordinates of the same particle at
the time t will be a+Jtidtf h-\-jvdt, respectively. Since the
squares of small quantities are omitted, it is immaterial whether
we conceive u and t; to be expressed in terms of a, 6, <, or in terms
of X, y, t; and, on the latter supposition, we may consider x and y
as constant in the integration with respect to L Since the varia-
ble terms in the expressions for the co-ordinates are supposed to
contain t under the form sin nt or cos nt, the same must be the cjise
with u and v. We may therefore assume

u = u^ sin nt 4- w, cos nt, v = r, sin nt 4- v^ cos nt,

where u^, u,, Vj, «, are functions of a: and y without t. Substituting
these values of u and t; in the general equations of motion, neglect-
ing the 8(inares of small quantities, and observing that the only
impressed force acting on the fluid is that of gravity, we get

■ : = — nu. cos nt 4- nu^ sm nt,
pax ' *

,i, f '""■

- _*^ =5 — /I —

pdy

g— m\ cos nt + ni\ sm nt

and the equation of continuity becomes

Eliminating p by differentiation from the two equations (10),
we get

\dy dxj ^^^ "^ \(iy

dt\\ , /du. dv^\ . ^ ,,^^

-^-jcoR«<-(^-^-)sm«« = (12);

ON WAVES. 231

and in order that this equation may be satisfied, we must have
separately

p-f*^0, P-P = (13).

ay ax ay dx

The first of these equations requires that u^dx + v^dy be an
exact differential dt^^y and is satisfied merely by this supposition.
Similarly the second requires that ujix + v^dy be an exact diflfer-
ential d^,. The functions ^j, ^, may be supposed not to contain
t, provided that in integrating equations (10) we express explicitly
an arbitrary function of t instead of an arbitrary constant. In
order to satisfy (11) we must equate separately to zero the coeflfi-
cients of sin w^ and cos ni. Expressing Wj, Vj, u^, v, in terms of
^j, 0^ in the resulting equations, we get

#-f-» <")■

with a similar equation for ^^ Integrating the value of dp
given by (10), we get

^=z= — (jry — n(^jCosni + w^, sin nt'\--^{t) (15).

It remains to form the equation of condition which has to
be satisfied at the free surface. If we suppose the atmospheric
pressure not to be included in p, we shall have p = at the free
surface ; and we must have at the same time (Note II.)

I-S-I-" <•«•

The second term in this equation is of the second order, and
in the thinl we may put for dpjdy its approximate value —gp.
Consequently at the free surface, which is defined by the

equation

jry + n^jCOST?^ — Ti^^sin nt— -^ (f) = (17),

we must have

??«<^^ sin nt + n'<^g cos nt 4- ^' (<) - g (-^' sin nt 4- ^^ cos ntj = (18) :

and we have the further condition that the motion shall vanish
at an infinite depth. Since the value of y given by (17) is a
small (quantity of the first order, it will be sufficient after differen-
tiation to put y = in (18).

232 NOTES ox HYDRODYNAMICS.

Equations (18), (14), and the corresponding equation for <f>^
shew that the functions <f>^, ^, are independent of each other;
and (15), (17) shew that the pressure at any point, and the
ordinate of the free surface are composed of the sums of the parts
due to these two functions respectively. Consequently we may
temporarily suppress one of the functions <f)^, which may be easily
restored in the end by writing t + 7r/2n for f, and changing the
arbitrary constants.

Equation (14) may be satisfied in the most general way by
an infinite number of particular solutions of the form ^e**''"'''"'',
where any one of the three constants A, 7/1', m may be positive
or negative, real or imaginary, and m, m are connected by the
equation m'* + m' = 0.* Now m cannot be wholly real, nor partly
real and partly imaginary, since in that case the corresponding
particular solution would become infinite either for a? = — 00 or
for a? = -f 00 , whereas the fluid is supposed to extend indefinitely
in the direction of x, and the expressions for the velocity, &c.
must not become infinite for any point of space occupied by the
fluid. Hence m must be wholly imaginary, and therefore m
wholly real. Moreover 7/1 must be positive, since otherwise the
expression considered would become infinite for y = — 00 . The
equation connecting m and m gives m'= ± 77iV(~l). Uniting
in one the two corresponding solutions with their different arbi-
trary constants, we have for the most general particular solution

which we are at liberty to take (^€'~^^"^> + ^6"*~^^"*^)€"*^ which
becomes, on replacing the imaginary exponentials by circular
functions, and changing the arbitrary constants,

{A sin 77?^ + B cos mx) e^^.

Hence we must have

<f)^ = 'E(A 8in7?ia?+ Bcosmx)€^^ (19),

the sign S denoting that we may take any number of positive
values of m with the corresponding values of A and B.

Substituting now in (18), supposed to be deprived of the
function <f>^, the value of <f)^ given by (19), and putting y = after
diflerentiation, we have

sin nt 2 (n* — rug) {A sin vix + B cos vix) + '^' (t) = 0.

• See Poisflon, Traits de M^caniqtiey Tom. 11. p. .347, or Thforie de la Chalfur,
Chap. V.

ON WAVES. 233

Since no two terms such as A sin mx or B cos inx can destroy
each other, or unite with the term yjt' (<), we must have sepa-
rately y^ (t) = 0, and

v^'-mg = (20).

The former of these equations gives yjr (t) = k, where k is a
constant; but (17) shews that the mean value of the ordinate
y of the free surface is k/g, inasmuch as ^, and ^, consist of
circular functions so far as a? is concerned, and therefore we must
have A:— 0, since we have supposed the origin of co-ordinates to
be situated in the mean surface of the fluid. The latter equation
restricts (19) to one particular value of m.

To obtain ^^ it will be suflBcient to take the expression for
(|>^ with new arbitrary constants. If we put <f> for

<f>^ sin nt + <f)^ cos nt, so that <f> = f{tid.T + vdy),

we see that if> consists of four terms, each consisting of the pro-
duct of an arbitrary constant, a sine or cosine of nt, a sine or
cosine of mx and of the same function e^^ of y. By replacing
the products of the circular functions by sines or cosines of sums
or differences, and changing the arbitrary constants, we shall get
four terms multiplied by arbitrary constants, and involving sines
and cosines of mx — nt and of mx-^-nt The terms involving
mx — nt will represent a disturbance travelling in the positive
direction, and those involving ma; + ni a disturbance travelling in
the negative direction. If we wish to consider only the disturb-
ance which travels in the positive direction, we must suppress the
terms involving Tnx-hnt, and we shall then have got only two
terras left, involving respectively sin (mx — nt) and cob (mx — nt).
One of these terms, whichever we please, may be got rid of by
altering the origin of x ; and we may therefore take

<f>^A8\n{mx-nt)€'^v (21);

and <f> determines, by its partial diff*erential coefficients with
respect to x and y, the horizontal and vertical components of the
velocity at any point. We have from (21), and the definitions of

<f>^ = — A cos mr . e*»* ', <f>^ = A sin mx . e^.

234 NOTES ON HYDRODYNAMICS.

Substituting in (15) and (17), putting y^{t) = 0, and replacing
y by in the second and third terms of (17), we get

^ = g{^y) + nA cos (7nx-7it)€^^ (22),

r

which gives the pressure at any point, and

y = — cos {mx — nt) (23)*,

if

which gives the equation to the free surface at any instant.

If X be the length of a wave, T its period, c the velocity of
propagation, we have m = 27r/\, n = 27r/7', n = cm; and therefore
from (20)

= y ©

(24).

Hence the velocity of propagation varies directly, and the period
of the wave inversely, as the scpiare root of the wave's length.
Equation (23) shews that a section of the surface at any instant
is the curve of sines.

It may be remarked that in consequence of the form of (f)
equation (18) is satisfied, not merely for y = 0, but for any value
of y; and therefore (16) is satisfied, not merely at the free surface,
but throughout the mass. Hence the pressure experienced by a
given particle is constant throughout the motion. This is not true
when the depth is finite, as may be seen from the value of ^
adapted to that case, which will be given presently; but it may bo
shewn to be tnie when the depth is infinite, whether the motion
take place in two, or three dimensions, and whether it be regular
or irregular, provided it be small, and be such that vcIjc 4- vJi/ 4- wJz
is an exact diflferential.

It will be interesting to determine the motions of the indi-
vidual particles. Let a? + f, y -f i; be the co-ordinates of the par-
ticle whose mean position has for co-ordinates or, ;/. Then we have

dt~''~~dx' dt~'^'^d^''
and in the values of ?/, v we may take .r, y to denote the actual

* EqnatlonH (22), (23) mfty be Rot at once from the eqnationB

ON WAVES. 233

co-ordinates of any particle or their mean values indiflferently, on
account of the smallness of the motion. Hence we get from (21)
after differentiation and integration

f = sin {vix — lit) €"**, r)= - cos (jfix — w^)e"**' . . . (25).

Hence the particles describe circles about their mean places, with a
uniform angular motion. Since 17 is a maximum at the same time
with y in (23), and d^/dt is then positive, any particle is in its
highest position when the crest of the wave is passing over it, and
is then moving horizontally forwards, that is, in the direction of
propagation. Similarly any particle is in its lowest position when
the middle of the trough is passing over it, and it is then moving
horizontally backwards. The radius of the circle described is equal
to mil/n. €*"^ and it therefore decreases in geometric progression as
the depth of the particle considered increases in arithmetic. The
rate of decrease is such that at a depth equal to X the displace-
ment is to the displacement at tho surface as e"*"" to 1, or as 1 to
535 nearly.

If the depth of the fluid be finite, the preceding solution may
of course be applied without sensible error, provided e"^ be insensi-
ble for a negative value of ^ equal to the depth of the fluid. This
will be equally true whether the bottom be regular or irregular,
provided that in the latter case we consider the depth to be repre-
sented by the least actual depth.

Let us now suppose the depth of the fluid finite and uniform.
Let h be the mean depth of the fluid, that is, its depth as unaffected
by the waves. It will be convenient to mea.sure y from the bottom
rather than from the mean surface. Consequently w^e must put
y = A, instead of 7/ = 0, in the values of <f>^, <f>^, and their differential
coefficients, in (17) and (18). The only essential change in the
equations of condition of the problem is, that the condition that
the motion shall vanish at an infinite depth is replaced by the
condition that the fluid shall not penetrate into, or separate from
the bottom, a condition which is expressed by the equation

^^ = when ?/ = () (20).

Everything is the same as in the preceding investigation till
we come to the selection of a particular integral of (14). As before,

236 NOTES ON HYDRODYNAMICS.

y must appear in an exponential, and x under a circular function ;
but both exponentials must now be retained. Hence the only
particular solution which wc are at liberty to take is of the form

Ae"^ cos mx + Be"** sin iiix + Cfe"'*" cos mx + De"^ sin mx,

or, which is the same thing, the coefBciente only being altered,

(f -f €"'"*') {A cos mx 4- B sin mx)
+ (€*"" — 6""") ( C cos mx 4- D sin mx).

Now (26) must be satisfied by ^^ and ^^ sepai^ately. Substituting
then in this equation the value of 0, which is made up of an infi-
nite number of particular values of the above form, we see that we
must have for each value of m in particular (7= 0, D = 0; so that

^, = 2 (e"^ + €"*') (-4 cos mx + B sin mx).

Substituting in equation (18), in which ^, is supposed to be
suppressed, and y put equal to A after differentiation, we get

??•(€"* + c"*^) - 7n5r (f"* - e"^) = (27),

and '^' (/) = 0, which gives -^ {t) = k. The equation (17) shews
that this constant h must be equal to A, which is the mean value
of y at the surface. It is easy to prove that equation (27), in
which 771 is regarded as the unknown quantity, has one and but
one positive root. For, putting mh = fi, and denoting by v the
function of fi defined by the equation

1/ (€^ + e-**) = fi (e^ - 6"**) (28),

we get by taking logarithms and differentiating

V dijL~^ €^ - €-** "" €^ H-e"** '

Now the right-hand member of this equation is evidently positive
when fi is positive; and since v is also positive, as appears from
(28), it follows that dp/d/i is positive; and therefore fi and v in-
crease together. Now (28) shews that v passes from to oo as /a
passes from to oo , and therefore for one and but one positive
value of fi, V is equal to the given quantity f^Vi/ff, which proves the
theorem enunciated. Hence as l>efore the most general value of (f>
corresponds to two series of waves, of determinate length, which
are propagated, one in the positive, and the other in the negative

ON WAVES. 237

direction. If c be the velocity of propagation, we get from (27),
since n = cfn = c, 27r/\,

"" \^Zirl + €-*^^^] ^'^^^•

If we consider only the series which is propagated in the posi-
tive direction, we may take for the same reason as before

<l)=^A{€'^ + €^')Hm{nuc-nt) (30);

which gives

^^g {h- y) -i-nA {e""^ + €'"') cos {mx -7it) (31),

r

and for the equation to the free surface

g(y-h) = nA (€** + e-**) cos (nix - nt) (32).

Equations (21), (22), (23) may be got from (30), (31), (32) by
writing y-hh for y, ^e"*** for A, and then making h infinite.
When X is very small compared with A, the formula (29) reduces
itself to (24): when on the contrary X is very great it reduces it-
self to (7). It should be observed however that this mode of prov-
ing equation (7) for very long waves supposes a section of tbe
surface of the fluid to be the curve of sines, whereas the equation
has been already obtained independently of any such restriction.

The motion of the individual particles may be determined, just
as before, from (30;. We get

f = - — ^ (e-^ + e-ir) sin (nix - nt),

^ = ^^*^ (e-i' -€"«') cos (77W? - ?t^) (33).

Hence the particles describe elliptic orbits, the major axes of
wLich are horizontal, and the motion in the ellipses is the same
as in the case of a body describing an ellipse under the action of a
force tending to the centre. The ratio of the minor to the major
axis is that of 1 — c""**" to 1 + €■*"^ which diminishes from the
surface downwards, and vanishes at the bottom, where the ellipses
pass into right lines.

The ratio of the horizontal displacement at the depth A — y
to that at the surface is equal to the ratio of e"^ + €"**' to €** -f- c"***.
The ratio of the vertical displacements is that of c**' — e"^ to
€^ — c"*^. The former of these ratios is greater, and the latter

238 NOTES ON HYDRODYNAMICS.

less than that of e"*^*"*' to 1. Hence, for a given length of wave,
the horizontal displacements decrease less, and the vertical dis-
placements more rapidly from the surface downwards when the
depth of the fluid is finite, than when it is infinitely great.

In a paper " On the Theory of Oscillatory Waves* " I have
considered these waves as mathematically defined by the character
of uniform propagation in a mass of fluid others ise at rest, so that
the waves are such as could be propagated into a portion of fluid
which had no previous motion, or excited in such a portion by
means of forces applied to the surface. It follows from the latter
character, by virtue of the theorem proved in Note IV, that
udx -f vd^ is an exact diflferential. This definition is equally
applicable whether the motion be or be not very small ; but in the
present note I have supposed the species of wave considered to be
defined by the character of periodicity, which perhaps forms a
somewhat simpler definition when the motion is small. In the
paper just mentioned I have proceeded to a second approximation,
and in the particular case of an infinite depth to a third approxima-
tion. The most interesting result, perhaps, of the second approxi-
mation is, that the ridges are steeper and narrower than the
troughs, a character of these waves which must have struck every-
body who has been in the habit of watching the waves of the
sea, or even the ripples on a pool or canal. It appears also from
the second approximation that in addition to their oscillatory
motion the particles have a progressive motion in the direction of
propagation, which decreases rapidly from the surface downwards.
The factor expressing the rate of decrease in the case in which
the fluid is very deep is €"*"'*', y being the depth of the particle
considered below the surface. The velocity of propagation is
the same as to a first approximation, as might have been seen
a prion, because changing the sign of the coefficient denoted by
A in equations (21) and (30) comes to the same thing as shifting
the origin of x through a space equal to ^X, which does not alter
the physical circumstances of the motion ; so that the expression
for the velocity of propagation cannot contain any odd powers of
A. The third approximation in the case of an infinite depth gives
an increase in the velocity of propagation depending upon the
height of the waves. The velocity is found to be equal to

* Cambridge Philoiopliical TrantactioiUf Vol. vin. p. 441. [AnU, Vol. l p. 197.]

ON WAVES.

239

c^ (1 + 27rV/X'), Cq being the velocity given by (24), and a the
height of the waves above the mean surface, or rather the coeffi-
cient of the first term in the equation to the surface.

A comparison of theory and observation with regard to the
velocity of propagation of waves of this last sort may be seen at
pages 271 and 274 of Mr Russell's second report. The following
table gives a comparison between theory and experiment in the
case of some observations made by Capt. Stanley, R.N. The
observations were communicated to the British Association at its
late meeting at Swansea*.

In the following table

A is the length of a wave, in fathoms ;

B is the velocity of propagation deduced from the observations,
expressed in knots per hour ;

C is the velocity given by the formula (24), the observations
being no doubt made in deep water ;

D is the ditferencc between the numbers given in columns
B and C.

In calculating the numbers in table C, I have taken (/ = 32*2
feet, and expressed the velocity in knots of 1000 fathoms or GOOO
feetf.

.1

B

C

2>

55

27

24-7

2-3

43

24-5

21-8

2-7

50

24-0

23-5

0-5

35 to 40

22-1

20-4

1-7

33

22-1

191

30

57

26-2

251

11

35

22-0

19-7

2-3

The mean of the numbers in column D is 1*94, nearly, which
is about the one-eleventh of the mean of those in column C, The
quantity 1*94 appeal^ to be less than the most probable error of
any one observation, judging by the details of the experiments ;
but as all the errors lie in one direction, it is probable that the

* Beport for 1848, Part n. p. 88.

t I have taken a knot to be 1000 fathoms rather than 2040 yards, becaxise the
former yaloe appears to have been used in oalcolating the numbers in eolmnn B,

240 NOTES ON HYDRODYNAMICS.

formula (24) gives a velocity a little too small to agree with obser-
vations under the circumstances of the experiments. The height
of the waves from crest to trough is given in experiments No. 1 ,
2, 3, 6, 7, by numbers of feet ranging from 17 to 22. I have
calculated the theoretical correction for the velocity of propagation
depending upon the height of the waves, and found it to be '5 or
•6 of a knot, by which the numbers in column C ought to be
increased. But on the other hand, according to theory, the par-
ticles at the surface have a progressive motion of twice that
amount ; so that if the ship's velocity, as measured by the log-
line, were the velocity relatively to the surface of the water, her
velocity would be under-estimated to the amount of 1 or 1*2 knot,
which would have to be added to the numbers in column B, or
which is the same subtracted from those in column C, in order to
compare theory and experiment ; so that on the whole '5 or G
would have to be subtracted from the numbers in column G,
But on account of the depth to which the ship sinks in the sea,
and the rapid decrease of the factor e^*"' from the surface down-
wards, the correction 1 or 1*2 for the "heave of the sea*" would
be too great; and therefore, on the whole, the numbers in column
C may be allowed to stand. If the numbers given in Capt.
Stanley's column, headed " Speed of Ship " already contain some
such correction, the numbers in column C must be increased, and
therefore those in column D diminished, by '5 or '6.

It has been supposed in the theoretical investigation that
the surface of the fluid was subject to a uniform pressure. But in
the experiments the wind was blowing strong enough to propel
the ship at the rate of from 5 to 7*8 knots an hour. There is
nothing improbable in the supposition that the wind might have
slightly increased the velocity of propagation of the waves.

There is one other instance of wave motion which may be
noticed before we conclude. Suppose that two series of oscillatory
waves, of equal magnitude, are prof)agated in opposite directions.
The value of ^ which belongs to the compound motion will be

(e"" + e~^') [A cos {pix - nt) + A cos [mx + n^ + a)],

* I have been told by a naval friend that an allowanoe for the "heave of
the aea" is sometimes actually made. As well as I recollect, this allowance
might have been about 10 knots a day for waves of the magnitude of those here
oonsidered.

] (35).

ON WAVES. 241

the squares of small quantities being neglected, as throughout this
note. Siuce

cos (mx - tU) + cos {mx + nt + a) = 2 cos {mx + Jx) cos {nt + |a),

we get by writing ^A for A, and altering tho origins of x and t, so
as to get rid of a,

<ff = A (e"^ + €"^) cos mx . cos nt (34).

This is in fact one of the elementary forms already considered,
from which two series of progressive oscillatory waves were derived
by merely replacing products of sines and cosines by sums and
differences. Any one of these four elementary forms corresponds
to the same kind of motion as any other, since any two may be
derived from each other by merely altering the origins of x and t;
and therefore it will be sufficient to consider that which has
just been written. We get from (34)

t* =s — mA (€"*" + e"^') sin mx cos nt]
V = mA (e"** — e'"^) cos vnx cos nt

We have also for the equation to the free surface

y — h^ — (e"^ + €"^) cos mx ainnt (36).

Equations (35) shew that for an infinite series of planes for
which ma? = 0, = ± tt, = ± 27r, &c., i.e. a? = 0, = ± JX, = ± X, &c.,
there is no horizontal motion, whatever be the value of t ; and for
planes midway between these the motion is entirely horizontal
When ^ = 0, (36) shews that the surface is horizontal ; the parti-
cles are then moving with their greatest velocity. As t increases,
the surface becomes elevated {A being supposed positive) from
a? = to a? = ^X, and depressed from a? = JX to a? = JX, which suffi-
ciently defines the form of the whole, since the planes whose
equations are a? = 0, a? = JX, are planes of symmetry. When
nt = Jtt, the elevation or depression is the greatest ; the whole
fluid is then for an instant at rest, after which the direction of
motion of each particle is reversed. When n^ becomes equal to 7r,
the surface again becomes horizontal ; but the direction of each
particle's motion is just the reverse of what it was at first, the
magnitude of the velocity being the same. The previous motion
of the fluid is now repeated in a reverse direction, those por-
tions of tlie surface which were elevated becoming depressed, and
vice versd. When nt^^Zir, everything is the same as at first.

s. II. V^

242 NOTES ON HYDRODYNAMICS.

Equations (35) shew that each particle moves backwards and
forwards in a right line.

This sort of wave, or rather oscillation, may be seen formed
more or less perfectly when a series of progressive oscillatory waves
is incident perpendicularly on a vertical wall. By means of this
kind of wave the reader may if he pleases make experiments
for himself on the velocity of propagation of small oscillatory
waves, without trouble or expense. It will be suflBcient to pour
some water into a rectangular box, and, first allowing the water
to come to rest, to set it in motion by tilting the box, turning
it round one edge. The oscillations may be conveniently counted
by watching the bright spot on the wall or ceiling occasioned
by the light of the sun reflected from the surface of the water,
care being taken not to have the motion too great. The time
of oscillation from rest to rest is half the period of a wave, and
the length of the interior edge parallel to the plane of motion is
half the length of a wave; and therefore the velocity of propaga-
tion will be got by dividing the length of the edge by the time of
oscillation. This velocity is then to be compared with the for-
mula (29).

[From the Tran.^actions of the Cainbridge Philosophical Society j

Vol. IX. p. 1.]

T. On the Dynamical Theory of Diffraction.

[Read Novfimher 26, 1849.]

When light is incident on a small aperture in a screen, the
illumination at any point in front of the screen is determined, on
the undulatory theory, in the following manner. The incident
waves are conceived to be broken up on arriving at the aperture ;
each element of the aperture is considered as the centre of an
elementary disturbance, which diverges spherically in all direc-
tions, with an intensity which does not vary rapidly from one
direction to another in the neighbourhood of the normal to the
primary wave ; and the disturbance at any point is found by
taking the aggregate of the disturbances due to all the secondary
waves, the phase of vibration of each being retarded by a quantity
corresponding to the distance from its centre to the point where
the disturbance is sought. The square of the coefficient of vibra-
tion is then taken as a measure of the intensity of illumination.
Let us consider for a moment the hypotheses on which this pro-
cess rests. In the first place, it is no hypothesis that we may
conceive the waves broken up on arriving at the aperture : it is
a necessary consequence of the dynamical principle of the superpo-
sition of small motions ; and if this principle be inapplicable to
light, the undulatory theory is upset from its very foundations.
The mathematical resolution of a wave, or any portion of a wave,
into elementary disturbances must not be confounded with a phy-
sical breaking up of the wave, with which it has no more to do
than the division of a rod of variable density into differential

16—2

244- ON THE DYNAMICAL THEORY OF DIFFRACTION.

elements, for the purpose of finding its centre of gravity, has to do
with breaking the rod in pieces. It w a hypothesis that we may
find the disturbance in front of the aperture by merely taking the
aggregate of the disturbances due to all the secondary waves, each
secondary wave proceeding as if the screen were away ; in other
words, that the eflfect of the screen is merely to stop a ceilaiu
portion of the incident light. This hypothesis, exceedingly pro-
bable a priori^ when we are only concerned with points at no
great distance from the normal to the primary wave, is confirmed
by experiment, which shews that the same appearances are pre-
sented, with a given aperture, whatever be the nature of the screen
in which the aperture is pierced, whether, for example, it consist
of paper or of foil, whether a small aperture be divided by a hair
or by a wire of equal thickness. It is a hypothesis, again, that
the intensity in a secondary wave is nearly constant, at a given
distance from the centre, in dififerent directions very near the
normal to the primary wave ; but it seems to me almost impossible
to conceive a mechanical theory which would not lead to this
result. It is evident that the difference of phase of the various
secondary waves which agitate a given point must be determined
by the difference of their radii; and if it should afterwards be
found necessary to add a constant to all the phases the results will
not be at all affected. Lastly, good reasons may be assigned why
the intensity should be measured by the square of the coefficient
of vibration ; but it is not necessary here to enter into them.

In this way we are able to calculate the relative intensities at
different points of a diffraction pattern. It may be regarded as
established, that the coefficient of vibration in a secondary wave
varies, in a given direction, inversely as the radius, and conse-
quently, we are able to calculate the relative intensities at differ-
ent distances from the aperture. To complete this part of the
subject, it is requisite to know the absolute intensity. Now it has
been shewn that the absolute intensity will be obtained by taking
the reciprocal of the wave length for the quantity by which to
multiply the product of a differential element of the area of the
aperture, the reciprocal of the radius, and the circular function
expressing the phase. It appears at the same time that the phase
of vibration of each secondary wave must be accelerated by a
quarter of an undulation. In the investigations alluded to, it is
supposed that the law of disturbance in a secondary wave is the

ON THE DYNAMICAL THEORY OF DIFFBACTION. 245

same in all directions ; but this will not affect the result, provided
the solution be restricted to the neighbourhood of the normal to
the primary wave, to which indeed alone the reasoning is appli-
cable; and the solution so restricted is sufficient to meet all
ordinary cases of diffraction.

Now the object of the first part of the following paper is, to
determine, on purely dynamical principles, the law of disturbance
in a secondary wave, and that, not merely in the neighbourhood of
the normal to the primary wave, but in all directions. The oc-
currence of the reciprocal of the radius in the coefficient, the
acceleration of a quarter of an undulation, and the absolute value
of the coefficient in the neighbourhood of the normal to the
primary wave, will thus appear as particular results of the general
formula.

Before attacking the problem dynamically, it is of course
necessary to make some supposition respecting the nature of that
medium, or ether, the vibrations of which constitute light, accord-
ing to the theory of undulations. Now, if we adopt the theory of
transverse vibrations — and certainly, if the simplicity of a theory
which conducts us through a multitude of curious and complicated
phenomena^ like a thread through a labyrinth, be considered to
carry the stamp of truth, the claims of the theory of transverse
vibrations seem but little short of those of the theory of universal
gravitation — if, I say, we adopt this theory, we are obliged to
suppose the existence of a tangential force in the ether, called into
play by the continuous sliding of one layer, or film, of the medium
over another. In consequence of the existence of this force, the
ether must behave, so far as regards the luminous vibrations, like
an clastic solid. We have no occasion to speculate as to the cause
of this tangential force, nor to assume either that the ether does,
or that it does not, consist of distinct particles ; nor are we directly
called on to consider in what manner the ether behaves with
respect to the motion of solid bodies, such as the earth and
planets.

Accordingly, I have assumed, as applicable to the luminiferous
ether in vacuum, the known equations of motion of an elastic
medium, such as an elastic solid. These equations contain two
arbitrary constants, depending upon thQ nature of the medium.
The argument which Green has employed to shew that the lumi-
niferous ether must be regarded as sensibly incompressible, in

246 ON THE DYNAMICAL THEORY OF DIFFRACTION.

treating of the motions which constitute light*, appears to me of
great force. The supposition of incompressibility reduces the two
arbitrary constants to one ; but as the equations are not thus
rendered more manageable, I have retained them in their more
general shape.

The first problem relating to an elastic medium of which the
object that I had in view required the solution was, to determine
the disturbance at any time, and at any point of an elastic medium,
produced by a given initial disturbance which was confined to a
finite portion of the medium. This problem was solved long ago by
Poisson, in a memoir contained in the tenth volume of the Memoirs
of the Academy of Sciences. Poisson indeed employed equations
of motion with but one arbitrary constant, which are what the
general equations of motion become when a certain numerical
relation is assumed to exist between the two constants which
they involve. This relation was tlie consequence of a particular
physical supposition which he adopted, but which has since been
shewn to be untenable, inasmuch as it leads to results which arie
contradicted by experiment. Nevertheless nothing in Poisson*s
method depends for its success on the particular numerical rela-
tion assumed; and in fact, to save the constant writing of a
radical, Poisson introduced a second constant, which made his
equations identical with the general equations, so long as the
particular relation supposed to exist between the two constants
was not employed. I might accordingly have at once assumed
Poisson's results. I have however begun at the beginning, and
given a totally dififerent solution of the problem, which will I hope
be found somewhat simpler and more direct than Poif^son's. The
solution of this problem and the discussion of the result occupy the
first two sections of the paper.

Having had occasion to solve ths problem in all its generality,
I have in one or two instances entered into details which have no
immediate relation to light. I have also occasionally considered
some points relating to the theory of light which have no imme-
diate bearing on diffraction. It would occupy too much room to
enumerate these points here, which will be found in their proper
place. I will merely mention one very general theorem at which
I have arrived by considering the physical interpretation of a

* Camh, Phil. Trans, Vol. vu. p. 2.

ON THE DYNAMICAL THEORY OF DIFFRACTION. 247

certain step of analysis, though, properly speaking, this theorem
is a digression from the main object of the paper. The theorem
may be enunciated as follows.

If any material system in which the forces acting depend only
on the positions of the particles be slightly disturbed from a
position of equilibrium, and then left to itself, the part of the
subsequent motion which depends on the initial displacements
may be obtained from the part which depends on the initial
velocities by replacing the arbitrary functions, or arbitrary con-
stants, which express the initial velocities by those which express
the corresponding initial displacements, and differentiating with
respect to the time.

Particular cases of this general theorem occur so frequently
in researches of this kind, that I think it not improbable that the
theorem may be somewhere given in all its generality. I have
not however met with a statement of it except in particular cases,
and even then the subject was mentioned merely as a casual re-
sult of analysis.

In the third section of this paper, the problem solved in the
second section is applied to the determination of the law of
disturbance in a secondary wave of light. This determination
forms the whole of the dynamical part of the theory of diffraction,
at least when we confine ourselves to diffraction in vacuum, or,
more generally, within a homogeneous singly refracting medium :
the rest is a mere matter of integration ; and whatever diflSculties
the solution of the problem may present for particular forms of
aperture, they are purely mathematical.

In the investigation, the incident light is supposed to be
plane-polarized, and the following results are arrived at. Each
diffracted ray is plane-polarized, and the plane of polarization is
determined by this law ; The plane of vibration of the diffracted
ray is parallel to the direction of vibration of the incident ray.
The expression plane of vibration is here used to denote the plane
passing through the ray and the direction of vibration. The
direction of vibration in any diffracted ray being determined by
the law above mentioned, the phase and coefficient of vibration
at that part of a secondary wave are given by the formulae of
Art. 33.

The law just enunciated seems to lead to a crucial experiment
for deciding between the two rival theories respecting the direc-

248 ON THE DYNAMICAL THEORY OF DIFFRACTION.

tions of vibration in plane-polarized light. Suppose the plane of
polarization, and consequently the plane of vibration, of the
incident light to be turned round through equal angles of say
5* at a time. Then, according to theory, the planes of vibration
of the diffracted ray will not be distributed uniformly, but will be
crowded towards the plane perpendicular to the plane of diffrac-
tion, or that which contains the incident and diffracted rays.
The law and amount of the crowding will in fact be just the
same as if the planes of vibration of the incident ray were repre-
sented in section on a plane perpendicular to that ray, and then
projected on a plane perpendicular to the diffracted ray. Now
experiment will enable us to decide whether the planes of polariza-
tion of the diffracted ray are crowded towards the plane of dif-
fraction or towards the plane perpendicular to the plane of dif-
fraction, and we shall accordingly be led to conclude, either that
the direction of vibration is perpendicular, or that it is parallel to
the plane of polarization.

In ordinary cases of diffraction, the light is insensible at such
a small distance from the direction of the incident ray produced
that the crowding indicated by theory is too small to be detected
by experiment. It is only by means of a fine grating that we
can obtain light of considerable intensity which has been diffracted
at a large angle.

On mentioning to my friend. Professor Miller, the result at
which I had arrived, and making some inquiries about the fine-
ness, &c. of gratings, he urged me to perform the experiment
myself, and kindly lent mo for the purpose a fine glass grating,
which he has in his possession. For the use of two graduated
instruments employed in determining the positions of the planes
of polarization of tho incident and diffracted rays I am indebted
to the kindness of my friend Professor O'Brien.

Thedesciiption of the experiments, and the discussion of the
results, occupies Part II. of this Paper. Since in a glass gratiug
the diffraction takes place at the common surface of two different
media, namely, air and glass, the theory of Part. I. does not quite
meet the case. Nevertheless it does not fail to point out where-
abouts the plane of polarization of the diflfracted ray ought to lie,
according as we adopt one or other of the hypotheses respecting
the direction of vibration. For theory assigns exact results on the
two extreme suppositions, first, that the diffraction takes place

ON THE DYNAMICAL THEORY OF DIFFRACTION. 2 49

before the light reaches the grooves; secondly^ that it takes place
after the light has passed between them ; and these results are
veiy different, according as we suppose the vibrations to be per-
pendicular or parallel to the plane of polarization. Most of the
experiments were made on light which was diffracted in passing
through the grating. The results appeared to be decisive in
favour of Fresnel's h3rpothesis. In fact, theory shews that diffrac-
tion at a large angle is a powerful cause of crowding of the planes
of vibration of the diffracted ray towards the perpendicular to the
plane of diffraction, and experiment pointed out the existence of a
powerful cause of crowding of the planes of polarization towards the
plane of diffraction ; for not only was the crowding in the contrary
direction due to refraction overcome, but a considerable crowding
was actually produced towards the plane of diffraction, especially
when the grooved face of the glass plate was turned towards the
incident light.

The experiments were no doubt rough, and are capable of
being repeated with a good deal more accuracy by making some
small changes in the apparatus and method of observing. Never-
theless the quantity with respect to which the two theories are
at issue is so large that the experiments, such as they were, seem
amply suflScient to shew which hypothesis is discarded by the
phenomena.

The conclusive character of the experimental result with
regard to the question at issue depends, I think, in a great
measure on the simplicity of the law which forms the only result
of theory that it is necessary to assume. This law in fact merely
asserts that, whereas the direction of vibration in the diffracted
ray cannot be parallel to the direction of vibration in the incident
ray, being obliged to be perpendicular to the diffracted ray, it
makes with it as small an angle as is consistent with the above
restriction. This law seems only just to lie beyond the limits of
the geometrical part of the theory of undulations. At the same
time I may be permitted to add that, for my own part, I feel very
great confidence in the equations of motion of the luminiferous
ether in vacuum, and in that view of the nature of the ether
which would lead to these equations, namely, that in the propa-
gation of light, the ether, from whatever reason, behaves like an
elastic solid. But when we consider the mutual action of the
luminiferous ether and ponderable matter, a wide field, as it

250 ON THE DYNAMICAL THEORY OF DIFFRACTION.

seems to me, is thrown open to conjecture. Thus, to take the
most elementary of all the phenomena which relate to the action
of transparent media on light, namely, the diminution of the
velocity of propagation, this diminution seems capable of being
accounted for on several different hypotheses. And if this elemen-
tary phenomenon leaves so much room for conjecture, much more
may we form various hypotheses as to the state of things at the
confines of two media, such as air and glass. Accordingly, con-
clusions in favour of either hypothesis which are derived from the
comparison of theoretical and expeiimental results relating to the
effects of reflection and refraction on the polarization of light,
appear to me much more subject to doubt than those to which we
are led by the experiments here described.

In commencing the theoretical investigation of diffraction, I
naturally began with the simpler case of sound. As, however, the
results which I have obtained for sound are of far less interest
than those which relate to light, I have here omitted them, more
especially as the paper has already swelled to a considerable size.
I may, perhaps, on some future occasion bring them before the
notice of this Society.

PART I.
THEORETICAL INVESTIGATION.

Section I. Preliminary Analysis.

1. In what follows there will frequently be occasion to ex-
press a triple integration which has to be performed with respect
to all space, or at least to all points of space for which the quantity
to be integrated has a value different from zero. The conception
of such an integration, regarded as a limiting summation, presents
itself clearly and readily to the mind, without the consideration of
co-ordinates of any kind. A system of co-ordinates forms merely
the machinery by which the integration is to be effected in par-
ticular cases ; and when the function to be integrated is arbitrary,
and the nature of the problem does not point to one system rather
than another, the employment gf Qome particular system, and the

PRELIMINARY ANALYSIS. 251

analytical expression thereby of the function to be integrated,
serves only to distract the attention by the introduction of a
foreign element, and to burden the pages with a crowd of un-
necessary symbols. Accordingly, in the case mentioned above, I
shall merely take d!F to represent an element of volume, and
write over it the sign jjj, to indicate that the integration to be
performed is in fact triple. Integral signs will be used in this
manner without limits expressed when the integration is to extend
to all points of space for which the function to be integrated dififers
from zero.

There will frequently be occasion too to represent a double
integration which has to be performed with reference to the sur-
face of a sphere, of radius r, described round the point which is
regarded as origin, or else a double integration which has to be
performed with reference to all angular space. In this case the
sign // will be used, and dS will be taken to represent an element
of the surface of the sphere, and da- to represent an elementary
solid angle, measured by the corresponding element of the surface
of a sphere described about its vertex with radius unity. Hence,
if dV, dS, da- denote corresponding elements, dS=^r^d(r, dV
= drdS =■ r^drda-. When the signs /// and //, referring to diflFeren-
tials which are denoted by a single symbol, come together, or
along with other integral signs, they will be separated by a dot, as
for example Jff . jjUd Vda,

jf j% Ji

2. As the operation denoted by ^3 + j-, + -3-5 will be per-

djj dy dz

petually recurring in this paper, I shall denote it for shortness
by y. This operation admits of having assigned to it a geometri-
cal meaning which is independent of co-ordinates. For if P be
the point {x, y, z), T a small space containing P, which will finally
be supposed to vanish, dn an element of a normal drawn outwards
at the surface of 1\ U the function which is the subject of the

JS JS J2

operation, and if v be defined as the equivalent of -7-5 4- -1-5 + v-^ ,
it is easy to prove that

Vf/'= limit of iiT^-dS (1),

the integration extending throughout the surface of T, of which

252 ON THE DYNAMICAL THEORY OF DIFFRACTION.

i£> is an element. In fact, if I, m, n be the direction-cosines of the
normal, we shall have

//S"'«-//('f-f-S)^

We have also, supposing the origin of co-ordinates to be at the
point P, as we may without loss of generality,

dx \dx) ^ U^y ^ \dxdy) y ^ [dxdzJ ^

-h terms of the 2nd order, &c (3),

where the parentheses denote that the differential coefficients
which are enclosed in them have the values which belong to the

point P. In the integral jj -j-dydz, each element must be

taken positively or negatively, according as the normal which
relates to it makes an acute or an obtuse angle with the positive
direction of the axis of x. If we combine in pairs the elements of
the integral which relate to opposite elements of the surface of T,

we must write \\ (-r-^ ""T") ^H^^* where the single and double

accents subscribed refer respectively to the first and second points
in which the surface of T is cut by an indefinite straight line
drawn parallel to the axis of x, and in the positive direction,
through the point (0, y, z). We thus get by means of (3), omitting
the terms of a higher order than the first, which vanish in the
limit,

//(t-f)''^*=(S)//c»-.)^^^-

But JJ {x^^ — x^ dy dz is simply the volume T. Treating in the
same manner the two other integrals which appear on the right-
hand side of equation (2), we get

Dividing by T and passing to the limit, and omitting the paren-
theses, which are now no longer necessary, we obtain the theorem
enunciated.

PRELIMINARY ANALYSIS. 253

If in equation (1) we take for T the elementary volume
r^ sin dr d0 d(t>, or rdrdOdz, according as we wish to employ
polar co-ordinates, or one of three rectangular co-ordinates com-
bined with polar co-ordinates in the plafie of the two others, we
may at once form the expression for y U, and thus pass from rect-
angular co-ordinates to either of these systems without the trouble
of the transformation of co-ordinates in the ordinary way.

3. Let / be a quantity which may be regarded as a function
of the rectangular co-ordinates of a point of space, or simply, with-
out the aid of co-ordinates, as having a given value for each point
of space. It will be supposed that / vanishes outside a certain
portion T of infinite space, and that within T it does not become
infinite. It is required to determine a function U by the conditions
that it shall satisfy the partial differential equation

VU=/. (4)

at all points of infinite space, that it shall nowhere become in-
finite, and that it shall vanish at an infinite distance.

These conditions are precisely those which have to be satisfied
by the potential of a finite mass whose density is — //47r ; and we
shall have accordingly, if be the point for which the value of U
is required, and r be the radius vector of any element drawn from 0,

^=-MK^^- <^>-

In fact, it may be proved, just as in the theory of potentials, that
the expression for U given by (5) does really satisfy (4) and the
given conditions ; and consequently, if U+ IT be the most general
solution, IT must satisfy the equation y CT = at all points, must
nowhere become infinite, and must vanish at an infinite distance.
But this being the case it is easy to prove that IT cannot be
different from zero.

The solution will still hold good in certain cases when / is
infinite at some points, or when it is not confined to a finite space
T, but only vanishes at an infinite distance. But such instances
may be regarded as limiting cases of the problem restricted as
above, and therefore need not be supposed to be excluded by those
restrictions.

254 ON THE DYNAMICAL THEORY OF DIFFRACTION.

4, Let Z7be a quantity depending upon the time t, as well as
upon the position of the point of space to which it relates, and
satisfying the partial differential equation

'3'=«'V?^. (6).

It is required to determine Uhy the above equation and the con-
ditions that when ^ = 0, U and dU/dt shall have finite values
given arbitrarily within a finite space T,and shall vanish outside T,

Let be the point for which the value of U is sought, r the
radius vector of any element drawn from ; /(r), F (r) the initial
values of U, dUjdt By this notation it is not meant that these
values are functions of r alone, for they will depend likewise upon
the two angles which determine the direction of r ; but there will
be no occasion to express analytically their dependence on those
angles. The solution of the problem is

U^^i!F(at)d<r + ^^^tfJ/{at)do- (7).

See a memoir by Poisson M^m, de VAcademie, Tom. ill. p. 130,
or Gregorys Examples^ p. 499.

5. Let 8 be a function which has given finite values within
a finite portion of space, and vanishes elsewhere ; and let it bo
required to determine three functions f , i;, f by the conditions

^-*?=^-^ = ^^-^=0 (8)

dy dz dz dx rfo; dy

dx dy dz ^ ''

The functions f , i;, f are further supposed not to become infinite,
and to vanish at an infinite distance. To save repetition, it will
here be remarked, once for all, that the same supposition will be
made in similar cases.

By virtue of equations (8), ^dx-^ rjdy-h ^dz is an exact diffe-
rential d-^, and (9) gives ^^ = 5. Hence we have by the
formula (5)

^-M/^^ •••••»

and '^ being known, ^, 17, 1^ will be obtained by mere differentia-

PRELIMINARY ANALYSIS. 255

tion. To differentiate yp* with respect to or, it will be suflScient to
differentiate 8 under tjie integral sign. For draw 0(7 parallel to
the axis of x, and equal to Ax, let P, P' be two points similarly
situated with respect to 0, O, respectively, and consider the part
of -^ and that of i^4- Ai^ due to equal elements of volume dV
situated at P, P' respectively. For these two elements r has the
same value, since OP = O'P'y and in passing from the first to the
second 8 is changed into S + A8, and therefore the increment of -^
is simply — AS/47rr . dF. To get the complete increment of -^ we
have only to perform the triple integration, an integration which
is always real, even though r vanishes in the denominator, as may
be readily seen on passing momentarily to polar co-ordinates.
Dividing now by Ax and passing to the limit, we get

^ dx'^ iirjlldx r (^^^•

By employing temporarily rectangular co-ordinates in the
triple integration, integrating by parts with respect to x, and
observing that the quantity free from the integral sign vanishes at
the limits, we get

f = -:^///pC09(rx)dF (12),

as might have been readily proved from (10), by referring to
a fixed origin, and then differentiating with respect to x. The
expressions for rj and (f may be written down from symmetry.

0. Let tj', tj", tj'" be three functions which have given finite
values throughout a finite space and vanish elsewhere ; it is re-
quired to determine three other functions, f, 17, (f, by the condi-
tions

^i'_^? = 2w' ^-^=2w" ^^-^ = 2tir"' (13)
dy dz * dz dx * dx dy '"^ ^'

^ + ^4.^?^0 (14)

It is to be observed that tj', w", w"' are not independent. For
differentiating equations (13) with respect to x, y, z, and adding,
we get

dm dm" dm'" __ ^ .

dx dy dz ~ ^ ^'

256 ON THE DYNAMICAL THEORY OF DIFFRACTION.

Hence m', w", «r"' must be supposed given arbitrarily only in so
far as is consistent with the above equation.

Eliminating f from (14), and the second of equations (13),
we get

dx \dx duj dz^ dz '

^^'^dvKdx rfj- dz '

^ "^^ dy \dx dyj

which becomes by the last of equations (13)

dm" dvr

-. ^ fdm" dfs "\

dz djf
Consequently, by equation (5),

^'2'7rJJj[dy dz) r'

Transforming this equation in the same manner as (11), sup-
posing X, y, z measured from 0, and writing down the two equa-
tions found by symmetry, we have finally.

(IG).

7. Let 8, «/, 'c/y w'" be as before ; and let it be required to
determine three functions f, rj, ^ from the equations (9) and (13).

From the linearity of the equations it is evident that we have
merely to add together the expressions obtained in the last two
articles.

8. Let fo, 17^,, gj be three functions given arbitrarily within
a finite space outside of which they are equal to zero : it is re-
quired to decompose these functions into two parts f,, 17,, fj and
f,, 17,, (T, such that ^idx + fj^dy '\- ^^dz may be an exact differential
rf^j, and f„ 17,, (; may satisfy (14).

PROPAGATION OF AN ARBITRARY DISTURBANCE. 257

Observing that f,=fo~fi> V^^Vo-Vi^ b, = ?o-S'i» expressing
fi» ^i» ?t ^^ terms of i/r^, and substituting in (14), we get

whore S^ is what S becomes when fo» ^o> ^ ^^® written for f, 17, (f.
The above equation gives

*.=-i///i<".

whence fj, i;^ J*!, and consequently f,, ^,, Jii are known.

Section II.
Propagation of an Arbitrary Disturbance in an Elastic Medium.

9. The equations of motion of a homogeneous uncrystallized
elastic medium, such as an elastic solid, in which the disturbance
is supposed to be very small, are well known. They contain two
distinct arbitrary constants, which cannot be united in one with-
out adopting some particular physical hypothesis. These equations
may be obtained by supposing the medium to consist of ultimate
molecules, but they by no means require the adoption of such a
hypothesis, for the same equations are arrived at by regarding the
medium as continuous.

Let Xy y, z be the co-ordinates of any particle of the medium in
its natural state; f, 1;, f the displacements of the same particle at
the end of the time t, measured in the directions of the three axes
respectively. Then the first of the equations may be put under
the form

where a", J', denote the two arbitrary constants. Put for shortness

MA-' W'

s. II. 17

258 ON THE DYNAMICAL THEORY OF DIFFRACTION.

and as before represent by vf ^^® quantity multiplied by V. Ac-
cording to this notation, the three equations of motion are

^J = h\r, + {a^-b*)f\ (18).

It is to be observed that h denotes the dilatation of volume of
the element situated at the point (a?, y, z). In the limiting case
in which the medium is regarded as absolutely incompressible 8
vanishes ; but in order that equations (18) may preserve their
generality, we must suppose a at the same time to become infinite,
and replace a'S by a new function of the co-ordinates. If we take
— p to denote this function, we must replace the last terms in these

equations by t v^ , — ;/ > — w » I'cspectively, and we shall thus

have a fourth unknown function, as well as a fourth equation,
namely that obtained by replacing the second member of (17) by
zero. But the retention of equations (18) in their present more
general form does not exclude the supposition of incompressibility,
since we may suppose a to become infinite in the end just as well
as at first.

10. Suppose the medium to extend infinitely in all directions,
and conceive a portion of it occupying the finite space T to receive
any arbitrary small disturbance, and then to be left to itself, the
whole of the medium outside the space T being initially at rest ;
and let it bo required to determine the subsequent motion.

DiflTercntiating equations (18) with respect to x, y, z, respec-
tively, and adding, we get by virtue of (17)

^ = a*vS (19).

Again, differentiating the third of equations (18) with respect to y,
and the second with respect to z^ and subtracting the latter of the
two resulting equations from the former, and treating in a similar

PROPAGATION OF AN ARBITRARY DISTURBANCE. 259

manner the first and third, and then the second and first of equa-
tions (18), we get

W^^^'^' -rf?^^^''' -5^-r=6V«r -(ZO).

where «r', «r", «r'" are the quantities defined by equations (13).
These quantities express the rotations of the element of the
medium situated at the point (ar, y, z) about axes parallel to the
three co-ordinate axes respectively.

Now the formula (7) enables us to express 8, tj', w", and «r'" in
terms of their initial values and those of their diflferential coeffi-
cients with respect to t, which are supposed known ; and these
functions being known, we shall determine f , ri^ and f as in Art. 7.
Our equations being thus completely integrated, nothing will
remain but to simplify and discuss the formulae obtained.

11. Let be the point of space at which it is required to
determine the disturbance, r the radius vector of any element
drawn from ; and let the initial values of S, dildt be represented
by /(r), F{r)y respectively, with the same understanding as in
Art. 4. By the formula (7), we have

8 = ^//^(«0 da + ^j^tfj/(at) da (21).

The double integrals in this expression vanish except when a
spherical surface described round as centre, with a radius equal
to at, cuts a portion of the space T, Hence, if be situated out-
side the space T^ and if r^, r, be respectively the least and greatest
values of the radius vector of any element of that space, there will
be no dilatation at until at = r^. The dilatation will then com-
mence, will last during an interval of time equal to a"' (r, — r^), and
will then cease for ever. The dilatation here spoken of is under-
stood to be either positive or negative, a negative dilatation being
the same thing as a condensation.

Hence a wave of dilatation will be propagated in all directions
from the originally disturbed space T, with a velocity a. To find
the portion of space occupied by the wave, we have evidently only
got to conceive a spherical surface, of radius at, described about
each point of the space T as centre. The space occupied by the
assemblage of these surfaces is that in which the wave of dilatation

17—2

260 ON THE DYNAMICAL THEORY OF DIFFRACTION.

is comprised. To find the limits of the wave, we need evidently
only attend to those spheres which have their centres situated in
the surface of the space T, When t is small, this system of spheres
will have an exterior envelope of two sheets, the outer of these
sheets being exterior and the inner interior to the shell formed by
the assemblage of the spheres. The outer sheet forms the outer
limit to the portion of the medium in which the dilatation is differ-
ent from zero. As t increases, the inner sheet contracts, and at
last its opposite sides cross, and it changes its character from being
exterior, with reference to the spheres, to interior. It then ex-
pands, and forms the inner boundary of the shell in which the
wave of condensation is comprised. It is easy to shew geometri-
cally that each envelope is propagated with a velocity a in a normal
direction.

12. It appears in a similar manner from equations (20) that
there is a similar wave, propagated with a velocity 6, to which are
confined the rotations xa\ m'\ o'". This wave may be called for
the sake of distinction, the wave of distoHion^ because in it the
medium is not dilated nor condensed, but only distorted in a man-
ner consistent with the preservation of a constant density. The
condition of the stability of the medium requiies that the ratio
of 6 to a be not greater than that of V3 to 2*.

13. If the initial disturbance be such that there is neither
dilatation nor velocity of dilatation initially, there will be no wave
of dilatation, but only a wave of distortion. If it be such that the
expressions ^dx + rjdy + ?rf^ and d^jdt . dx -f drjjdt . dy + d^^jdt . dz
are initially exact differentials, there will be no wave of distortion,
but only a wave of dilatation. By making 6 = we pass to the
case of an elastic fluid, such as air. By supposing a = x we pass
to the case of an incompressible elastic solid. In this case wo
must have initially 8 = and dB/dt = 0; but in order that the
results obtained by at once putting a = oo may have the same
degree of generality as those which would be obtained by retaining
a as a fiuite quantity, which in the end is supposed to increase
indefinitely, we must not suppose the initial disturbance confined

* See a memoir by Orecn On the reflection and refraction of Light, Camh, Phil,
Tram. Vol. vii. p. 2. See also Camb, Phil. Trans, Vol. viii. p. 319. [Ante, Vol. i.
p. 128.]

PROPAGATION OF AN ARBITRARY DISTURBANCE. 261

to the space T, but only the initial rotations and the initial
angular velocities. Consequently, outside T the expression

must be initially an exact differential cZ^, where i/r satisfies the
equation yi/r = derived from (14), and the expression

must be initially an exact differential d^^, where -^^ satisfies the
equation yi^r^rsO. So long as a is finite, it comes to the same
thing whether we regard the medium as animated initially by
certain velocities given arbitrarily throughout the space T, or as
acted on by impulsive accelerating forces capable of producing
those velocities ; and the latter mode of conception is equally
applicable to the case of an incompressible medium, for which a
is infinite, although wo cannot in that case conceive the initial
velocities as given arbitrarily, but only arbitrarily in so far as is
compatible with their satisfying the condition of incompressibility.
It is not so easy to see what interpretation is to be given, in the
case of an incompressible medium, to the initial displacements
which are considered in the general case, in so far as these dis-
placements inyolve dilatation or condensation. As no simplicity
worth mentioning is gained by making a at once infinite, this
constant will be retained in it^ present shape, more especially as
the results arrived at will thus have greater generality.

14. The expressions for the disturbance of the medium at the
end of the time i are linear functions of the initial displacements
and initial velocities; and it appears from (21), and the corre-
sponding equations which determine «r', «r", and «r"', that the part
of the disturbance which is due to the initial displacements may
be obtained from the part which is due to the initial velocities by
differentiating with respect to t, and replacing the arbitrary func-
tions ^hich represent the initial velocities by those which represent
the initial displacements. The same result constantly presents
itself in investigations of this nature : on considering its physical
interpretation it will be found to be of extreme generality.

Let any material system whatsoever, in which the forces acting
depend only on the positions of the particles, be slightly disturbed

262 ON THE DYNAMICAL THEORY OF DIFFRACTION.

from a position of equilibrium, and then left to itself. In order
to represent the most general initial disturbance, we must suppose
small initial displacements and small initial velocities, the most
general possible consistent with the connexion of the parts of the
system, communicated to it. By the principle of the superposition
of small motions, the subsequent disturbance will be compounded
of the disturbance due to the initial velocities and that due to
the initial displacements. It is immaterial for the truth of this
statement whether the equilibrium be stable or unstable; only,
in the latter case, it is to be observed that the time t which has
elapsed since the disturbance must be sufficiently small to allow
of our neglecting the square of the disturbance which exists at
the end of that time. Still, as regards the purely mathematical
question, for any previously assigned interval t, however great, it
will be possible to find initial displacements and velocities so
small that the disturbance at the end of the time t shall be as
small as we please ; and in this sense the principle of superposi-
tion, and the results which flow from it, will be equally true
whether the equilibrium be stable or unstable.

Suppose now that no initial displacements were communicated
to the system we are considering, but only initial velocities, and
that the disturbance has been going on during the time t Let
f{t) be the type of the disturbance at the end of the time t, where
f (t) may represent indifferently a displacement or a velocity,
linear or angular, or in fact any quantity whereby the disturbance
may be defined. In the case of a rigid body, or a finite number
of rigid bodies, there will be a finite number of functions / (t) by
which the motion of the system will be defined : in the cases of
a flexible string, a fluid, an clastic solid, &c., there will be an
infinite number of such functions, or, in other words, the motion
will have to be defined by functions which involve one or more
independent variables besides the time. Let v^ be in a similar
manner the type of the initial velocities, and let t be an incre-
ment of- 1, which in the end will be supposed to vanish. The
disturbance at the end of the time t + r will be rcpreserffed by
fit + r); but since by hypothesis the forces acting on the system
do not depend explicitly on the time, this disturbance is the same
as would exist at the end of the time t in consequence of the
system of velocities v^ communicated to the material system at the
commencement of the time — t, the system being at that instant

PROPAGATION OF AN ARBITRARY DISTURBANCE. 263

in its position of equilibrium. Suppose then the system of velo-
cities Vq communicated in this manner, and in addition suppose
the system of velocities — v^ communicated at the time 0. On
account of the smallness of the motion, the disturbance produced
by the system of velocities v^ will be expressed by linear functions
of these velocities ; and consequently, if / (t) represent the dis-
turbance due to the system of velocities v^, ^f(t) will represent
the disturbance due to the system — v^. Hence the disturbance
at the end of the time t will be represented by /(^ + t)— /(O.
Now we may evidently regard the state of the material system
immediately after the communication of the system of velocities
— Vq as its initial state, and then seek the disturbance which would
be produced by the initial disturbance. The velocities v^ going on
during the time t will have produced by the end of that time a
system of displacements represented by rv^. By hypothesis, the
system was in a position of equilibrium at the commencement of
the time — t; and since the forces are supposed not to depend
on the velocities, but only on the positions of the particles, the
effective forces during the time t vary from zero to small quan-
tities of the order t, and therefore the velocities generated by the
end of the time — t are small quantities of the order t*. Hence
the velocities — v^ communicated at the time destroy the pre-
viously existing velocities, except so far as regards small quantities
of the order t', which vanish in the limit, and therefore we have
nothing to consider but the system of displacements rv^. Hence
the disturbance produced by a system of initial displacements rv^
is represented by /(^ + t) — /(^), ultimately; and therefore the
disturbance produced by a system of initial displacements v^ is
represented by the limit of {/ {t + t) -/ (01 /r, or by y* (t). Hence,
to get the disturbance due to the initial displacements from that
due to the initial velocities, we have only to differentiate with
respect to t, and to replace the arbitrary constants or arbitrary
functions which express the initial velocities by those which
express the corresponding initial displacements. Conversely, to
get the disturbance due to the initial velocities from that due to
the initial displacements, we have only to change the arbitrary
constants or functions, and to integrate with respect to t, making
the integral vanish with t if the disturbance is expressed by dis-
placements, or correcting it so as to give the initial velocities when
f = if the disturbance is expressed by velocities.

264 ON THE DYNAMICAL THEORY OF DIFFRACTION.

The reader may easily, if be pleases, verify this theorem on
some dynamical problem relating to small oscillations.

13. Let us proceed now to determine the general values of
f, tf, f in terms of their initial values, and those of their differential
cocflScients with respect to t. By the formulje of Section I., f, 77, ?
are linear functions of S, w\ «r", and w"\ and we may therefore
first form the part which depends upon 8, and afterwards the part
which depends upon «r', «r", tj'", and then add the results together.
Moreover, it will be unnecessary to retain the part of the expres-
sions which depends upon initial displacements, since this can be
supplied in the end by the theorem of the preceding article.

Omitting then for the present tj', tj", «r'", as well as the
second term in equations (21), we get from equations (10) and (21),

t=-T^f//^^-^' (^^)-

To understand the nature of the integration indicated in this
equation, let be the point of space for which the value of yjr is
sought ; from draw in an arbitrary direction OP equal to r, and
from F draw, also in an arbitrary direction, PQ equal to at Then
F (at) denotes the value of the function F, or the initial rate of
dilatation, at the point Q of space, and we have first to perform a
double integration referring to all such points as Q, P being fixed,
and then a triple integration referring to all such points as P. To
fticilitate the transformation of the integral (22), conceive PQ
produced to Q\ let PQ' =8, let d K' be an clement of volume,
and replace the double integral // F. da- by the triple integral
h'^ JfJF ,8'^ dV\ taken between the limits defined by the imparities
ai<8<at-\-hy which may be done, provided A be finally made to
vanish. We shall thus have two triple integrations to perform,
each of which we may conceive to extend to all space, provided we
regard the quantity to be integrated as equal to zero when PQ',
(or as it may now be denoted PQyQ being a point taken generally,)
lies beyond the limits at and at + h, as well as when the point Q
falls outside the space T, to which the disturbance was originally
confined. Now perform the first of the two triple integrations on
the supposition that Q remains fixed while Pis variable, instead
of supposing P to remain fixed while Q is variable. We shall thus
have F constant and r variable, instead of having F variable and r.

PROPAGATION OF AN ARBITBAHY DISTURBANCE. 265

constant. This first triple integration must evidently extend
throughout the spherical shell which has Q for centre and at, at + h
for radii of the interior and exterior surfaces. We get, on making
h vanish,

dS being an element of the surface of a sphere described with Q
for centre and at for radius. Now if OQ = r', the integral jj r~* dS,
which expresses the potential of a spherical shell, of radius at and
density unity, at a point situated at a distance r from the centre,
is equal to ^irat or 47raV/r', according as r' <> at Substituting
in (22), and omitting the accents, which are now no longer necessary,
we get

t = -^Jjj^'dV(r<at)-^jjj^.dV(r>at)...i2Z),

where the limits of integration are defined by the imparities written
after the integrals, as will be done in similar cases.

16. Let Uq, Vjp Wf^, be the initial velocities; then

dx dy dz

Substituting in the first term of the right-hand member of equation
(23), and integrating by parts, exactly as in Art. 5, we get

where the S denotes that we must take the sum of the expression
written down and the two formed from it by passing from a; to y
and from y to z, and the single and double accents refer respectively
to the first and second point in which the surface of a sphere
having for centre, and at for radius, is cut by an indefinite line
drawn parallel to the axis of a;, and in the positive direction, through
the point (0, y, 2:). Treating the last term in equation (23) in the
same way, and observing that the quantities once integrated Vitnish
at an infinite distance, or, to speak more properly, at the limits of
the space T, we get

~ 4^ /// ^"'^ "*' *°^ ■*" "'*^^ ~7 ^'' ^ "'^"

266 ON THE DYNAMICAL THEORY OF DIFFRACTION.

The double integrals arising from the transformation of the
second member of equation (23) destroy one another, and we get
finally

'^ = -:^jjj(:uj^ + vj/ + tc^)-^{r>at) (24).

17. To obtain the part of the displacement f due to the
initial velocity of dilatation, we have only to differentiate -^ with
respect to x, and this will be effected by differentiating m^, r^, w^
under the integral signs, as was shewn in Art. 5. Treating the
resulting expression by integration by parts, as before, and putting
i, m, n for the direction-cosines of the radius vector drawn to the
point to which the accents, refer, and f ^ for the part of f due to F,
we get

t

ri]J(^oip-^^^i?^ + ^£?)^^(^>^^>-

Let q^ be the initial velocity resolved along the radius vector,
so that go = ^"o + ^^o + '*^o> ^^^ (?©)«« ^^ the value of q^ at a dis-
tance at from ; then

, d X d y d z u^ — Slg^

Substituting in the expression for f^, we get finally

18. Let us now form the part of f which depends on the
initial rotations and angular velocities, and which may be denoted
by f,. The theorem of Art. 14 allows us to omit for the present
the part due to the initial rotations, which may be supplied in the
end. Let (o^', (o^\ (o"' be the initial angular velocities. Then f,
is given in terms of vr" and vr'" by the first of equations (16), and
w", vr'" are given in terms of o)^", o)/" by the formula (7), in which
however h must be put for a. We thus get

PROPAGATION OF AN ARBITRARY DISTURBANCE. 267

The integrations in this expression are to be understood as in
Art. 15, and cOq', (o^" are supposed to have the values which belong
to the point Q, but P^ is now equal to ht instead of at. The
quintuple integral may be transformed into a triple integral just
as before. We get in the first place

f.-s^a/k'"^'//^'-///--'""//^) »

The double integration in this expression refers to all angular
space, considered as extending round Q; x, y, z are the co-ordi-
nates, measured from 0, of a point P situated at a distance bt from
Q, and r = OP. If dS = {Uyda, the expressions for the integrals

jjxr-'dS, S!yr"d8, Hzr-'dS

may be at once written down by observing that these integrals
express the components of the attraction of a spherical shell, of
radius ht and density 1, having Q for centre, on a particle situated
at 0, Hence if a?', y\ z' be the co-ordinates of Q, measured from
0, and r = OQ, the integrals vanish when r' < ht^ and are equal to

47r(60Vr-', ^Tr[htfy'r-\ ^ir {btyzY-^,

respectively, when r > bt. Hence we get from (26), omitting the
accents, which are now no longer necessary, since we have done
with the point P,

^.-^jjj{<"y-<^)^ir>bt) (27).

Now

9.,'-^^o_^. <^,''_du^ dw^, 9,ro__dv, du,
^"^^ " dy dz ' ^"^^ " dz dx' ^"^^ ^ dx'"jbj'

Substituting in (27), and adding and subtracting x . dujdx under
the integral signs, we get

^« = 4l///{-(^i + 2^| + ^S"'

/ du^ dvo . dwX\dV, ..

But X . d/dx + y . d/dy + z . d/dz is the same thing as r . d/dr, and
we get accordingly

m4-4,^4h-^^^>^'>

'Nl'^Jr * ''■' <"■* "' - -// W-^'-

268 ON THE DYNAMICAL THEORT OF DIFFRACTION.

The second part of f, is precisely the expression transformed in
the preceding article, except that the sign is changed, and b put
for a. Hence we have

19. Adding together the expressions for f^ and f^, we get for
the disturbance due to the initial velocities

+ 4^/// (3^0 - ««) -J(bt<r<at) (29).

The part of the disturbance due to the initial displacements
may be obtained immediately by the theorem of Art. 14. Let f^,
Vo* So ^^ ^^^ initial displacements, p^ the initial displacement
resolved along a radius vector drawn from 0. The last term in
equation (29j, it will be observed, involves t in two ways, for t
enters as a coeflScient, and likewise the limits depend upon t To
find the part of the differential coeflScient which relates to the
variation of the limits, we have only to replace dVhy r^Jr da-, and
treat the integral in the usual way. We get for the part of the
disturbance due to the initial displacements

+ ^jjli3lp,-f:i^(.U<r«U) (30).

It is to be recollected that in this and the preceding ec^uation I
denotes the cosine of the angle between the axis of x and an arbi-
trary radius vector drawn from 0, whose direction varies from one
element da of angular space to another, and that the at or U sub-
scribed denotes that r is supposed to be equjd to at or It niter
differentiation. To obtain the whole displacement parallel to x
which exists at the end of the time t at the point 0, we have only
to add together the second members of equations (29) and (30).
The expressions for rj and f may be written down from symmetry,
or rather the axis of x may be supposed to be measured in the
direction in which we wish to estimate the displacement.

PROPAGATION OF AN ARBITRARY DISTURBANCE. 269

20. The first of the double integrals in equations (29), (30)
vanishes outside the limits of the wave of dilatation, the second
vanishes outside the limits of the wave of distortion. The triple
integrals vanish outside the outer limit of the wave of dilatation,
and inside the inner limit of the wave of distortion, but have finite
values within the two waves and between them. Hence a particle
of the medium situated outside the space T does not begin to move
till the wave of dilatation reaches it. Its motion then commences,
and does not wholly cease till the wave of distortion has passed,
after which the particle remains absolutely at rest.

21. If the initial disturbance be such that there is no wave of
distortion, the quantities m\ «r", tj'", cd', oj", o)'" must be separately
equal to zero, and the expression for f will be reduced to fj, given
by (25), and the expression thence derived which relates to the
initial displacements. The triple integral in the expression for fj
vanishes when the wave of dilatation has passed, and the same is
the case with the corresponding integral which depends upon the
initial displacements. Hence the medium returns to rest as soon
as the wave of dilatation has passed ; and since even in the general
case each particle remains at rest until the wave of dilatation
reaches it, it follows that when the initial disturbance is such that
no wave of distortion is formed the disturbance at any time is con-
fined to the wave of dilatation. The same conclusion might have
been arrived at by transforming the triple integral.

22. When the initial motion is such that there is no wave of
dilatation, as will be the case when there is initially neither dilata-
tion nor velocity of dilatation, f will be reduced to f,, given by
(28), and the corresponding expression involving the initial dis-
placements. By referring to the expression in Art. 17, from which
the triple integral in equation (28) was derived, we get

Now

=l0h''-lll^?'^'^^'

270 ON THE DYNAMICAL THEORY OF DIFFRACTION.

the parentheses denoting that the (Quantity enclosed in them is to
be taken between limits. By the condition of the absence of initial
velocity of dilatation we have

Substituting in the second member of equation (31), and writing
down for the present only the terms involving v^, we obtain

which, since djdx . y/r* = d/dy . ar/r", becomes

Treating the tenns involving w^ in the same manner, and substitu-
ting in (31), we get

Now the integration is to extend from r = 6^ to r = oo . The
quantities once integrated vanish at the second limit, and the first
limit relates to the surface of a sphere described round as centre
with a radius equal to 6^. Putting dS or Vf da for an element of
the surface of this sphere, we obtain for the value of the second
member of the last equation

- (to "* // (^"o + ^^^'o + nw^IdS, or - jjl {qX^<^ \

and therefore the triple integral in equation (28) destroys the
second part of the double integral in the same equation. Hence,
writing down also the terms depending upon the initial displace-
ments, we obtain for f the very simple expression

This expression might have been obtained at once by applying
the formula (7) to the first of equations (18), which in this case
take the form (6), since 8 = 0.

23. Let us return now to the general case, and consider
especially the terms which alone are important at a great distance
from the space to which the disturbance was originally confined ;

PROPAGATION OF AN ARBITRARY DISTURBANCE. 271

and, first, let us take the part of f which is due to the initial
velocities, which is given by equation (29).

Let the three parts of the second member of this equation be
denoted by f., ^4, f^, respectively, and replace d<r by {aty*d8
or (Ity^dSy as the case may be ; then

f- = 4l^/^<^'>«'^^ <32)-

Let Oj be a fixed point, taken within the space T, and regarded as
the point of reference for all such points as 0. Then when is at
such a distance from 0^ that the radius vector, drawn from 0, of
any element of T makes but a very small angle with 00^, we may
regard I as constant in the integration, and equal to the cosine of
the angle between 00^ and the direction in which we wish to
estimate the displacement at 0. Moreover the portion of the
surface of a sphere having for centre which lies within T will be
ultimately a plane perpendicular to 00^, and q^ will be ultimately
the initial velocity resolved in the direction 00^. Hence we have
ultimately

where, for a given direction of Ofi, the integral receives the same
series of values, as at increases through the value 00^^ whatever
be the distance of from 0^. Since the direction of the axis of x
is arbitrary, and the component of the displacement in that direc-
tion is found by multiplying by i a quantity independent of the
direction of the axes, it follows that the displacement itself is in
the direction 00^^ or in the direction of a normal to the wave. For
a given direction of 0^ 0, the law of disturbance is the same at one
distance as at another, and the magnitude of the displacements
varies inversely as at, the distance which the wave has travelled in
the time t

We get in a similar manner

?. = 4^jrJ/ («o - ^.)« d^f (33),

where I, and the direction of the resolved part, q^^ of the initial
velocity are ultimately constant, and the surface of which d8 is an
element is ultimately plane. To find the resolved part of the dis-
placement in the direction 00^, we must suppose x measured in

272 ON THE DYNAMICAL THEORY OF DIFFRACTION.

that direction, and therefore put Z = 1, Jo'^^o* which gives ^^= 0.
Hence the displacement now considered takes place in a direction
perpendicular to 00 ^t or is transversal.

For a given direction of Ofi^ the law of disturbance is constant,
but the magnitude of the displacements varies inversely as ht^ the
distance to which the wave has been propagated. To find the dis-
placement in any direction, OE, perpendicular to 00^^ we have
only to take OE for the direction of the axis of j*, and therefore
put ?= 0, and suppose u^ to refer to this direction.

Consider, lastly, the displacement, f^, expressed by the last
term in equation (29). The form of the expression shews that
f^ will be a small quantity of the order tli^ or 1/r', since t is of the
same order as r ; for otherwise the space T would lie outside the
limits of integration, and the triple integral would vanish. But
f. and fft are of the order 1/r, and therefore f^ may be neglected,
except in the immediate neighbourhood of T.

To see more clearly the relative magnitudes of these quantities,
let V be a velocity which may be used as a standard of comparison
of the initial velocities, R the radius of a sphere whose volume is
equal to that of the space T, and compare the displacements f^, f^,
f, which exist, though at diflFerent times, at the same point 0,
where 0^0 = r. These displacements are comparable with

v^ vj^ vm

ar ' hr ' i^ '

which are proportional to

1 I ^ t
a b r r

But, in order that the triple integral in (29) may not wholly vanish,
tjr must lie between the limits 1/a and 1/6, or at most lie a very
little outside these limits, which it may do in consequence of the
finite thickness of the two waves. Hence the quantity neglected
in neglecting f^ is of the order Rjr compared with the quantities
retained.

The important terms in the disturbance due to the initial dis-
placements might be got from equation (30), but they may be
deduced immediately from the corresponding terms in the disturb-
ance due to the initial velocities by the theorem of Art. 14.

PROPAGATION OF AN ARBITRARY DISTURBANCE. 273

24. If we confine our attention to the terms which vary
ultimately inversely as the distance, and which alone are sensible
at a great distance from T, we shall be able, by means of the
ibrraulse of the preceding article, to obtain a clear conception of
the motion which takes place, and of its connexion with the initial
disturbance.

From the fixed point 0^, draw in any direction the right line
Oj equal to r, r being so large that the angle subtended at by
any two elements of T is very small ; and let it be required to
consider the disturbance at 0. Draw a plane P perpendicular to
00^, and cutting 00^ produced at a distance p from 0^. Let —p^,
+ Pj be the two extreme values of ^ for which the plane P cuts
the space T. Conceive the displacements and velocities resolved
in three rectangular directions, the first of these, to which f and u
relate, being the direction 00^. ^^^/u {p)fA {p)> /*> (P) ^® three
functions oi p defined by the equations

fJp)=II%dS. /.(p)=ShdS, fAp)=iKdS, (34).

and/^(p),/, (/?),/^ (p) three other functions depending on the initial
displacements as the first three do on the initial velocities, so that

/*(?)=//?. rf-sf, Mp)=JIvM fs{p)^S5?^s (35).

These functions, it will be observed, vanish when the variable lies
outside of the limits —p^ and +p^. They depend upon the direc-
tion Ofi, so that in passing to another direction their values
change, as well as the limits of the variable between which they
diflFer firom zero. It may be remarked however that in passing
from any one direction to its opposite the functions receive the same
values, as the variable decreases from +p^ to — p,, that they before
received as the variable increased from —p^ to +p,, provided the
directions in which the displacements are resolved, as well as the
sides towards which the resolved parts are reckoned positive, are
the same in the two cases.

The medium about remains at rest until the end of the time
(r'-p^)/a, when the wave of dilatation reaches 0. During the
passage of this wave, the displacements and velocities are given by
the equations

f = Jar/- ('^*-'-> + i^/*' <«'-»•)• " = «' ^=T..(36).
s. n. 18

274 ON THE DYNAMICAL THEOEY OF DIFFRACTION.

The first term in the right-hand member of the first of these
equations is got from (32) by putting Z = 1, introducing the func-
tion /^, and replacing a^ in the denominator by r, which may
be done, since a t differs from r only by a small quantity depending
upon the finite dimensions of the space T. The second term is
derived from the first by the theorem of Art. 14, and u is of
course got from f by differentiating with respect to t Had t
been retained in the denominator, the differentiation would have
introduced terms of the order ^■*, and therefore of the order r"*,
but such terms are supposed to be neglected.

The wave of dilatation will have just passed over at the end of
the time {r-\'p^)la. The medium about will then remain
sensibly at rest in its position of equilibrium till the wave of
distortion reaches it, that is, till the end of the time {r—p^lh.
During the passage of this wave, the displacements and velocities
will be given by the equations

(37).

«'=i/:(t*-'-)+i^,//'(i«-r)

After the passage of the wave of distortion, which occupies an
interval of time equal to (i>i+p,)/6, the medium will return
absolutely to rest in its position of equilibrium.

25. A caution is here necessary with reference to the em-
ployment of equation (30). If we confine our attention to the
important terms, we get

Now the initial displacements and velocities are supposed to have
finite, but otherwise arbitrary, values within the space T^ and to

PROPAGATION OF AN ARBITRARY DISTURBANCE. 27^

vanish outside. Consequently we cannot, without unwarrantably
limiting the generality of the problem, exclude from considera-
tion the cases in which the initial displacements and velocities
alter abruptly in passing across the surface of T. In particular,
if we wish to determine the disturbance at the end of the time t
due to the initial disturbance in a part only of the space through-
out which the medium was originally disturbed, we are obliged
to consider such abrupt variations; and this is precisely what
occurs in treating the problem of difiraction. In applying equa-
tion (38) to such a case, we must consider the abrupt variation as
a limiting case of a continuous, but rapid, variation, and we shall
have to add to the double integrals found by taking for dpjdr
and d^Jdr the finite values which refer to the space T, certain
single integrals referring to the perimeter of that portion of the
plane P which lies within T. The easiest way of treating the
integrals is, to reserve the diflferentiation with respect to t from
which the diflferential coefficients just written have arisen until
after the double integration, and we shall thus be led to the for-
mulae of the preceding article, where the correct values of the
terms in question were obtained at once by the theorem of
Art. 14.

26. It appears from Arts. 11 and 12, that in the wave of
distortion the density of the medium is strictly the same as in
equilibrium ; but the result obtained in Art. 23, that the displace-
ments in this wave are transversal, that is, perpendicular to the
radius of the wave, is only approximate, the approximation
depending upon the largeness of the radius, r, of the wave
compared with the dimensions of the space T, or, which comes
to the same, compared with the thickness of the wave. In fact,
if it were strictly true that the displacement at due to the
original disturbance in each element of the space T was trans-
versal, it is evident that the crossing at of the various waves
corresponding to the various elements of T under finite, though
small angles, would prevent the whole displacement from being
strictly perpendicular to the radius vector drawn to from an
arbitrarily chosen point, Oj, within T. But it is not mathemati-
cally true that the disturbance proceeding from even a single
point Oj, when a disturbing force is supposed to act, or rather
that part of the disturbance which is propagated with the velocity

18—2

276 ON THE DYNAMICAL THEORY OF DIFFRACTION.

6, is perpendicular to 00^, as will be seen more clearly in the next
article. It is only so nearly perpendicular that it may be re-
garded as strictly so without sensible error. As the wave grows
larger, the inclination of the direction of displacement to the
wave's front decreases with great rapidity.

Thus the motion of a layer of the medium in the front of a
wave may be compared with the tidal motion of the sea, or rather
with what it would be if the earth were wholly covered by water.
In both cases the density of the medium is unchanged, and there
is a slight increase or decrease of thickness in the layer, which
allows the motion along the surface to take place without change
of density : in both cases the motion in a direction perpendicular
to the surface is very small compared with the motion along the
surface.

27. From the integral already obtained of the equations of
motion, it will be easy to deduce the disturbance due to a given
variable force acting in a given direction at a given point of the
medium.

Let 0^ be the given point, T a space comprising Oj. Let the
time t be divided into equal intervals t ; and at the beginning of
the n^ interval let the velocity tF (n t) be communicated, in the
given direction, to that portion of the medium which occupies
the space T. Conceive velocities communicated in this manner at
the beginning of each interval, so that the disturbances produced
by these several velocities are superposed. Let D be the den-
sity of the medium in equilibrium ; and let F{nT) = {DTy^f(n t),
so that Tf(n r) is the momentum communicated at the beginning
of the n"* interval. Now suppose the number of intervals
T indefinitely increased, and the volume T indefinitely dimin-
ished, and we shall pass in the limit to the case of a moving
force which acts continuously.

The disturbance produced by given initial velocities is ex-
pressed, without approximation, by equation (29), that is, without
any approximation depending on the largeness of the distance
OOi', for the square of the disturbance has been neglected all
along. Let OO^^r; refer the displacement at to the rect-
angular axes of ajfi/.z; let Z, m, n be the direction-cosines of
00^; tf m\ n those of the given force, and put for shortness k for

PROPAGATION OF AN ARBITRARY DISTURBANCE. 277

the cosine of the angle between the direction of the force and the
line OOj produced, so that

Jc^lV + mm' + nn\

Consider at present the first term of the right-hand side of
(29). Since the radius vector drawn from to any element of T
ultimately coincides with 00,, we may put I outside the integral
signs, and replace da by r^dS. Moreover, since this term vanishes
except when at lies between the greatest and least values of the
radius vector drawn from to any element of T, we may replace t
outside the integral signs by rja. Conceive a series of spheres,
with radii ar^ 2aT...naT,,.. described round 0, and let the n^ of
these be the first which cuts T. Let flf,, 8^.., be the areas of the
surfaces of the spheres, beginning with the n^, which lie within T;
then

jj{qXdS = kTF{t-nT)8, + krF[t^{n + l)T}8,+ ...

But F{t — nT), F{t'-{n-\-l)T} ... are ultimately equal to each
other, and to

F(t-^^, or iDTrf(t-^^;

and arS^ + arS^ + ... is ultimately equal to T, Hence we get, for
the part of f which arises from the first of the double integrals,

Ik

-/('-3-

4!TrDd

The second of the double integrals is to be treated in exactly the
same way.

To find what the triple integral becomes, let us consider first
only the impulse which was communicated at the beginning of the
time t — nr, where nr lies between the limits r/a and r/6, and is
not so nearly equal to one of these limits that any portion of the
space T lies beyond the limits of integration. Then we must
write nr for t in the coeflScient, and Slq^ — u^ becomes ultimately
{Slk — r) TF(t — nr), and, as well as r, is ultimately constant in the
triple integration. Hence the triple integral ultimately becomes

^—^-^nT.TF{t^nT),
and we have now to perform a summation with reference to

278 ON THE DYNAMICAL THEORY OF DIFFRACTION.

different values of w, which, in the limit becomes an integration.
Putting nr = t\ we have ultimately

T = dt\ ^nT.TF(t''nT)^j't'F{t-t')dt\

a

It is easily seen that the terms arising from the triple integral
when it has to be extended over a part only of the space T vanish
in the limit. Hence we have, collecting all the terms, and express-
ing F{t) in terms of/(i),

a

To get 1) and f, we have only te pass from Z, T te wi, m' and
then to n, n. If we take 00, for the axis of a?, and the plane
passing through 00^ and the direction of the force for the plane
xZy and put a for the inclination of the direction of the force to
00^ produced, we shall have

Z=l, m = 0, n=0, T = k^ cos cr, m =0, n = sin i ;

whence

f-4^v/('-s)-S'/;w-0'«-

a
17=0

(40).

In the investigation, it has been supposed that the force began
to act at the time 0, before which the fluid was at rest, so that
f(^t)=0 when t is negative. But it is evident that exactly the
same reasoning would have applied had the force begun to act at
any past epoch, as remote as we please, so that we are not obliged
to suppose /(O equal to zero when t is negative, and we may even
suppose /(<) periodic, so as to have finite values from ^ = — 00 to
« = + 00.

By means of the formula (39), it would be very easy to write
down the expressions for the disturbance due to a system of forces
acting throughout any finite portion of the medium, the disturbing

PROPAQATION OF AN ARBITRARY DISTURBANCE. 279

force varying in any given manner, both as to magnitude and
direction, from one point of the medium to another, as well as
from one instant of time to another.

The first term in ^ represents a disturbance which is propa-
gated from Oj with a velocity a. Since there is no corresponding
term in 17 or ^, the displacement, as far as relates to this disturb-
ance, is strictly normal to the front of the wave. The first term in
f represents a disturbance which is propagated from 0^ with a
velocity b, and as far as relates to this disturbance the displace-
ment takes place strictly in the front of the wave. The remaining
terms in ^ and ^ represent a disturbance of the same kind as that
which takes place in an incompressible fluid in consequence of the
motion of solid bodies in it. If/ (t) represent a force which acts
for a short time, and then ceases, /(t—f) will differ from zero
only between certain narrow limits of t, and the integral contained
in the last terms of f and f will be of the order r, and therefore
the terms themselves will be of the order r"*, whereas the leading
terms are of the order r"\ Hence in this case the former terms
will not be sensible beyond the immediate neighbourhood of 0^.
The same will be true if / (t) represent a periodic force, the mean
value of which is zero. But if/ {t) represent a force always acting
one way, as for example a constant force, the last terras in f and f
will be of the same order, when r is large, as the first terms.

28. It has been remarked in the introduction that there is
strong reason for believing that in the case of the luminiferous
ether the ratio of a to 6 is extremely large, if not infinite. Conse-
quently the first term in f, which relates to normal vibrations, will
be insen^ble, if not absolutely evanescent. In fact, if the ratio of
a to 6 were no greater than 100, the denominator in this term
would be 10000 times as great as the denominator of the first
term in fl Now the molecules of a solid or gas in the act of com-
bustion are probably thrown into a state of violent vibration, and
may be regarded, at least very approximately, as centres of disturb-
ing forces. We may thus see why transversal vibrations should
alone be produced, unaccompanied by normal vibrations, or at
least by any which are of sufficient magnitude to be sensible. If
we could be sure that the ether was strictly incompressible, we
should of course be justified in asserting that normal vibrations
are impossible.

r

280 ON THE DYNAMICAL THEORY OF DIFFRACTION.

29. If we suppose a = oo , and f{t) = c sin 27rbt/\ we shall get
from (40)

i; =

^ c sin a . 27r ,, ^ . cX sin a 27r ,, , . ^ (*^/»

. cX'sina . -Trr 27r/,^ r\

and we see that the most important term in f is of the order X/trr
compared with the leading term in f, which represents the trans-
versal vibrations properly so called. Hence f , and the second and
third terms in f, will be insensible, except at a distance from 0,
comparable with X, and may be neglected ; but the existence of
terms of this nature, in the case of a spherical wave whose radius
is not regarded as infinite, must be borne in mind, in order to
understand in what manner transversal vibrations are compatible
with the absence of dilatation or condensation.

30. The integration of equations (18) might have been eflfected
somewhat difibrently by first decomposing the given functions f^,
tJq, fo, and u^, v^, w^ into two parts, as in Art. 8, and then treating
each part separately. We should thus be led to consider separately
that part of the initial disturbance which relates to a wave of dila-
tation and that part which relates to a wave of distortion. Either
of these parts, taken separately, represents a disturbance which is
not confined to the space T, but extends indefinitely around it.
Outside Ty the two disturbances are equal in magnitude and oppo-
site in sign.

Section III.

Determination of the Law of the Disturbance in a Secondary

Wave of Light

31. Conceive a series of plane waves of plane-polarized light
propagated in vacuum in a direction perpendicular to a fixed
mathematical plane P. According to the undulatory theory of
light, as commonly received, that is, including the doctrine of

LAW OF DISTURBANCE IN A SECONDARY WAVE. 281

transverse vibrations, the light in the case above supposed consists
in the vibrations of an elastic medium or ether, the vibrations
being such that the ether moves in sheets, in a direction perpen-
dicular to that of propagation, and the vibration of each particle
being symmetrical with respect to the plane of polarization, and
therefore rectilinear, and either parallel or perpendicular to that
plane. In order to account for the propagation of such vibrations,
it is necessary to suppose the existence of a tangential force, or
tangential pressure, called into play by the continuous sliding of
the sheets one over another, and proportional to the amount of the
displacement of sliding. There is no occasion to enter into any
speculation as to the cause of this tangential force, nor to entertain
the question whether the luminiferous ether consists of distinct
molecules or is mathematically continuous, just as there is no
occasion to speculate as to the cause of gravity in calculating the
motions of the planets. But we are absolutely obliged to suppose
the existence of such a force, unless we are prepared to throw over-
board the theory of transversal vibrations, as usually received, not-
withstanding the multitude of curious, and otherwise apparently
inexplicable phenomena which that theory explains with the ut-
most simplicity. Consequently we are led to treat the ether as an
elastic solid so far as the motions which constitute light are con^
cemed. It does not at all follow that the ether is to be regarded
as an elastic solid when lar^e displacements are considered, such
as we may conceive produced by the earth and planets, and solid
bodies in general, moving through it. The mathematical theories
of fluids and of elastic solids are founded, or at least may be
founded, on the consideration of internal pressures. In the case
of a fluid, these pressures are supposed normal to the common sur-
face of the two portions whose mutual action is considered : this
supposition forms in fact the mathematical definition of a fluid.
In the case of an elastic solid, the pressures are in general oblique,
and may even in certain directions be wholly tangential The
treatment of the question by means of pressures presupposes the
absence of any sensible direct mutual action of two portions of the
medium which are separated by a small but sensible interval The
state of constraint or of motion of any element affects the pressures
in the surrounding medium, and in this way one element exerts an
indirect action on another from which it is separated by a sensible
interval.

282 ox THE DYNAMICAL THEORY OF DIFFRACTION.

Now the absence of prismatic colours ia the stars, depending
upon aberration, the absence of colour in the disappearance and
reappearance of Jupiter's Satellites in the case of eclipses, and, still
more, the absence of change of colour in the case of certain periodic
stars, especially the star Algol, shew that the velocity of light of
different colours is, if not mathematically, at least sensibly the
same. According to the theory of undulations, this is equivalent
to saying that in vacuum the velocity of propagation is independ-
ent of the length of the waves. Consequently the direct action of
two elements of ether separated by a sensible interval must be
sensibly if not mathematically equal to zero, or at least must be
independent of the disturbance; for, were this not the case, the
expression for the velocity of propagation would involve the length
of a wave. An interval is here considered sensible which is com-
parable with the length of a wave. We are thus led to apply to
the luminiferous ether in vacuum the ordinary equations of motion
of an elastic solid, provided we are only considering those disturb-
ances which constitute light.

Let us return now to the case supposed at the beginning of
this section. According to the preceding explanation, we must
regard the ether as an elastic solid, in which a series of rectilinear
transversal vibrations is propagated in a direction perpendicular to
the plane P, The disturbance at any distance in front of this
plane is really produced by the disturbance continually transmitted
across it; and, according to the general principle of the superposi-
tion of small motions, we have a perfect right to regard the dis-
turbance in front as the aggregate of the elementary disturbances
due to the disturbance continually transmitted across the several
elements into which we may conceive the plane P divided. Let it
then be required to determine the disturbance corresponding to an
elementary portion only of this plane.

In practical cases of diffraction at an aperture, the breadth of
the aperture is frequently sensible, though small, compared with
the radius of the incident waves. But in determining the law of
disturbance in a secondary wave we have nothing to do with an
aperture; and in order that we should be at liberty to regard the
incident waves as plane all that is necessary is, that the radius of
the incident wave should be very large compared with the wave*s
length, a condition always fulfilled in experiment.

LAW OF DISTURBANCE IN A SECONDARY WAVE. 283

32. Let Oj be any point in the plane P; and refer the medium
to rectangular axes passing through 0^ x being measured in the
direction of propagation of the incident light, and z in the direc-
tion of vibration. Let / (bt — x) denote the displacement of the
medium at any point behind the plane P, x of course being nega-
tive. Let the time t be divided into small intervals, each equal
to T, and consider separately the effect of the disturbance which is
transmitted across the plane P during each separate interval. The
disturbance transmitted during the interval t which begins at the
end of the time i! occupies a film of the medium, of thickness frr,
and consists of a displacement / (6^') and a velocity bf {ht'). By
the formulae of Section II. we may find the effect, over the whole
medium, of the disturbance which exists in so much only of the
film as corresponds to an element dS of P adjacent to Oj. By
doing the same for each interval t, and then making the number
of such intervals increase and the magnitude of each decrease
indefinitely, we shall ultimately obtain the effect of the disturb-
ance which is continually propagated across the element dS.

Let be the point of the medium at which the disturbance is
required; Z, 7/1, n the direction-cosines of 0^0 measured from 0,,
and therefore — Z, — m, — n those of 00^ measured from ; and
let OO^^r. Consider first the disturbance due to the velocity of
the film. The displacements which express this disturbance are
given without approximation by (29) and the two other equations
which may be written down from symmetry. The first terms in
these equations relate to normal vibrations, and on that account
alone might be omitted in considering the diffraction of light.
But, besides this, it is to be observed that t in the coefficient of
these terms is to be replaced by r/a. Now there seems little
doubt, as has been already remarked in the introduction, that in
the case of the luminiferous ether a is incomparably greater
than 6, if not absolutely infinite*; so that the terms in question
are insensible, if not absolutely evanescent. The third terms are
insensible, except at a distance from 0^ comparable with X, as has
been already observed, and they may therefore be omitted if we
suppose r very large compared with the length of a wave. Hence
it will be sufficient to consider the second terms only. In the

* I have explained at fuU my views on this sabject in a paper On the eonstitution
of the luminiferout ether, printed in the 82nd Tolome of the Philaophical Magazine,
p. 849. [Ante, p. 12.]

284 ON THE DYNAMICAL THEORY OF DIFFUACTION.

coefficient of these tenns we must replace t by r/b ; we must put
V'o = 0, 1^0 = 0, w^^hf (J)t — r\ write — Z, — m, — n for /, m, n\ and

put ?o "= "" ^^"'o ~ ■" '*^' (^^ "■ 0- "^^^ integral signs are to be
omitted, since we want to get the disturbance which corresponds
to an elementary portion only of the plane P.

It is to be observed that da represents the elementary solid
angle subtended at by an element of the riband formed by that
portion of the surface of a sphere described round 0, with radius r,
which lies between the plane yz and the parallel plane whose
abscissa is hr. To find the aggregate disturbance at correspond-
ing to a small portion, 8, of the plane P lying about 0^, we must
describe spheres with radii . . . r — 26t, r -hr^ r, r + 6t, r + 26t . . . ,
describing as many as cut 8. These spheres cut 8 into ribands,
which are ultimately equal to the corresponding ribands which lie
on the spheres. For, conceive a plane drawn through 00^ per-
pendicular to the plane yz. The intersections of this plane by two
consecutive spheres and the two parallel planes form a quadrilate-
ral, which is ultimately a rhombus ; so that the breadths of corre-
sponding ribands on a sphere and on the plane are equal, and their
lengths are also equal, and therefore their areas are equal. Hence
we must replace da by r^dS, and we get accordingly

^Jlr-2-fibt-r) (42).

4nrr

Since l^-^-mrj-^-n^^O, the displacement takes place in a plane
through perpendicular to Ofi. Again, since ^ : i; :: Z : m, it
takes place in a plane through Ofi and the axis of z. Hence
it takes place along a line drawn in the plane last mentioned
perpendicular to 00^. The direction of displacement being known,
it remains only to determine the magnitude. Let ^ be the dis-
placement, and the angle between 0^ and the axis of z, so that
n = cos <^. Then ^ sin <^ will be the displacement in the direction
of z, and equating this to ^ in (4*2) we get

r. = 4^8in^/(6«-r) (43).

The part of the disturbance due to the successive displace-
ments of the films may be got in the same way from (30) and the

LAW OF DISTUKBANCE IN A SECONDAKY WAVE. 283

two other equations of the same system. The only terms which it
will be necessary to retain in these equations are those which
involve the differential coefficients of f^, i/^, g,, and p^ in the second
of the double integrals. We must put as before r for bt, and write
r^dS for d<r. Moreover we have for the incident vibrations

f=0, 17=0, i^f{bt'^x\ p nf{M^x).

To find the values of the differential coefficients which have to be
used in (30) and the two other equations of that system, we must
differentiate on the supposition that f, ^, f, p are functions of r in
consequence of being functions of x, and after differentiation we
must put a: = 0, t' — t — r/b. Since d/dr = - i . d/dx, we get

whence wo get, remembering that the signs of I, m, n in (30) have
to be changed,

The displacement represented by these equations takes place along
the same line as before; and if we put (f, for the displacement,
and write cos B for i, we get

r. = 4^003^ sin 0/'(6<-r) (44).

33. By combining the partial results obtained in the preceding
article, we arrive at the following theorem.

Let f = 0, ^ = 0, ?=/(6^ — ic) be the displacements correspond-
ing to the incident light ; let 0^ be any point in the plane P, dS
an element of that plane adjacent to 0^ ; and consider the disturb-
ance due to that portion only of the incident disturbance which
passes continually across dS. Let be any point in the medium
situated at a distance from the point 0^ which is large in compari-
son with the length of a wave ; let Ofi^r^ and let this line make
angles 6 with the direction of propagation of the incident lights or
the axis of a;, and ^ with the direction of vibration, or the axis of
z. Then the displacement at will take place in a direction per-

286 ON THE DYNAMICAL THEORY OF DIFFRACTION.

pendicular to O^Oy and lying in the plane zOfi\ and if f be the
displacement at 0, reckoned positive in the direction nearest to
that in which the incident vibrations are reckoned positive,

?' = ^^(l+cos^)sin^/(i«-r)* (45).

In particular, if

27r
f{bt — x)^c sin — (6^ — x),

we shall have

?' = |^(l+cos^)sin^cos^(ie-r) (46).

34. On finding by means of this formula the aggregate dis-
turbance at due to all the elements of the plane P, being
supposed to be situated at a great distance from P, we ought to
arrive at the same result as if the waves had not been broken up.

To verify this, let fall from the perpendicular 00' on the
plane P, and let 00' =p, or = — p, according as is situated in
front of the plane P or behind it. Through 0' draw OV, y',
parallel to O^x, O^y, and let OV, = r', Ofiy = ©. Then

dS = r'drdco = rdrday,

since r* = jo' + r", and p is constant. Let 4' = ^ sin <f). The dis-
placement 5" takes place in the plane zOfi, and perpendicular to
0^0; and resolving it along and perpendicular to O^z, we get for
resolved parts 8 sin' (f>, s sin cos <f), of which the latter is estimated
in the direction OM, where M is the projection of 0, on O'y'. Let
MOO' = X, X being reckoned positive when M falls on that side of
0' on which y is reckoned positive ; then, resolving the displace-
ment along OM parallel to OV, O'y, we get for resolved parts
— 8 sin <f> cos <l> cos x, s sin <f> cos (f) sin x- Hence we get for the dis-
placements f , ^, f at

f = — « sin ^ cos cos x» 17 = « sin ^ cos ^ sin x, ?= * si^' ^«

Now produce O'O^ to 0,, and refer 0,a?, O^y, O^z, Ofi^, Ofi to a
sphere described round 0, with radius unity. Then zOfi forms a
spherical triangle, right-angled at 0„ and

* The correspondiiig expression which I have obtained for sound differs from
this only in having cos in pUoe of sin 0, provided we suppose 6 to be the velocity
of propagation of sound, and ( to represent a displacement in the direction OiO,

LAW OF DISTURBANCE IN A SECONDARY WAVE. 287

whence we get from spherical uigonometry,

cos = — sin d sin <i), sin <f> cos ;^ = cos 9,
sin ^ sin ;^ = cos 6 tan ;^ = sin ^ cos <o.

We have therefore

f = « sin ^ cos ^ sin G), ri^ — s sin' sin o) cos w,

f = « (1 — sin' ^ sin* ©).

To find the aggregate disturbance at 0, we must put for 8 its
value, and perform the double integrations, the limits of <o being
and 27r, and those of r being tjp^ and x . The positive and nega-
tive parts of the integrals which give f and ^ will evidently destroy
each other, and we need therefore only consider f. Putting for 8
its value, and expressing 6 in terms of r, we get

f = - - //(r + p) (r* cos' (o^-p^ sin* (o) cos -— {bt — ?*) — ,- . . ..(47).

Let us first conceive the integration performed over a large area
A surrounding (7, which we may afterwards suppose to increase
indefinitely. Perform the integration with respect to r first, put
for shortness F (r) for the coefficient of the cosine under the inte-
gral signs, and let R, a function of o), be the superior limit of r.
We get by integration by parts

27r
jF (r) cos — Q)t - r) dr

Now the terms after the first must be neglected for consistency's
sake, because the formula (46) is not exact, but only approximate,
the approximation depending on the neglect of terms which are of
the order X compared with those retained. The first term, taken
between limits, gives

where the upper or lower sign has to be taken according as lies
in front of the plane P or behind it We thus get from (47)

C=|(l±l)8in^(6«T;>)-^J'V(iJ)8in^(6<-iJ)da,.

VA Oy THE DTyXMICAL TH190KT OF DDTBACTIOX.

WLea i* ]«3cr^2i*?? iiiSiiit's, /* /£; reduces iiself lo cos* «, a&i iLc

Sapp<we that no finite p rtk.n of tbe perimeter of J is a circular
arc with </ for <*titre, aD'3 I'rt this perimeter l<r conceive^i to ci-
paijd irj<J<;fiD;tely, remamiri;^ rinsilar to itselt TL^-n, for any finite
interval, how^rver >:rnalL in the iategrati >n -sriih re?peci to «. the
furi/irtion rin Sir a.'* ''^ — iZ, will change si 211 an infinite number of
timefl, liaving a mean vahie which i* uh:mate!j zer^. and the limit
of the above expression will be ri^'or>u<Iy zcr •. Hence we get in
the limit

^^csiu "- 'hi — p;, or == 0,

according aj{ p is positive or negative. Hence the disturbance
a/ntinijally tran^smitto^l acr<^iS5 the plane P produces the same
disturbance in front of that plane as if the wave had not been
broken np, and does not prrjduce any back wave, which is what
it was re^juired to verify.

It may l>e objecte<l that the supposition that the perimeter of
A is free from circular arcs having (/ for centre is an arbitrary
restriction. The reply to this objection is, that we have no right
to assume that the disturbance at which corresponds to an area
A approaches in all cases to a limit as A expands, remaining
similar to itself. All we have a ri^ht to assert a priori is, that
if it approaches a limit that limit must be the disturbance which
would exist if the wave had not been broken up.

It is hardly necessary to observe that the more general formula
(45) might have been treated in precisely the same way as (46).

35. In the third Volume of the Cambridge Mathematical
Journal^ p. 46, will be found a short paper by Mr Archibald Smith,
of which the object is to determine the intensity in a secondary
wave of light. In this paper the author supposes the intensity
at a given distance the same in all directions, and assumes the
coefficient of vibration to vary, in a given direction, inversely as
the radius of the secondary wave. The intensity is determined
on the i)rinciplo that when an infinite plane wave is conceived to
bo broken up, the aggregate effect of the secondary waves must

LAW OF DISTUBBANCE IN A SECONDARY WAVE. 289

be the same as that of the primary wave. In the investigation,
the difference of direction of the vibrations corresponding to the
various secondary waves which agitate a given point is not taken
into account, and moreover a term which appears under the form
cos 00 is assumed to vanish. The correctness of the result arrived
at by the latter assumption may be shewn by considerations simi-
lar to those which have just been developed. If we suppose the
distance from the primary wave of the point which is agitated by
the secondary waves to be large in comparison with X, it is only
those secondary waves which reach the point in question in a
direction nearly coinciding with the normal to the primary wave
that produce a sensible effect, since the others neutralize each
other at that point by interference. Hence the result will be
true for a direction nearly coinciding with the normal to the
primary wave, independently of the truth of the assumption that
the disturbance in a secondary wave is equal in all directions^
and notwithstanding the neglect of the mutual inclination of
the directions of the disturbances corresponding to the various
secondary waves. Accordingly, when the direction considered is
nearly that of the normal to the primary wave, cos 6 and sin ^
in (46) are each nearly equal to 1, so that the coefficient of the
circular function becomes cdS (Xr)"S nearly, and in passing from
the primary to the secondary waves it is necessary to accelerate
the phase by a quarter of an undulation. This agrees with Mr
Smith's results.

The same subject has been treated by Professor Eelland in a
memoir On the Theoretical Investigation of the Absolute Intensity
of Interfering Light, printed in the fifteenth Volume of the
Transactions of the Royal Society of Edinburgh, p. 315. In this
memoir the author investigates the case of a series of plane
waves which passes through a parallelogram in front of a lens,
and is received on a scieen at the focus of the lens, as well as
several other particular cases. By equating the total illumination
on the screen to the area of the aperture multiplied by the illu-
mination of the incident light, the author arrives in all cases at
the conclusion that in the coefficient of vibration of a secondary
wave the elementary area dS must be divided by Xr. In con-
sequence of the employment of intensities, not displacements, the
necessity for the acceleration of the phase by a quarter of an
undulation does not appear from this investigation*

S. IL V^

290 ON THE DYNAMICAL THEOBY OF DIFFRACTION.

In the investigations of Mr Smith and Professor Eelland, as
well as in the verification of the formula (46) given in the last
article, we are only concerned with that part of a secondary wave
which lies near the normal to the primary. The correctness of
this formula for all directions must rest on the dynamical theory.

36. In any given case of diffraction, the intensity of the
illumination at a given point will depend mainly on the mode of
interference of the secondary waves. If however the incident
light be polarized, and the plane of polarization be altered, every
thing else remaining the same, the mode of interference will not
be changed, and the coefficient of vibration will vary as sin (f>,
so that the intensity will vary between limits which are as 1 to
cos* 0. If common light of the same intensity be used, the inten-
sity of the difl^ted light at the given point will be proportional
to i (1 + cos* ^).

PART II.

EXPERIMENTS ON THE ROTATION OF THE PLANE OF
POLARIZATION OF DIFFRACTED LIGHT.

Section I.

Description of the Experiments.

If a plane passing through a ray of plane-polarized light, and
containing the direction of vibration, be called the plane of vibra-
tion, the law obtained in the preceding section for the nature of
the polarization of diffracted light, when the incident light is
plane-polarized, may be expressed by saying, that any diffracted
ray is plane-polarized, and the plane of vibration of the diffracted
ray is parallel to the direction of vibration of the incident ray.
Let the angle between the incident ray produced and the difiracted
ray be called the angle of diffraction, and the plane containing
these two rays the plane of diffraction; let a,, a^ be the angles
which the planes of vibration of the incident and diffracted rays
respectively make with planes drawn through those rays perpen-

DESCRIPTION OF THE EXPEBIMENTS. 291

dicular to the plane of diffiraction, and the angle of diffraction.
Then we easily get by a spherical triangle

tan a^ = cos tan a^

If then the plane of vibration of the incident ray be made to
turn round with a uniform velocity, the plane of vibration of the
diffracted ray will turn round with a variable velocity, the law
connecting these velocities being the same as that which connects
the sun's motions in right ascension and longitude, or the motions
of the two axes of a Hook's joint. The angle of diffraction
answers to the obliquity of the ecliptic in the one case, or the
supplement of the angle between the axes in the other. If we
suppose a series of equidifferent values given to a,, such as 0®, 5*^,
10V..355^ the planes of vibration of the diffracted ray will not be
distributed uniformly, but will be crowded towards the plane
perpendicular to the plane of diffraction, according to the law
expressed by the above equation.

Now the angles which the planes of polarization of the inci-
dent and diffracted rays, (if the diffracted ray prove to be really
plane-polarized,) make with planes perpendicular to the plane of
diffraction can be measured by means of a pair of graduated
instruments furnished with Nicol's prisms. Suppose the plane of
polarization of the incident light to be incUned at the angles
0^ 5*, 10°..., successively to the perpendicular to the plane of
diffraction ; then the readings of the instrument which is used as
the analyzer will shew whether the planes of polarization of the
diffracted ray are crowded towards the plane of diffraction or
towards the plane perpendicular to the plane of diflfraction. If «r,
a be the azimuths of the planes of polarization of the incident and
diffracted rays, both measured from planes perpendicular to the
plane of diffraction, we should expect to find these angles con-
nected by the equation tan a = sec tan «r in the former event,
and tan a = cos tan «r in the latter. If the law and amount of
the crowding agree with theory as well as could reasonably be
expected, some allowance being made for the influence of modify-
ing causes, (such as the direct action of the edge of the diffracting
body,) whose exact effect cannot be calculated, then we shall be
led to conclude that the vibrations in plane-polarized light are
perpendicular or pai*allel to the plane of polarization, according as
the crowding takes place towards or from the plane of diffraction.

19— ^

292 ON THE DYNAMICAL THEORY OF DIFFRACTION.

In all ordinary cases of diffraction, the light becomes insensible
at such a small angle from the direction of the incident ray pro-
duced that the crowding indicated by theory is too small to be
sensible in experiment, except perhaps in the mean of a very
great number, of observations. It is only by means of a fine
grating that we can obtain strong light which has been difiracted
at a large angle. I doubt whether a grating properly so called,
that is, one consisting of actual wires, or threads of silk, has ever
been made which would be fine enough for the purpose. The
experiments about to be described have accordingly been performed
with the glass grating already mentioned, which consisted of a
glass plate on which parallel and equidistant lines had been ruled
with a diamond at the rate of about 1300 to an inch.

Although the law enunciated at the beginning of this section
has been obtained for diffraction in vacuum, there is little doubt
that the same law would apply to diflfraction within a homogeneous
uncrystallized medium, at least to the degree of accuracy that we
employ when we speak of the refractive index of a substance,
neglecting the dispersion. This is rendered probable by the
simplicity of the law itself, which merely asserts that the vibra-
tions in the difiracted light are rectilinear, and agree in direction
with the vibrations in the incident light as nearly as is consistent
with the necessary condition of being perpendicular to the dif-
fracted ray. Moreover, when dispersion is neglected, the same
equations of motion of the luminiferous ether are obtained, on
mechanical theories, for singly refracting media as for vacuum; and
if these equations be assumed to be correct, the law under con-
sideration, which is deduced from the equations of motion, will
continue to hold good. In the case of a glass grating however the
diffraction takes place neither in air nor in glass, but at the
confines of the two media, and thus theory fails to assign exact
values to a. Nevertheless it does not fail to assign limits within
which, or at least not far beyond which, a must reasonably be
supposed to lie ; and as the values comprised within these limits
are very different according as one or other of the two rival
theories respecting the direction of vibration is adopted, experi-
ments with a glass grating may be nearly as satisfactory, so far as
regards pointing to one or other of the two theories, as experiments
would be which were made with a true grating.
^ The glass grating was mounted for me by Prof. Miller in a

DESCRIPTION OF THE EXPERIMENTS. 293

small frame fixed on a board which rested on three screws, by
means of which the plane of the plate and the direction of the
grooves could be rendered perpendicular to the plane of a table on
which the whole rested.

The graduated instruments lent to me by Prof. O'Brien con-
sisted of small graduated brass circles, mounted on brass stands, so
that when they stood on a horizontal table the planes of the circles
were vertical, and the zeros of graduation vertically over the
centres. The circles were pierced at the centre to admit doubly
refracting prisms, which were fixed in brass collars which could be
turned round within the circles, the axes of motion being perpen-
dicular to the planes of the circles, and passing through their
centres. In one of the instruments, which I used for the polarizer,
the circle was graduated to degrees from 0® to 360®, and the collar
carried simply a pointer. To stop the second pencil, I attached a
wooden collar to the brass collar, and inserted in it a Nicol's
prism, which was turned till the more refracted pencil was extin-
guished. In a few of the latest experiments the Nicol's prism was
dispensed with, and the more refracted pencil stopped by a screen
with a hole which allowed the less refracted pencil to pass. In the
other instrument, which I used for the analyzer, the brass collar
carried a vernier reading to 5'. In this instrument the doubly
refracting prism admitted of being removed, and I accordingly
removed it, and substituted a Nicol's prism, which was attached
by a wooden collar. The Nicol's prism was usually inserted into
the collar at random, and the index error was afterwards deter-
mined from the observations themselvea

The light employed in all the experiments was the sun light
reflected from a mirror placed at the distance of a few feet from
the polarizer. On account of the rotation of the earth, the mirror
required re-adjustment every three or four minutes. The continual
change in the direction of the incident light was one of the chief
sources of difficulty in the experiments and inaccuracy in the
results ; but lamplight would, I fear, be too weak to be of much
avail in these experiments.

The polarizer, the grating, and the analyzer stood on the same
table, the grating a few inches from the polarizer, and the analyzer
about a foot from the grating. The plane of diflfraction was as-
sumed to be parallel to the table, which was nearly the case;
but the change in the direction of the incident light produced

294 ON THE DYNAMICAL THEORY OP DIFFRACTION.

continual small changes in the position of this plane. In most
experiments the grating was placed perpendicular to the incident
light, by making the light reflected from the surface go back into
the hole of the polarizer. The angle of diffraction was measured
at the conclusion of each experiment by means of a protractor,
lent to me for the purpose by Prof. Miller. The grating was
removed, and the protractor placed with its centre as nearly as
might be under the former position of the bright spot formed on
the grating by the incident light. The protractor had a pair of
opposite verniers moveable by a rack ; and the directions of the
incident and diffiracted light were measured by means of sights
attached to the verniers. The angle of diffraction in the different
experiments ranged from about 20® to 60^

The deviation of the less refracted pencil in the doubly re-
fracting prism of the polarizer, though small, was very sensible,
and was a great source both of difficulty and of error. To under-
stand this, let AB be a ray incident at i? od a slip of the surface of
the plate contained between two consecutive grooves, BC a dif-
fracted ray. On account of the interference of the light coming
from the different parts of the slip, if a small pencil whose axis is
AB be incident on the slip, the diffiracted light will not be sensible
except in a direction BC, determined by the condition that AB 4-
BC shall be a minimum, A and C being supposed fixed. Hence
AB, BC must make equal angles with the slip, regarded as a line,
the acute angles lying towards opposite ends of the slip, and there-
fore G must lie in the surface of a cone formed by the revolution
of the produced part of AB about the slip. If AB represent the
pencil coming through the polarizer, it will describe a cone of
small angle as the pointer moves round, and therefore both the
position of the vertex and the magnitude of the vertical angle of
the cone which is the locus of C will change. Hence the sheet of
the cone may sometimes fall above or below the eye-hole of the
analyzer. In such a case it is necessary either to be content to
miss one or more observations, corresponding to certain readings of
the polarizer, or else to alter a little the direction of the incident
light, or, by means of the screws, to turn the grating through a
small angle round a horizontal axis. The deviation of the light
which passed through the polarizer, and the small changes in
the direction of the incident light, I regard as the chief causes
of error in my experiments. In repeating the experiments so

DESCRIPTION OF THE EXPERIMENTS. 295

as to get accurate results^ these causes of error would have to
be avoided.

At first 1 took for granted that the instrument-maker had
inserted the doubly refracting prism in the polarizer in such a
manner that the plane of polarization of the less refracted pencil
was either vertical or horizontal, (the instrument being supposed
to stand on a horizontal table,) when the pointer stood at 0^, having
reason to know that it was not inserted at random ; and having
determined which, by an exceedingly rough trial, I concluded it
was vertical. Meeting afterwards with some results which were
irreconcileable with this supposition, I was led to make an actual
measurement, and found that the plane of polarization was vertical
when the pointer stood at 25°. Consequently 25® is to be regarded
as the index error of the polarizer, to be subtracted from the
reading of the pointer. The circumstance just mentioned accounts
for the apparently odd selection of values of «r in the earlier
experiments, the results of which are given in the tables at the end
of this section.

On viewing a luminous point or line through the grating, the
central colourless image was seen accompanied by side spectra,
namely, the spectra which Fraunhofer called Spectra of the second
class. After a little, these spectra overlapped in such a manner
that the individual spectra could no longer be distinguished, and
nothing was to be seen but two tails of light, which extended, one
on each side, nearly 90® from the central image. On viewing the
flame of a spirit lamp through the grating, the individual spectra
of the second class could be seen, where, with sun-light, nothing
could be perceived but a tail of light. The tails themselves were
not white, but exhibited very broad impure spectra ; about two
such could be made out on each side. These spectra are what
were called spectra of the first class by Fraunhofer, who shewed
that their breadth depended on the smaller of the two quantities,
the breadth of a groove, and the breadth of the polished interval
between two consecutive grooves. In the grating, the breadth of
the grooves was much smaller than the breadth of the intervals
between*.

* On viewing the grating nnder a microscope, the grooyes were easily seen to be
much narrower than the intervals between; their breadth was too small to be
measured. On looking at the flame of a spirit lamp through the grating, I counted
sixteen images on one side, then several images were too faint to be seen, and

296 ON THE DYNAMICAL THEORY OF DIFFRACTION.

In the experiments, the diflfracted light observed belonged to a
bright, though not always the brightest, part of a spectrum of the
first class. The compound nature of the light was easily put in
evidence by placing a screen with a vertical slit between the
grating and the eye, and then viewing the slit through a prism
with its edge vertical*. A spectrum was then seen which con-
sisted of bright bands separated by dark intervals, strongly resem-
bling the appearance presented when a pure spectrum is viewed
through a pinhole, or narrow slit, which is half covered by a plate
of mica, placed on the side at which the blue is seen. At a con-
siderable angle of diffraction as many as 15 or 20 bands might
be counted.

In the first experiment the grating was placed with its
plane perpendicular to the light which passed through the pola-
rizer, the grooved face being turned from the polarizer. The
light observed was that which was diffracted at emergence
from the glass. It is only when the eye is placed close to
the grating, or when, if the eye be placed a few inches oflf, the
whole of the grating is illuminated, that a large portion of a tail of
light can be seen at once. When only a small portion of the
grating is illuminated, and the eye is placed at the distance of
several inches, as was the case in the experiments, it is only a
small portion of a tail which can enter the pupil. The appearance
presented is that of a bright spot on the grooved face of the glass.
The angle of diffraction in the first experiment was large, 57® 5' by
measurement Besides the principal image, or bright spot, a row
of images were seen to the left: the regularly transmitted light
lay to the right, right and left being estimated with reference to
the position of the observer. These images were due to internal
diffraction and reflection, as will be better understood further on.

farther still the images again appeared, though they were fainter than before.
I estimated the direction of zero illumination to be situated about the eighteenth
image. If we take this estimation as correct, it follows from the theory of these
gratings that the breadth of a groove was the eighteenth part of the interval
between any point of one groove and the corresponding point of its consecutive, an
interval which in the case of the present grating was equal to the l-1300th part of
an inch. Hence the breadth of a groove was equal to the l-28400th part of an
inch.

* To separate the different spectra, Fraimhofer used a small prism with an
angle of about 2(fi, fixed with its edge horizontal in front of the eye-piece of the
telescope through which, in his experiments, the spectra were viewed.

DESCRIPTION OF THE EXPERIMENTS. 297

They were separated by small angles, depending on the thickness
of the glass, but sufficient to allow of one image being observed by
itself. The observations were confined to the principal or right-
hand image.

In the portion of a spectrum of the first class which was
observed there was a predominance of red light. In most posi-
tions of the pointer of the polarizer the diffracted light did not
wholly vanish on turning round the analyzer, but only passed
through a minimum. In passing through the minimum the light
rapidly changed colour, being blue at the minimum. This shews
that the different colours were polarized in different planes, or
perhaps not strictly plane-polarized. Nevertheless, as the intensity
of the light at the minimum was evidently very small compared
with its intensity at the maximum, and the change of colour was
rapid, it is allowable to speak in an approximate way of the plane
of polarization of the diffracted light, just as it is allowable to
speak of the refractive index of a substance, although there is
really a different refractive index for each different kind of light.
It was accordingly the angular position of the plane which was the
best representative of a plane of polarization that I sought to
determine in this and the subsequent experiments.

In the first experiment the plane of polarization of the dif-
firacted light was determined by six observations for each angle at
which the pointer of the polarizer was set. This took a good deal
of time, and increased the errors depending on changes in the
direction of the light. Accordingly, in a second experiment, I
determined the plane of polarization by single observations only,
setting the pointer of the polarizer at smaller intervals than
before. Both these experiments gave for result that the planes
of polarization of the diffiracted light were distributed very
nearly uniformly. This result already points very decidedly
to one of the two hypotheses respecting the direction 'of
vibration. For according to theory the effect of diffraction alone
would be, greatly to crowd the planes either in one direction or in
the other. It seems very likely that the effect of oblique emer-
gence alone should be to crowd the planes in the manner of
refraction, that is, towards the perpendicular to the plane of dif-
fraction. If then we adopt FresneVs hypothesis, the two effects
will be opposed, and may very well be supposed wholly or nearly
to neutralize each other. But if we adopt the other hypothesis we

^

298 ON THE DTNAMICAL THEORY OF DIFFRACTION.

shall be obliged to suppose that in the oblique emergence from the
glass, or in something else, there exists a powerful cause of crowd-
ing towards the plane of dififraction, that is, in the manner of re-
flection, suflScient to neutralize the great crowding in the contrary
direction produced by diflBraction, which certainly seems almost
incredible.

The nearly uniform distribution of the planes of polarization of
the diffracted light shews that the two streams of light, polarized
in and perpendicular to the plane of diffraction respectively, into
which the incident light may be conceived to be decomposed, were
diffracted at emergence from the glass in very nearly the same
proportion. This result appeared to offer some degree of vague
analogy with the depolarization of light produced by such sub-
stances as white paper. This analogy, if borne out in other cases,
might seem to throw some doubt on the conclusiveness of the
experiments with reference to the decision of the question as to
the direction of the vibrations of plane-polarized light. For the
deviation of the light from its regular course might seem due
rather to a sort of scattering than to regular diffiraction, though
certainly the fact that the observed light was very nearly plane-
polarized does not at all harmonize with such a view. Accord-
ingly, I was anxious to obtain a case of diffraction in which the
planes of polarization of the diffracted light should be decidedly
crowded one way or other. Now, according to the explanation
above given, the approximate uniformity of distribution of the
planes of polarization in the first two experiments was due to
the antagonistic effects of diffraction, (according to Fresnel's
hypothesis respecting the direction of vibration), and of oblique
emergence from the glass, or irregular refraction, that is, refraction
produced wholly by diffraction. If this explanation be correct,
a very marked crowding towards the plane of diffraction ought
to be produced by diffraction at reflection, since diffraction
alone and reflection alone would crowd the planes in the same
manner.

To put this anticipation to the test of experiment, I placed the
grating with its plane perpendicular to the incident light, and the
grooved face towards the polarizer, and observed the light which
was diffracted at reflection. Since in this case there would be no
crowding of the planes of polarization in the regularly reflected
light, any crowding which might be observed would be due either

DESCRIPTION OF THE EXPERIMENTS. 299

to diflfraction directly, or to the irregular reflection due to diflfrac-
tioD, or, far more probably, to a combination of the two.

The experiments indicated indeed a marked crowding towards
the plane of diflfraction, but the light was so strong at the mini-
mum, for most positions of the pointer of the polarizer, that the
observations were very uncertain, and it was evidently only a
rough approximation to regard the diflfracted light as plane-pola-
rized. The reason of this was evident on consideration. Of the
light incident on the grating, a portion is regularly reflected,
forming the central image of the system of spectra produced by
diflfraction at reflection, a portion is diflfracted externally at such
an angle as to enter the eye, a small portion is scattered, and the
greater part enters the glass. Of the light which enters the glass,
a portion is diflfracted internally at such an angle that after regular
reflection and refraction it enters the eye, a portion diflfracted at
other angles, but the greater part falls perpendicularly on the
second surface. A portion of this is reflected to the first surface,
and of the light so reflected a portion is diflfracted at emergence
at such an angle as to enter the eye. Thus there are three princi-
pal images, each formed by the light which has been once diffracted
and once reflected, the externally diflfracted light being considered
as both diflfracted and reflected, namely, one which has been dif-
fracted internally, and then regularly reflected and refracted, a
second in which the light has been regularly refracted and reflected,
and then diflfracted at emergence, and a third in which the light
has been diflfracted externally. Any other light which enters the
eye must have been at least twice diflfracted, or once diflfracted and
at least three times reflected, and therefore will be comparatively
weak, except perhaps when the angle of incidence, or else the
angle of diflfraction, is very large. Now when the grating is per-
pendicular to the incident light the second and third of the
principal images are necessarily superposed; and as they might be
expected to be very differently polarized, it was likely enough that
the light arising from the mixture of the two should prove to be
very imperfectly polarized.

To separate these images, I placed a narrow vertical slit in
front of the grating, between it and the polarizer, and inclined
the grating by turning it round a vertical axis so that the normal
fell between the polarizer and the analyzer. As soon as the
grating was inclined, the image which had been previously

300 ON THE DYNAMICAL THEORY OF DIFFRACTION.

observed separated into two, and at a certain inclination the
three principal images were seen equidistant. The middle image,
which was the second of those above described, was evidently
the brightest of the three. The three images were found to be
nearly if not perfectly plane-polarized, but polarized in different
planes. The third image, and perhaps also the first, did not
wholly vanish at the minimum. This might have been due
to some subordinate image which then appeared, but it was more
probably due to a real defect of polarization.

The planes of polarization of the side images, especially the
first, were greatly crowded towards the plane of diffraction, or,
which is the same, the plane of incidence. Those of the middle
image were decidedly crowded in the same direction, though
much less so than those of the side images. The light of the
fiirst and second images underwent one regular refraction and
one regular reflection besides the diffraction and the accompany-
ing irregular refraction. The crowding of the planes of polari-
zation in one direction or the other produced by the regular
refraction and the regular reflection can readily be calculated
from the known formulae*, and thus the crowding due to diffrac-
tion and the accompanying irregular refraction can be deduced
from the observed result

The crowding of the planes of polarization of the third image
is due solely to diffraction and the accompanying irregular
reflection. The crowding in one direction or the contrary, ac-
cording as one or other hypothesis respecting the direction of
vibrations is adopted, is readily calculated from the dynamical
theory, and thus is obtained the crowding which is left to be
attributed to the irregular reflection. In the absence of an exact
theory little or no use can be made of the result in the way of
confirming either hjrpothesis; but it is sufiicient to destroy the
vague analogy which might have been formed between the effects
of diffraction and of irregular scattering.

The crowding of the planes of polarization of the middle
image, after the observations had been reduced in the manner
which will be explained in the next section, appeared somewhat

* It is here supposed that the regalarlj reflected or refracted light which foixna
the central colourless image hclonging to a system of spectra is affected as to its
polarization in the same way as if the surface were free from grooves.

DESCRIPTION OP THE EXPERIMENTS. 301

greater than was to have been expected from the first two
experiments. This led me to suspect that the crowding in the
manner of reflection produced by diflfraction accompanying the
passage of light from air, across the grooved surface, into the
glass plate, might be greater than the crowding had proved to
be which was produced by diffraction accompanying the passage
from glass, across the grooved surface, into air. I accordingly
placed the grating with its plane perpendicular to the incident
light, and the grooved face towards the polarizer, and placed the
analyzer so as to receive the light which was difiracted in passing
across the first surface, and then regularly refracted at the second.
I soon found that the planes of polarization were very decidedly
crowded towards the plane of diffraction, and that, notwithstand-
ing the crowding in the contrary direction which must have been
produced by the regular refraction at the second surface of the
plate, and the crowding, likewise in the contrary direction, which
might naturally be expected to result from the irregular refraction
at the first surface, considered apart from diffraction. This result
seemed to remove all doubt respecting the hypothesis as to the
direction of vibration to which the experiments pointed as the
true one.

On account of the decisive character of the result just men-
tioned, I took several sets of observations on light diffracted in
this manner at different angles. I also made two more careful
experiments of the same nature as the first two. The result
now obtained was, that there was a very sensible crowding
towards the plane of diffraction when the grooved face was turned
from the polarizer, although there was evidently a marked differ-
ence between the two cases, the crowding being much less than
when the grooved face was turned towards the polarizer. Even
the first two experiments, now that I was aware of the index
error of the polarizer, appeared to indicate a small crowding in
the same direction.

Before giving the numerical results of the experiments, it may
be as well to mention what was observed respecting the defect
of polarization. I would here remark that an investigation of
the precise nature of the diffracted light was beside the main
object of my experiments, and only a few observations were taken
which belong to such an investigation. In what follows, w
denotes the inclination of the plane of polarization of the light

\

302 ON THE DYNAMICAL THEORY OF DIFFRACTION.

incident on the grating to a vertical plane passing through the
ray, that is, to a plane perpendicular to the plane of diflfraction.
It is given by the reading of the pointer of the polarizer corrected
for the index error 25", and is measured positive in the direction
of revolution of the hands of a watch placed with its back towards
the incident light.

Whether the diffraction accompanied reflection or refraction,
external or internal, the dififracted light was perfectly plane-
polarized when «r had any one of the values 0^ 90^ 180°, or
270^ The defect of polarization was greatest about 45** from any
of the above positions. When the dififracted light observed was
red or reddish, on analyzation a blue light was seen at or near
the minimum; when the dififracted light was blue or blueish,
a red light was seen at or near the minimum. When the angle
of dififraction was moderately small, such as 15° or 20°, the defect
of polarization was small or insensible; when the angle of
dififraction was large, such as 50° or 60°, the defect of polarization
was considerable. For equal angles of dififraction, the defect of
polarization was much greater when the grooved face was turned
towards the polarizer than when it was turned in the contrary
direction. By the term angU of diffraction^ as applied to the
case in which the grooved face was turned towards the polar-
izer, is to be understood the angle measured in air, from which
the angle of dififraction within the glass may be calculated, from
a knowledge of the refractive index.

The grating being placed perpendicularly to the incident light,
with the grooved face towards the polarizer, the light dififracted at
a considerable angle, (59° 52' by measurement,) to the left of the
regularly transmitted light was nearly white. When the pointer
of the polarizer stood at 70°, so that w = -f 45^ on turning the
NicoFs prism of the analyzer in the positive direction through the
position of minimum illumination, the light became in succession
greenish yellow, blue, plum colour, nearly red. When w was
equal to - 45°, the same appearance was presented on reversing
the direction of rotation. Since the colours appeared in the order
blue, red, when -07 = + 45°, and in the order red, blue, when
tir = — 45°, the analyzer being in both cases supposed to turn in the
direction of the hands of a watch, the deficiency of colour took
place in the order red, blue, when vr — + 45°, and in the order blue,
red, when «r = - 45°. Hence the planes of polarization, or approxi-

DESCBIFTION OF THE EXPERIMENTS. 303

mate polarization, of the blue were more crowded towards the
plane of diflfraction than those of the red.

On placing a narrow slit so as to allow a small portion only of
the diflBracted light to pass, and decomposing the light by a prism,
in the manner already described, so as to get a spectrum consisting
of bright bands with dark intervals, and then analyzing this spec-
trum with a Nicol's prism, it was found that at a moderate angle
of difiraction all the colours were sensibly plane-polarized, though
the planes of polarization did not quite coincide. At a large angle
of diffraction the bright part of the spectrum did not quite dis-
appear on turning round the NicoFs prism, while the red and blue
ends, probably on account of their less intensity, appeared to be
still perfectly plane-polarized, though not quite in the same plane.
On treating in the same manner the diflPracted light produced
when the grooved face of the glass plate was turned from the
polarizer, all the colours appeared to be sensibly plane-polarized.
In the former case the light of the brightest part of the spectrum
waa made to disappear, or nearly so. by using a thin plate of mica
in combination with the NicoFs prism, which shews that the defect
of plane polarization was due to a slight elliptic polarization.

The numerical results of the experiments on the rotation of the
plane of polarization are contained in the following table. In this
table "BJ is the reading of the polarizer corrected for the index
error 25°. A reading such as 340° is entered indiflferently in the
column headed "tsr" as +315° or -45°, that is, 340° -25° or
— (360° — 340°) — 25°. a is the reading of the analyzer, determined
by one or more observations. The analyzer was graduated only
from — 90° to -f 90°, and any reading such as — 20° is entered
indififerently as — 20°, -f 160°, or + 340°, being entered in such a
manner as to avoid breaking the sequence of the numbers. On
account of the light left at the minimum, the determination of a
was very uncertain when the angle of dififraction was large, except
when vr had very nearly one of the values 0°, 90°, 180°, or 270°.
In the most favourable circumstances the mean error in the deter-
mination of a was about a quarter of a degree. In some of the
experiments a red glass was used to assist in rendering the obser-
vations more definite. This had the advantage of stopping all
rays except the red, but the disadvantage of considerably diminish-
ing the intensity of the light. The minutes in the given value of
0, the angle of diffiraction, cannot be trusted ; in fact^ during any

304 ON THE DYNAMICAL THEORY OF DIFFRACTION.

experiment was liable to changes to at least that extent in con-
sequence of the changes in the direction of the light The same
remark applies to i, the angle of incidence, in experiments 11 and
12. In these experiments the three principal images already
described were observed separately. The angle of dififraction is
measured from the direction of the regularly reflected ray, so that
t is the angle of incidence, and i + d the angle of reflection, or, in
the case of the images which sufiered one internal reflection, the
angle of emergence.

The eleven experiments which are not found in the following
tables consist of five on dififraction by reflection, which did not
appear worth giving on account of the supei-position of diflferent
images ; one on dififraction by refraction, to which the same remark
applies, the grating having been placed at a considerable distance
from the polarizer, so that the spot illuminated was too large to
allow of the separate observation of dififerent images; one on
dififraction by reflection, in which the grating was placed perpen-
dicularly to the incident light, with the grooved face turned from
the polarizer, but the errors of observation, though much smaller
than the whole quantity to be observed, were so large on account
of the large angle of dififraction, (about 75^,) with which the obser-
vations were attempted, that the details are not worth giving ; one
on dififraction by refraction, in which the diflferent observations
were so inconsistent that the experiment seemed not worth reduc-
ing ; one which was only just begun ; and two qualitative experi-
ments, the results of which have been already given. I mention
this that I may not appear to have been biassed by any particular
theory in selecting the experiments of which the numerical results
are given.

The following remarks relate to the particular experiments :

No. 1. In this experiment each value of a was determined by
six observations, of which the mean error* ranged from about 15'

* The difference between each individual observation and the mean of the six is
regarded as the error of that observation, and the mean of these differences taken
positively is what is here called the mean error. When two observations only are
taken, the mean error is the same thing as the semi-difference between the observa-
tions. Since, for a given position of the pointer of the polarizer, the readings
of the analyzer were usually taken one immediately after another, the mean error
famishes no criterion by which to judge of the errors produced by the small
ohanges in the direction of the light incident on the grating, but only of those
which arise from the va^eness of the object observed. The reader will be maoh

DESCRIPTION OF THE EXPERIMENTS. 305

to 55'. So far the experiment was very satisfactory, but it was
vitiated by changes in the direction of the light, sufficient care not
having been taken in the adjustment of the mirror.

No. 2. a determined by single observations.

No. 13. a determined by two observations at least, of which
the mean error ranged from about 10' to nearly 1", but was usually
decidedly less than 1**. At and about the octants, that is to say,
when w was nearly equal to 45^ or an odd multiple of 45^ the
light was but very imperfectly polarized in one plane.

No. 14. a determined by two observations. Marked in note
book as " a very satisfactory experiment." The mean of the mean
errors was only 11'.

No. 15. a determined by three observations at least. The
light was very imperfectly polai'ized, except near the standard
points, that is to say when «r was equal to 0" or 90^ or a multiple
of 90°. This rendered the observations very uncertain. About
the octants the mean error in a set of observations taken one
immediately after another amounted to near 2^

No. 17. a determined by two observations. The light was
very imperfectly polarized, except near the standard points. Yet
the observations agreed very fairly with one another. The mean
of the mean errors was 25', and the greatest of them not quite V.

No. 18. a determined by two observations, which, generally
speaking, agreed well with one another. For «r = — 90° and
«r = + 225° the light observed was rather scattered than regularly
diffracted, the sheet of the cone of illumination having fallen above
or below the hole of the analyzer.

No. 21. a determined by two observations at least. In this
experiment the polarizer was covered with red glass.

No. 22. a determined by two observations. Marked in note
book as " a very satisfactory experiment, though the light was not
perfectly polarized."

No. 23. a determined by two observations at least. The hole
in a screen placed between the polarizer and the grating was
covered with red glass. This appears to have been a good experi-
ment.

better able to judge of the amount of probable error from all causes after examining
the reduction of the cxperimentp given in the next section.

S. II. 20

306 ON THE DTNAUICAL THEOBT OF DIPFBACTION.

No. 11. o determined by two observatiooe, which agreed well
with one aoother. In the table, a (1), a (2), a (3) refer respec-
tively to the first, second, and third of the three principal im^es
already mentioned. In this experiment the polarizer was reversed,
that face being tvimed towards the mirror which in the other
experimenta was turned towards the grating, which is the reason
why a and w increase together, although the light observed
suETered one reflection. The same index error as before, namely
25°, is supposed to belong to the polarizer in its reversed position.

Ko. 12. a determined by three observations. The largeness
of the angle of diffraction rendered the determination of a very
uncertain.

. 1 a

^-

a

^ i »

OT a

H:. IS, codllBuod.

EipcrlmoDt. No. L

N.=.«,.«.ued.

'^""l-^^"/'""

- 30"

- 116°5.7

OrooTod fue from

6"

+ 2o»ao'

^ 40"

-121»25'

J-oUrUvr.
ff=67°5'.

i

6"
15"
25"

+ SO'oS'
+ WoS'
+ 60"46'

iJ°57'.
+ 22"25'

- 5(f>

- 60"

- 70°

-133"4l'
-140"29'
-14«"18'

- 70"il'

- oVaff

- 6"52'

+
+

SK-
IS"
65"

+ CI "43'
+ 70°55'
+ 82"15'

- 40"

- 30"

- 20°

- 10"

0°

+ 31"16'
+ 4F40'
+ 61°5S'

+ oa"37'

+ 71°10'

- 80»

- ilO"

-159'50'
-158°30'

GFoi>vi>d («» lovurdi

- 25"

+ 14°31'

+ 10"

+ HIHT

i-oUrim.

" Ji'

+ 37"5r

Eiperim

nt, No. IJ.

+ 30"

+ m-iT

e=50°15'.

+ 30"

■1- 61" 6'

jj

hM UmotJi

+ 30°

+ I03°10'

- 00"

+ 77^5'

+ ltlB''J6'

6=

89"S0'.

+ 10°
+ 60°

+ 113"15'

+ i2a"ia'

- 800

- 70"

+ 86°S0'
+ 93°12'

+ 60"

+ 1S2"43'

- 60°

+ 101"15'

-

80"
60"
40"

- 6" 6'

+ 70"
+ 80"

+ n;i°
+ 153"47'

- 600

- 10°

+ 109"47'
+ 117"ia'

lulartKr.

+ 4"53'
+ IS'o?

+ 90"
+ 100"

+ 16P6r
+ 171°d2'

- 30*

+ 129*5r

-

30°

+ 36"

+ 110"

+ 18a''69'

Eipnimcnl. No. tS.

fl = 50°23'.

20°

+ 33""i3'

+ 120"

+ 191"47'

''""'Pi.X'r™'*

10"

+ 46" 6'

+ 130"

+ 202"12'

-105'
~ 85"
- 85"

- 80°

- 70^'

- 6I°lfi'

0"
10"

ao"

+ 56"35'
*- 67"50'
+ 76"5e'

+ 140"

+ 211"J2'

fi = 21'39'.

- 90°

~103"28'

- 75"

- 51"3l>'

30"

+ H7"5S'

Eiperiiuedl, No. IS.

- 15"

- o9"58'

- 65"

- ll'lff

Hfi

+ 99"27'

e = 69"52'.

0"

- 12*68'

- 65"

- 2U"I5'

60"

+ 10a°30'

+ 45"

+ as^sT"

- *6"

- 20" 5'

GO"

+ J20"35'

+ 90»

+ 77W

- ar,»

- 9''55'

70°

+ 139" r

0° - 6fl"10'

+ VAC"

+iao»2'

- 26"

+ O-iO'

80°

+ l»7"l-2'

- 10° - 81"

+ 190°

+ lB7»fi7'

- 15"

+ 10»1S'

W

+ H6°57'

- 20° - 92-23'

+ 225" 1 +214"10' 1

DISCUSSION OF THE NUBfERICAL BESULTS.

307

Table I. (corUinusd).

«r

Experiment, No. SI.

Grooved face towards
Polviier.

Red glau used.

+

+

+
+

900
750
600
450
300
160
00
160
300
460
600
750
900

290
160 V
2012'
12035'
27052'

44^47'
6I04O'
78025'
92018'

+ 107025'

+ I2203O'

+ 1370

+ 161032'

+
+
+
+
+
+

Experiment, No. 22.

Grooved face /rom
Polariier.

d= 55038'.

1800
1650
1600

- 1870 2*
- 170037'
- I6403O'

No. 22, continued.
1350

1200
1060
900
750
600
450
800
150

- 140026'
- 124045'
-IIO04O'

- 96056'

- 83032'

- 690 r

- 54050'

- 88056'

- 22060'

Experiment, Na 2S.

Grooved face tawatdt
Polarizer.

Red glaas uaed.

5=54058',

00
160
300
460
600
750
900
+ 1050
+ 1200
+ 1860

+
+
+
+
+
+

- 603O'
+ 110 6'
+ 27055'
+ 42030^
+ 68022'
+ 710 5'
+ 83022'
+ 96012'
+ 108*^30'
+ 122<*45'

vr

a(l) a (2)

«(3)

Experiment, No. IL
»=«14060'; tf =22030'*

-1050

- 850

- 660

- 460

- 250

- 50
150
350
650

+
+
+

- 118086'
- 1030 6'

- 900

- 78040'

- 6805O'

- 250 6'
18015'
88035'
6305O'

+
+
+

-II705O'
-1010

- 880 5/

- 63066'

- 440

- 21010*
1025'

240 5'
4301O'

+
+
+

+
+
+

102020^
890
7406O'
68019'
2301O'
7066'
320
61030*

260

450

900

1350

Bxperimeut, No. 12.
t=901'; 5=63039',

+ 5035'

+ 150

+ 26016'

+ 34030'

—

820

—

13045'

—

904O'

+

20

+

26015'

+

26015'

+

650

+

61015'

Section II.

Discussion of the numerical results of ike experimeniSi with

referefnce to theory.

According to the known formulae which express the laws of the
rotation of the plane of polarization of plane-polarized light which
has undergone reflection or refraction at the surface of a trans-
parent uncrystalliz^d medium, if «r, a' be the azimuths of the
planes of polarization of the incident and reflected or refracted
light, both measured from planes perpendicular to the plane of
incidence, they are connected by the equation

tan a' = m tanw (48),

where m is constant, if the position of the surface and the direc-
tions of the rays be given, but is a different constant in the two
cases of reflection and refraction. According to the theory de*

20—2

308 ON THE DYNAMICAL THEORY OF DIFFRACTION.

veloped in this paper, the same law obtains in the case of diffrac-
tion in air, or even within an uncrystallized medium, but m has a
value distinct from the two former. It seems then extremely
likely that the same law should hold good in the case of that
combination of diffraction with reflection or refraction which exists
when the diffraction takes place at the common surface of two
transparent uncrystallized media, such as air and glass. If this be
true, it is evident that by combining all the observations belonging
to one experiment in such a manner as to get the value of m which
best suits that experiment, we shall obtain the crowding of the
planes of polarization better than we could from the direct obser-
vations, and we shall moreover be able in this way easily to
compare the results of different experiments. It seems reasonable
then to try in the fii'st instance whether the formula (48) will
represent the observations with sufficient accuracy.

In applying this formula to any experiment, there are two
unknown quantities to be determined, namely, m, and the index
error of the analyzer. Let e be this index error, so that a = a' + 6.
The regular way to determine e and m would no doubt be to
assume an approximate value e^ of e, put e = Cj + Ae^, where Ac, is
the small error of €^, form a series of equations of which the

type is

tan (a — e^) — sec' (a — ej Acj = m tan «r,

and then combine the equations so as to get the most probable
values of Aej and m. But such a refinement would be wholly
unneciessary in the case of the present experiments, which are
confessedly but rough. Moreover e can be determined with accu-
racy, except so far as relates to errors produced by changes in the
direction of the light, by means of the observations taken at the
standard points, the light beiug in such cases perfectly polarized.
By accuracy is here meant such accuracy as experiments of this
sort admit of, where a set of observations giving a mean error of a
quarter of a degree would be considered accurate. Besides, when-
ever the values of -cr selected for observation are symmetrically
taken with respect to one of the standard points, a small error in €
would introduce no sensible error into the value of m which would
result from the experiment, although it might make the formula
appear in fault when the only fault lay in the index error.

Accordingly I have determined the index error of the analyzer
in a way which will be most easily explained by an example.

DISCUSSION OF THE NUMERICAL RESULTS. 309

Suppose the values of a to have been determined by experiment
corresponding to the following values of w, — 15®, 0", + 15®,.,. + 75®,
+ 90^ + 105^ The value of a for isr = 0®, and the mean of the
values for — 15® and «r = + 15®, furnish two values of e; and the
value of a for or = + 90®, and the mean of the values for at = + 75®
and w = + 105®, furnish two values of e + 90®. The mean of the
four values of e thus determined is likely to be more nearly
correct than any of them. In some few experiments no two
values of tsr were symmetrically taken with respect to the stand-
ard points. In such cases I have considered it sufficient to take
proportional parts for a small interval. Thus if a^, a, be the
readings of the analyzer for w = — 10®, «r = + 5®, assuming

a^ = €- 10® - 2a:, a, = e + 5® + ir, we get 3a? = a, - a^ - 15®,

whence e, which is equal to a, — 5® — x, is known. The index
error of the analyzer having been thus determined, it remains to
get the most probable value of m from a series of equations of the
form (48). For facility of numerical calculation it is better to put
this equation under the form

logm = log tana — log tan «r (49),

where it is to be understood that the signs of a and vr are to be
changed if these angles should lie between and — 90®, or their
supplements taken if they should lie between + 90® and + 180®.
Now the mean of the values of log m determined by the several
observations belonging to one experiment is not at all the most
probable value. For the error in log tan a' produced by a small
given error in a' increases indefinitely as a' approaches indefinitely
to 0® or 90®, so that in this way of combining the observations an
infinite weight would be attributed to those which were taken
infinitely close to the standard points, although such observations
are of no use for the direct determination of log m, their use being
to determine e. Let a + Aa be the true angle of which a' is the
approximate value, a! being deduced from the observed angle a
corrected for the assumed index error e. Then, neglecting (Aa')',
we get for the true equation which ought to replace (49),

2lfAa'
log m = log tan a + -: — rr - log tan "bj,

M being the modulus of the common system of logarithms. Since
the eflfect of the error Aa is increased by the division by sin 2i', a
quantity which may become very small, in combining the equatio\^

810 ON THE DYNAMICAL THEORY OF DIFFRACTION.

such as (49) I have first multiplied the several equations by
sin 2a', or the sine of 2 (a — c) taken positively, and then added
together the equations so formed, and determined log m from the
resulting equation. Perhaps it would have been better to have
used for multiplier sin* 2^', which is what would have been given
by the rule of least squares, if the several observations be supposed
equally liable to error ; but on the other hand the use of sin 2a for
multiplier instead of sin* 2a' has the effect of diminishing the
comparative weight of the observations taken about the octants,
where, in consequence of the defect of polarization, the observa-
tions were more uncertain.

The following table contains the result of the reduction of the
experiments in the way just explained. The value of € used in
the reduction, and the resulting value of log m, are written down
in each case. The second column belonging to each experiment
gives the value of a' — «r calculated from (49) with the assumed
value of log m, and is put down for the sake of comparison with
the value of a — «r deduced from the difference, a — «r, of the
observed angles a, vr, corrected for the assumed index error e. In
the table, the experiments are arranged in classes, according to
their nature, and those belonging to the same class are arranged
according to the values of 0. The first three experiments in the
table relate to diffraction at refraction, in which the grooved face
of the grating was turned from the polarizer, the next six to
diffraction at refraction, in which the grooved face was turned
towards the polarizer, and the last two to the experiments in
which the grating was a little inclined, and the three principal
images were observed. The result of Experiment No. 1, is here
given separately, on account of the different values of vr there
employed,.
Experiment No. h 0^ 5To ; assumed index error e = 40*5'.

^ lis* - 1''46'

- 92J* - (ysi'

- 47i' + 0"33'

- 25' - 0"14'

- 2J* + O'W
+ 20" + 1"

+ 42^" + O'lS'

+ 65" + 1"41'

DISCUSSION OF THE NUMEBICAL BESULTS.

311

The values of a' for «r = - 115" and isr = + 65® ought to difier by
180®, whereas they differ by 3*27' more. This angle is so large
compared with the angles a — ■cj given just above, that it seems
best to reject the experiment. The experiment is suflBcient how-
ever to shew that the crowding of the planes of polarization, be it
in what direction it may, is very small. On combining all the
observations belonging to this experiment in the manner already
described, a small positive value of log m, namely + *002, appeared
to result This value, if exact, would indicate an extremely small
crowding in the manner of reflection.

TABLE II.

Experiment, No. 14

>.

1

Experiment, No. 2

•

d= 29057'

^=60023'

€=+72023'

€= + 24012'

logm=+*009

logw = + -010

a' — tar

a —tar

w

calc.

obs.

diff.

tor

calo.

obs.

diflf.

- 600

-00-6

00-0

+ 00-6

^.

1060

+ 00-8

-00-3

-00-6

- 400

-00-6

-10-1

-00-5

—

960

+00-1

-00-7

-00-8

- 300

-00-5

-00-7

-00-2

—

860

-00-1

-10-5

-10-4

- 200

-00-4

-00-5

-00-1

—

750

-00-3

-10-8

-10-6

- 100

-00-2

+ 00-2

+ 00-4

—

650

-00-6

-10-5

-10-0

00

00-0

-10-2

-10 2

~

650

-00-6

+ 00-4

+ 10-0

+ 100

+ 00-2

-00-6

-00-8

—

450

-00-7

-00-4

+ 00-3

+ 200

+ 00-4

+ 10-4

+ 10-0

—

350

-00-6

-00-2

+ 00-4

+ 300

+ 00-5

+ 00-8

+ 00-3

—

250

-00-5

00-0

+ 00-6

+ 400

+ 00-6

+ 00-9

+ 00-3

—

150

-00-3

00-0

+ 00-3

+ 600

+ 00-6

+ 00-3

-00-3

—

60

-00-1

00-0

+ 00-1

+ 600

+ 00-5

+ 00-3

-00-2

+

50

+ 00-1

+ 00-6

+ 00-5

+ 700

+ 00-4

+ 00-6

+ 00-2

+

150

+ 00-3

+ 00-6

+ 00-3

+ 800

+ 00-2

+ 00-4

+ 00-2

+

250

+ 00-6

+ 00-4

-00-1

+ 900

00-0

_0<»-4

-00-4

+

350

+ 00-6

?

?

+ 1000

- 00-2

-00-6

-00-3

+

4.-/

+ 00-7

+ 00-4

-00-3

+ 1100

-00-4

+ 00-5

+ 00-9

+

550

+ 00-6

+ 10-9

+ 10-3

+ 1200

-00-5

-00-6

-00-1

+ 1300

-00-6

-00-2

+ 00-4

+ 1400

-00-6

-00-7

-00-1

312

ON THE DYNAMICAL THEOEY OF DIFFRACTION.

Table II. (catitinued).

Experiment, No. 22.
logi»= +'03o

tor

180«

1660

1600

1360

1200

1050

900

750

600

450

SQO

160

a — -07

calo.

obs.

00-0

00-0

+ 10-2

+ 10-4

+ 20-0

+ 20-6

+ 20-3

+ 10-6

+ 20-0

+20-2

+ 10-1

+ 10-3

00-0

+ 00-1

-10-1

-10-6

-20-0

-20-1

-20-3

-20-8

-20-0

-10-9

-10-2

-00-8

difF.

00-0
+ 00-2
+00-5
-00-7
+ 00-2
+ 00-2
+ 00-1
-00-4
-00-1
-00-5
+ 00-1
+ 00-4

Experiment, No. 18.
d= 21089'
e=- 12044'
logw=+-029

tor

calo.

900
460
00
460
900
+ 1360
+ 1800
+ 2250

+
+

00-0
-10-9

00-0
+ 10-9

00-0
-10-9

00-0
+ 10-9

a — w
obs.

di£F.

-00-6
-20-1
-00-2
+ 10-3
+ 00-2
-20-2
+ 00-7
+ 10-9

-00-6
-00-2
-00-2
-00-6
+ 00-2
-00-3
+ 00-7
00-0

Experiment, No. 21.
^=28026'
€=60049'
logm=+-039

tor

\ 1

+
+
+
+
+
+

900
760
600
460
300
160
00
160
300
450
600
760
900

calo.

a -Iff
obs.

00-0
-10-2
-20-2
-20-6
-20-3
- 1^-3

0»0
+ 10-3
+ 20-3
+ 20-6
+ 20-2
+ 10-2

QO-O

+ 00-2
-10-6
-30-0
-30-2
-20-9
-10-0

+0P-8
+ 20-6
+ 10-6
+ 10-6
+ 10-7
+ 10-2
+00-7

diff.

+ 00-2
-00-4
-00-8
-00-6
-00-6
+ 00-3
+ 00-8
+ 10-3
-00-8
-10-0
-00-6
00-0
+ 00-7

Experiment, No. 13.
^=39060^
6=66060'
loR m = f '034

tor

+
+
+
+
+
+
+
+
+

600
600
400
800
200
100
00
100
200
300
400
600
600
700
800
900

a— tar

calo.

obs.

-10-9

-00-9

-20-2

-10-9

-20-2

-10-0

-20-0

-10-8

-10-6

-30-4

-00-8

-00-7

00-0

-00-2

+ 00-8

+ 10-0

+ 10-6

+00-1

+ 20-0

+ 10-1

+ 20-2

+ 20-6

+ 20-2

+ 10-7

+ 10-9

+ 30-7

+ 10-4

+ 20-2

+ 00-7

+ 00-9

00-0

+ 00-1

di£F.

+ 10-0
+ 00-3
+ 10-2
+00-2
-10-9
+00-1
-00-2
+ 00-2
-10-4
-00-9
+ 00-4
-00-6
+ 10-8
+ 00-8
+ 0'»-2
+ 00-1

Experiment, No. 17.
^=60046'
€=+167016'
logm=+-122

tor

900
800
700
600
600
400
300

a! "XB

calo.

obs.

00-0

00-0

-20-4

- 10.7

-40-6

- 400

-60-4

- 60-0

-70-6

- 70-6

-80-0

-100-0

-70-4

- 70-3

diff.

00-0
+00-7
+ 00-6
+ 0O-4
+00-1
-20-0
+ 00-1

Experiment, No. 23.
^=54053'
6 = - 7027'
logm = + -082

ts

00
150
300
450
600
760
900
+ 1050
+ 1200
+ 1860

+

+
+
+
+
+

a -tar

calo.

obs.

00-0

+ 00-2

+ 20-9

+ 20-7

+ 403

+ 40-6

+ 60-4

+ 40-2

4.40.4

+ 60-0

+ 20-6

+ 305

00-0

00-0

-20-5

-80-1

-40-4

-40-8

-60-4

-60-6

diff.

+ 00-2
-00-2
+ 00-3
-10-2
+ 00-6
+ 10-0
00-0
-00-6
-00-4
-00-2

DISCUSSION OF THE NUMERICAL RESULTS.

313

Table II. {continuecT).

Experiment, No. 15.

^=59052'

€=-68015'

logw=+-226

o'-tar

vr

calo.

obs.

difif.

00

00-0

+ 00-1

+ 00-1

- 100

- 60-5

- 20-7

+ 30-8

- 200

- 110-4

- 40-1

+ 70-3

- 300

- 140-1

- 170-7

-30-6

- 400

- 140-6

- 160-2

-10-6

- 500

- 130-4

- 160-4

-20-0

- 600

- 110-0

- 120-2

-10-2

- 700

- 70-8

-100-0

-20-2

- 800

- 40-0

- 40-6

-00-6

- 900

00-0

- 00-2

-00-2

Experiment, No. 11.

1=14050'; ^=22030'; €=-15030'.

First Image.

Second Image.

Third Image.

log m = + -289.

log m = + -061.

logm= + -209.

0' — tar

0' — tir

tar

calo. obB. difif.

calc. obs. difif.

calo. obs. difif.

-1060

+ 70-1 + 60-9 -00-2

+ 10-9 +20-7 +00-8

- 860

_ 20-4 - 20-6 -00-2

-00-7 -00-5 +00-2

- 10.9 _ 10-8 +00.1

- 650

-110-5 - 90-6 +20-0

-20-9 -20-6 +00-3

- 80-9 - 80-5 +00-4

- 450

-170-8 -180-2 -00-4

-400 -30-4 +00-6

-130-3 -140-3 -10-0

- 250

-170-2 -180-3 -10-1

-30-2 -30-6 -00-3

-120-0 -120-8 -00-8

- 50

_ 40-6 _ 4o.g (|0.o

-00-7 -00-7 00-0

- 30-0 - 20-7 +00-3

+ 16»

+ 120-5 +130-7 +10-2

+ 20-1 +10.9 -00-2

+ 80-4 + 80-4 00-0

+ 350

+ 180-7 +190-1 +00-4

+ 30-9 +40-6 +00-7

+ 130-6 +120-5 -10-1

+ 650

+ 150-2 +140-3 -00-9

+ 30-7 +30-7 00-0

+ 110-6 +120-0 +00-4

Experiment, No. 12.

t = 901'; ^ = 53039'; 6^-63045'.

First Image.

Second Image.

Third Image.

logm = +-756.

logm = + -122.

logm = + -366.

a— to*

a' — tar

a— tar

Iff

cole. obfl. difif.

oalc. obs. difif.

calc. obs. difif.

+ 260

+ 440-4 +440-3 -00-1

+ 60-7 +60-7 00-0

+ 220-3 +250-0 +20-7

+ 450

+ 350-1 +330-7 -10-4

+ 709 +90-1 +10-2

+ 210-7 +210-7 00-0

+ 900

Qo-O 00-0 00-0

00-0 00-0 00-0

00-0 00-0 00-0

+ 1360

-860-1 -360-7 -10-6

-70-9 -60-2 +10-7

-210-7 -200-0 +10-7

314 ox THE DTKAJaCAL TH£ORT OF DIFFRACTION.

A nearly constant error appearing in the table of differences
would indicate merely that the value of e used in the reduction
was slightly erroneous. A slight error in e, it is to be remembered,
produces no sensible error in log m, whenever the observations are
balanced with respect to one of the standard points.

In the first two experiments entered in the table, the crowding
of the planes of polarization is so small that it is masked by errors
of observation, and it is only by combining all the observations
that a slight crowding towards the plane of diffraction can be
made out. In all the other experiments, however, a glance at
the numbers in the third column is sufficient to shew in what
direction the crowding takes place. From an inspection of the
numbers found in the columns headed *'diff" it seems pretty
evident that if the formula (49) be not exact the error cannot be
made out without more accurate observations. In the case of
experiment No. 15, the errors are imusually lai^e, and moreover
appear to follow something of a regular law. In this experiment
the observations were extremely uncertain on account of the large
angle of diffraction and the great defect of polarization of the light
observed, but besides this there appears to have been some con-
fusion in the entry of the values of «r. This confusion affecting
one or two angles, or else some unrecorded change of adjustment,
was probably the cause of the apparent break in the second column
between the third and fourth numbers. Since the value of log m
is deduced from all the observations combined, there seems no
occasion to reject the experiment, since even a large error affecting
one angle would not produce a large error in the value of log m
resulting from the whole series. In the entry of experiment
No. 12 the signs of «r have been changed, to allow for the reversion
produced by reflection. This change of sign was unnecessary in
No. 11, because in that experiment the polarizer was actually
reversed. The results of experiment No. 12 would be best satisfied
by using slightly different values of the index error of the analyzer
for the three images, adding to the assumed index error about
— IJ^ +li°, +2°, for the first, second, and third images respec-
tively. The largest error in the third columns. 2*7", is for vr
= +25^ third image. The three readings by which a was deter-
mined in this case were —15^ —13^30', —12'*? Hence the error
+ 2*7®, even if no part of it were due to an index error, would
hardly be too large to be attributed to errors of observation.

DISCUSSION OF THE NUMERICAL RESULTS. 315

Since the formula (49), even if it be not strictly true, repre-
sents the experiments with sufficient accuracy, we may consider
the value of log m which results from the combination of all the
observations belonging to one experiment as itself the result of
direct observation, and proceed to discuss its magnitude. Let us
consider first the experiments on diffraction at refraction, in which
the light was incident perpendicularly on the grating.

Although the theory of this paper does not meet the case in
which diffraction takes place at the confines of air and glass, it
leads to a definite result on each of the three following suppo-
sitions :

First, that the diffraction takes place in air, before the light
reaches the glass :

Second, that the diffraction takes place in glass, after the light
has entered the first surface perpendicularly :

Third, that the diffraction takes place in air, after the light has
passed perpendicularly through the plate.

On the first supposition let a j, a,, a be the azimuths of the
plane of polarization of the light after diffraction, after the first
refraction, and after the second refraction respectively, and 0* the
angle of refraction corresponding to the angle of incidence 0, so
that sinO — fi sin ff, fi being the refractive index of the plate : and
first, let us suppose the vibrations of plane-polarized light to be
perpendicular to the plane of polarization. Then by the theory of
this paper we have tan «, = sec tan cr, and by the known formula
applying to refraction we have tana, = cos (^ — ^) tan a^, tana
= cos {0 — &) tan a,, whence tan a = m tan «r, where

m = sec ^ cos' {0 - 0').

On the second supposition, if a, be the azimuth after diffraction
at au angle 0^ within the gla.ss, we have tan a^ = sec ^ tan w,
tan a = cos {0 — ff) tan a^, whence tan a = m tan cr, where

m = sec ^ cos {0 — 0').

On the third supposition we have tan a = m tan w, where

m = sec 0.

If we suppose the vibrations parallel to the plane of polarization,
we shall obtain the same formulae except that cos^, cos^ will
come in place of seed, sec^, the factor cos {0 — 0') being un-
altered.

316

ON THE DYNAMICAL THEORY OF DIFFRACTION.

Theory would lead us to expect to find the value of logm
deduced from observations in which the grooved face was turned
from the polarizer lying between the values obtained on the
second and third of the suppositions respecting the place of diffrac-
tion, or at most not much differing from one of these limits.
Similarly, we should expect from theory to find the value of log m
deduced from observations in which the grooved face was turned
towards the polarizer lying between the values obtained on the
first and second suppositions, or at most not lying far beyond one
of these values.

The following table contains the values of logm calculated
from theory on each of the hypotheses respecting the direction of
vibration, and on each of the three suppositions respecting the
place of diffraction. The numerals refer to these suppositions.
The table extends from = to = 90", at intervals of 5^ When

= 0, m = 1, and log m = 0, in all cases. In calculating the table,

1 have supposed /a = 1'52, or rather equal to the number, (1'5206,)
whose common logarithm is *182. This table is followed by an-
other containing the values of log m deduced from experiment.

TABLE III. Values of log m from theorj', /x being supposed

equal to 1*5206.

YibnUioDB supposed

Vibrations supposed

perpendicular to the plane

parallel to the plane of

5*

of polarizati

on.
TTT

polarixation.

I

n

I

n

III

+ •001

+•001

+ -002

- ^002

-•001

- ^002

10«

+ •006

+•002

+ -007

- -008

-•004

- -007

lo«

+ 011

+ •004

+ •Olo

- ^019

-•008

- -015

200

+ •020

+ •008

+ ^027

- -034

-•016

- -027

26«

+ •032

+ •012

+ ^043

- -063

-•023

- -048

300

+ •047

+ •017

+ ^062

- -078

-•033

- ^062

350

+ •066

+ •022

+ ^087

- -109

-•044

- -087

400

+ •086

+ •028

+ ^116

- -146

-•068

- -116

450

+ •111

+ •033

+ ^160

- -190

-•073

- -160

600

+ •139

+ 037

+ ^192

- -244

-•090

- -192

560

+ 178

+ •040

+ 241

- -310

-•109

- 241

600

+ •214

+ •040

+ ^301

- -388

-•129

- ^301

660

+ •262

+ •039

+ -374

- ^486

-•161

- -374

700

+ •324

+ •034

+ ^466

- •eos

-•176

- -466

750

+ •408

+ •022

+ ^687

- ^766

-•202

- -687

800

+ •633

+ •006

+ ^760

- -987

-•231

- -760

860

+ •773

-•022

+ 1060

- 1847

-•266

-1060

900

+ 00

-069

+ 00

- 00

-•306

- 00

DISCUSSION OF THE NUMERICAL RESULTS.

317

TABLE IV. Values of logm from observatioiL

Nature of Experiment

No.

e

logm

Diinrmction at refrftctlon.
Inddoice perpendicular.
OrooTed face of glau
plate turned Jirom the
incklent light

14

2

22

29*67'
60*23'
64*38'

+ •009
+ •010
+ •036

DiflVactlon at refraction.
Inddenoe perpendicular.
Grooved &ce of ghus
plate turned towurdt the
faiddent light

18
21
13
17
23
16

21*39'
28*26'
39*50'
60*45'
64*63'
69*52'

+ 029
+ •039
+ 034
+ 122
+ •082
+ •226

A comparison of the two tables will leave no reasonable doubt
that the experiments are decisive in favour of Fresnel's hypo-
thesis, if the theory be considered well founded. In considering
the conclusiveness of the experiments, it is to be remembered
that on either the first or second supposition respecting the place
of diffraction, (and the third certainly cannot apply to the case
in which the grooved face is turned towards the incident light,)
the planes of polarization of the diffracted light are crowded by
refraction towards the perpendicular to the plane of diffraction,
and therefore the observed crowding towards the plane of diffrac-
tion does not represent the whole effect of the cause, be it what
it may, of crowding in that direction.

If /8 be the value of a' - «r for «r = 45", fi = V when log m =
•015, nearly; and when log m is not large, /8 is nearly propor-
tional to log m. In this case /8 is nearly the maximum value
of a'— -BT. Hence the greatest value of a' — «r, expressed in degrees,
may be obtained approximately from Table IV, and, within the
range of observation, from Table III, by regarding the decimals
as integers and dividing by 15. Thus, for log m = — '388 the
real maximum is 24"-8, and the approximate rule gives 25"*9, so
that this rule is abundantly sufficient to allow us to judge of the
magnitude of the quantity by which the two theories differ. For
^=60^ the two columns in Table III headed "I", as well as
those headed "III", differ by 602, and those headed "II", differ
by 169, so that the values assigned to /8 by the two theories differ
by about 40® or 11", according as we suppose the diffraction to
take place in air or in glass. For ^ = 40", the corresponding
differences are 15* and 6", nearly. These differences, even those

318

ON THE DYNAMICAL THEOBT OF DIFTRACTION.

which belong to ditrniction vitbin the glass plate, are large com-
pared with the errors of observation ; for the probable cause of
the large errors in ezperiDient No. 15, has been already mentioDed.
In the following figure the absciasie of the curves represent
the angle of diffraction, and the ordioates the values of log m
calculated from theory. The numerals refer to the three supposi-
tions respecting the place of diffraction, and the letters E, A,
(the first vowels in the words perpendicular and parcdlel,) to the
two hypotheses respecting the direction of vibration. The dots
represent the results of the experiments in which the grooved
&ce of the glass plate was turned towards the polarizer, and
the crosses those of the experiments in which it was turned in
the contrary direction.

DISCUSSION OF THE NUMERICAL RESULTS. 319

The smallness of log m in experiment No. 23, to which the
6th dot belongs, is probably due in part to the use of the red
glass, since, as has been already remarked, the planes of polariza-
tion of the blue were more crowded towards the plane of diffrac-
tion than those of the red. On this account the dot ought to
be slightly raised to make this experiment comparable with its
neighbours. On the other hand it will be seen by referring to
Table II, that No. 23 was a much better experiment than No. 15,
which is represented by the 6th dot, and apparently also better
than No. 17, which is represented by the 4th dot. No. 21,
represented by the 2nd dot, seems to have been decidedly better
than No. 13, which is represented by the 3rd. Nos. 14 and 22,
represented by the 1st and 3rd crosses respectively, were probably
much better, especially the latter of them, than No. 2, which is
represented by the 2nd cross. Now, bearing in mind the cha-
racter of the experiments, conceive two curves drawn with a free
hand, both starting from the origin, where they touch the axis,
and passing, the one among the dots, and the other among the
crosses. The former of these would apparently lie a little below
the curve mai'ked I. E, and the latter a very little below the
curve 11. E,

9

Hence the observations are very nearly represented by adopting
Fresnel's hypothesis respecting the direction of vibration, and,
whether the grooved face be turned towards or from the incident
light, supposing the wave broken up before it reaches the grooves.

I think a physical reason may be assigned why the supposition
of the wave's being broken up before it reaches the grooves shoidd
be a better representation of the actual state of things than the
supposition of its being broken up after it has passed between
them. Till it reaches the grooves, the wave is regularly propa-
gated, and, according to what has been already remarked in the
introduction, we have a perfect right to conceive it broken up at any
distance we please in front of the grooves.
Let the figure represent a section of the J

grooves, &c., by the plane of diffraction. ^^^^...y..^ ^^//.

Let aA,bB be sections of two consecutive
grooves, AB being the polished interval
Let eh be the plane at which a wave in-
cident in the direction represented by the
arrow is conceived to be broken up. Let be any point in eh,

320 ON THE DYNAMICAL THEORY OF DIFFRACTION.

and from draw ORS in the direction of a ray proceeding regu-
larly from and entering the eye ; so that OJB, MS are inclined to
the normal at angles Oy 6\ or ff^ 0, according as the light is passing
from air into glass or from glass into air. The latter case is repre-
sented in the figure. Of a secondary wave diverging spherically
from 0, which is only partly represented in the figure, those
rays which are situated between the limits OA^ OB, and are
not inclined at a small angle to either of these limiting di-
rections, may be regarded as regularly refracted across AB,
In a direction inclined at a small angle only to OA or OB,
it would be necessary to take account of the diflraction at the
edge A or JS. Let 7 be a small angle such that if OR be inclined
to OA and OB at angles greater than 7 the ray OR may be
regarded as regularly refracted, and draw Ae, Bg inclined at angles
7 to OR, and Af, Bh inclined at angles — 7. Then, in finding the
illumination in the direction R8, all the secondary waves except
those which come from points situated in portions such as ef, gh
of the plane eh may be regarded as regularly refracted, or else com-
pletely stopped, those which come from points in fg and similar
portions being regularly refracted, and those which come from
points to the left of e, between e and the point which bears to a the
same relation that h bears to 6, as well as those which come from
similar portions of the plane eh, being completely stopped. Now
the whole of the aperture -4i5 is not effective in producing illu-
mination in the direction R8, For let C be the centre of AB^
and through C draw a plane perpendicular to R8, and then draw
a pair of parallel planes each at a distance \\ from the former
plane, cutting AB in M^, N^, another pair at a distance X, and
cutting -4J5 in M^, N^, and so on as long as the points of section
fall between A and B, Let M, N be the last points of section.
Then the vibrations proceeding from MN in the direction R8
neutralize each other by interference, so that the effective portions
of the aperture are reduced to AM, NB, Now the distance
between the feet of the perpendiculars let fall from A, M on RS
may have any value from to \\, and for the angle of diffraction
actually employed AM was equal to about twice that distance on
the average, or rather less. Hence AM may be regarded as
ranging from to \ ; and since for the brightest part of a band
forming that portion of a spectrum of the first class which belongs
to light of given refrangibility AM has just half its greatest value.

DISCUSSION OF THE NUMERICAL RESULTS. 821

we may suppose ^ if = JX. But if the distance between the planes
eh, ab be a small multiple of X, and 7 be small, ef will be small
compared with X, and therefore compared with AM. Hence the
breadth of the portions of the plane eh, such as ef, for which we
are not at liberty to regard the light as first diffracted and then
regularly refracted, is small compared with the breadth of the
portions of the aperture, such as AM, which are really effective;
and therefore, so far as regards the main part of the illumination,
we are at liberty to make the supposition just mentioned. But
we must not suppose the wave to be first regularly refracted and
then diffracted, because the regular refraction presupposes the
continuity of the wave.

The above reasoning is not given as perfectly satisfactory, nor
could we on the strength of it venture to predict with confidence
the result; but the result having been obtained experimentally,
the explanation which has just been given seems a plausible way
of accounting for it. According to this view of the subject, the
result is probably not strictly exact, but only a very near approxi-
mation to the fact. For, if we suppose the distance between the
planes eh, ab to he only a small multiple of X, we cannot apply the
regular law of refraction, except as a near approximation. More-
over, the dynamical theory of diffraction points to the existence of
terms which, though small, would not be wholly insensible at the
distance of the plane ab. Lastly, when the radius of a secondary
wave which passes the edge -4 or jB is only a small multiple of X,
we cannot regard 7 as exceedingly small.

Let us consider now the results of experiments Nos. 11 and 12.
In diffraction at refraction, the amount of crowding with respect
to which the theory leaves us in doubt vanishes along with fi — 1;
and although this amount is far from insensible in the actual
experiments, it is still not suflSciently large to prevent the results
from being decisive in favour of one of the two hypotheses re-
specting the direction of vibration. Thus the curves marked "J."
in the first figure are well separated from those marked *'!!'*, and
if fi were to approach indefinitely to 1, the curves I. A and II. A
would approach indefinitely to HI. A, and I. E, and II. E to
III. E. In diffraction at reflection, however, the case is quite
different, and in the absence of a precise theory little can be made
of the experiments, except that they tend to confirm the law
expressed by the equation (49). In the case of the first and second

s. II. 21

322 ON THE DYNAMICAL THEORY OF BIFFRACTION.

images the diffraction accompanied refraction, and so far the
experiments were of the same nature as those which have been
just discussed, but the angle of incidence was not equal to zero,
and in that respect they differ.

Let a, p be the angles of refraction correspondiug to the angles
of incidence, i, i + 0, Then in the case of the first image the
tangent of the azimuth of the plane of polarization is multiplied
by cos {%-{■ — p) sec (i + ^ + p) in consequence of reflection, and
by cos (i + ^ — p) in consequence of refraction; and in the case of
the second image by cos (i — i) in consequence of refraction, and
by cos (t — %) sec (i + i') in oensequence of reflection. Hence if
m' be the factor corresponding to diffraction and the accompany-
ing refraction, m the factor got from observation, and regarded
as correct, we hswe

forlst image, log m' = log m + log cos (i + ^ 4- p) — 2 log cos (i +d — p),
for 2nd image, log m! = log m + log cos {% + i) — 2 log cos (i — %).

In the case of the first image, m! relates to diffraction at refrac-
tion from air into glass, where i is the angle of incidence in air,
and p — i tlie angle of diffraction in glass. In the case of the
second image, m relates to diffraction from glass into air, where i
is the angle oi incidence in glass, and the angle of diffraction in
air.

In experiment No. 11, 1st image, we have from Table II, log
m= + '289; for the 2nd image logm = + '061. In this experi-
ment i = 14'^50', ^ = 22^30', whence i' = 9°4r, p = 23'30'. We
thus get

for Ist image, log m' = + '289 - -286 = + '003,
for 2nd image, log m' = + '061 - '037 = + '024.

The positive values of log rn! which result from these ex;peri-
ments, notwithstanding the refraction which accompanied the
diffraction, bear out the results of the experiments already dis-
cussed, and confirm the hypothesis of Fresnel. It may be re-
marked that log m comes out larger for the second image, in
which diffraction aocompanied refraction from air into glass, than
for the first image, in which diffraction accompanied refraction
from glass into air. This also agrees with the experiments just re-
ferred to.

In experiment No. 12, the light which entered the eye came
in a direction not much different from that in which light regu-
larly reflected would have been perfectly polarized. Since in

DISCUSSION OF THE NUMERICAL RESULTS. 323

regularly reflected light the amount of crowding of the planes of
polarization changes rapidly about the polarizing angle, it is pro-
bable that small errors in fi, i, and would produce lai^e errors in
m. Hence little can be made of this experiment beyond confirm-
ing the formula (49).

I will here mention an experiment of Fraunhofer's, which,
when the whole theory is made out, will doubtless be found to
have a most intimate connexion with those here described. In
this experiment the light observed was reflected from the grooved
face of a glass-grating; the reflection from the second surface was
stopped by black varnish. In Fraunhofer's notation e is the
interval from one groove to the corresponding point of its consecu-
tive, and is measured in parts of a French inch, a is the angle of
incidence, t the inclination of the light observed to the plane of
the grating, (Er) the value of r for the fixed line E, and the
numerals mark the order of the spectrum, reckoned from the axis,
or central colourless image, the order being reckoned positive on
the side of the aeute angle made by the regularly reflected light
with the plane of the grating. The followipg is a translation of
Fraunhofer's description of the experiment.

" It is very remarkable that, under a certain angle of incidence,
a part of a spectrum arising from reflection consists of perfectly
polarized light This angle of incidence is very different for the
different spectra, and even very sensibly different for the different
colours of one and the same spectrum. With the glass-grating
€ = 0'0001223 there is polarized : {EtY*^^, that is, the green part of
this first spectrum, when <r = 49'; (EtY^^^^ or the green part in
the second spectrum lying on the same side of the axis, when
<r = 40^; lastly, (jFt)^"^^ or the green part of the first spectrum
lying on the opposite side of the axis, when <r = 69^ When
{ErY*^^ is polarized perfectly, the remaining colours of this spec-
trum are still but imperfectly polarized. This is less the case
with {ErY*^^\ and a can be sensibly changed while this colour still
remains polarized, {Et)^'^^ is under no angle of incidence so com-
pletely polarized (so ganz vollstandig polarisirt) as {Er^^^K With
a grating in which e is greater than in that here spoken of. the
angle of incidence would have to be qujte different in order that
the above-mentioned spectra should be polarized*."

* Gilbert's AnnaUn der Phynk, B. ziv. (1823) S. 364.

21—2

e

<r + ^

- 18" 13'

50' 47'

58»40'

+iri'

66n'

+ 33" 52'

73» 52'.

^

824 ON THE DYNAMICAL THEORY OF DIFFRACTION.

If we suppose o-y a function of v such that <r_j = 69, o*^, = 49,
<r^, = 40, WQ get by interpolation a^ = 58*33; so that if we suppose
the central colourless image, which arises from light reflected
according to the regular law, to have been polarized at the polar-
izing angle for light reflected at a surface free from grooves, we
get /i = tan 58® 40' = 1*64, from which it would result that the
grating was made of flint glass. The inclination of E in the spec-
trum of the order v to the plane of the grating may be calculated
from the formula cos t = sin <r + vX/e*, given by Fraunhofer, and
obtained from the theory of interference; and ^ = 90' — T — <r,
where is the angle of difiraction. We thus get for green light
polarized by reflection and the accompanying diffraction,

order of spectrum a

-1 69®

58® 40'

+ 1 49®

+ 2 40®

li we suppose the formula (49) to hold good in this case, m
becomes infinite for the angles of incidence <r and the correspond-
ing angles of reflection <r + d contained in the preceding table.

Another observation of Fraunhofer s described in the same
paper deserves to bo mentioned in connexion with the present
investigation, because at first sight it might seem to invalidate the
conclusions which have been built on the results of the experi-
ments. On examining the spectra produced by refraction in
another glass-grating on which the light was incident perpendicu-
larly, Fraunhofer found that the spectra on one side of the axis
were more than twice as bright as those on the other -f*. To
account for this phenomenon, he supposed that in ruling the
grating the diamond had had such a position with respect to the
plate that one side of each groove was sharp, the other less defined.
This view was confirmed by finding that a glass plate covered with
a thin coat of grease, and purposely ruled in such a manner, gave
similar results. Now with reference to the present investigation
the question might naturally be asked, If such material changes in
intensity are capable of being produced by such slight modifications
in the difiracting edge, how is it possible to build any certain con-

* In Fraunhofer's notation the wave length is denoted by tr.
t Gilbert's AnnaUn der Pkysik^ B. xiv. p. 353.

DISCUSSION OF THE NUMEKICAL RESULTS.

325

elusions on an investigation in whieh the nature of the diffracting
edge is not taken into account ?

To facilitate the explanation of the apparent cause of the
above-mentioned want of symmetry, suppose the diffraction pro-
duced by a wire grating in which the section of each wire is a
right-angled triangle, with one side of the right angle parallel to
the plane of the grating, and perpendicular to the incident light,
and the equal acute angles all turned the same way. The tri-
angles ABC, DEF in the figure repre-
sent sections of two consecutive wires,
and OB, HD, IE represent incident
rays, or normals to the incident waves,
which are supposed plane. Let BE=€,
and BD: BE ::n :1- n. Draw BK,
BL, EM parallel to one another in the
direction of the spectrum of the order v
on the one side of the axis, so that vk is the retardation of the
ray EM relatively to BK, and therefore sin = vk/e, being the
angle of diffraction, or the inclination of BK to GB produced.
Draw BN, FO, EP at an inclination on the other side of the
axis, and let ^ DBF=a. Then the retardation of DL relatively
to BK is equal to nv\ or ne sin 0, and that of BN" relatively to
FO is equal to we sin ^ + we tan a cos ^ — Tie tan a, so that if we
denote these retardations by

JBj, iJj, iJj = ne sin 0, iJ, = ne sin — ne tan a versin 0.

Let Pj, p^ be the greatest integers contained in the quotients of
R^,R^ divided by \ and let -Ki = jPiX + rj, B^^p^-^-r^. Then
the relative intensities of the two spectra of the order + v and — v
depend on r^yV^: in fact, we find for the ratio of intensities, on
the theory of interference, sin* irrjX : sin' irrjk. Now this ratio
may have any value, and we may even have a bright spectrum on
one side of the axis answering to an evanescent spectrum on the
other side. It appears then in the highest degree probable that
the want of symmetry of illumination in Fraunhofer's experiment
was due to a different mode of interference on opposite sides of the
axis. But this has nothing whatsoever to do with the nature of
the polarization of the incident light, and consequently does not
in the slightest degree affect the ratio of the intensities, or rather
the ratio of the coeflScients of vibration, of the two streams of

326 ON THE DYNAMICAL THEORY OF DIFFRACTION.

light belonging to the same spectrum corresponding to the two
streams of oppositely polarized light into which we may conceive
the incident light decomposed, and consequently does not affect
the law of the rotation of the plane of polarization of the diffracted
light.

P. S. Since the above was written, Professor Miller has de-
termined for me the refractive index of the glass plate by means
of the polarizing angle. Four observations, made by candle-light,
of which the mean error was only 1'^, gave for the double angle
113' 20', whence /i. = tan 56^40'= 1-52043, which agrees almost
exactly with the value I had assumed. In two of these obser-
vations the light was reflected at the ruled, and in two at the
plane surface. The accordance of the results bears out the sup-
position made in Part II, that the light belonging to the central
colourless image, which is reflected or refracted according to the
regular laws, is also afiected as to its polarization in the same
manner as if the surface were free from grooves. The refractive
index of the plate being now known for certain, the experiments
described in this paper render it probable that the crowding of
the planes of polarization which actually takes place is rather less
than that which results from theory on the supposition (which is
in a great measure empiiical), that the diffraction takes place
before the light reaches the grooves. The difference is however so
small that more numerous and more accurate experiments would
be required before we could aflSrm with confidence that such is
actually the case.

When a stream of light is incident obliquely on an aperture,
it is sometimes necessary to conceive each wave broken up as its
elements arrive in succession at the plane of the aperture. In
applying the formula (46) to such a case, it will be suflScient to
substitute for dS the projection of an element of the aperture on
the wave's front, 6 being measured as before from the normal to
the wave> which no longer coincides with the normal to the plane
of the aperture;

Before concluding, it will be right to say a few words re-
specting M. Cauchy's dynamical investigation of the problem of
diffraction, if it be only to shew that I have not been anticipate

DISCUSSION OF THE NUMERICAL RESULTS. 327

in the results which I here lay before the Society. This investi-
gation is referred to in Moigno's Repertoire d^Optique modeme^
p. 190, and will be found in the fifteentb Volume of the Comptea
Rendv^, where two short memoirs of M. Carucliy's on the subject
are printed, the first of which begins at p. 605, and the second at
p. 670. The first contains the analysis which M. Cauchy had
some years before applied to the problem. This solution he after-
wards, as it appears, saw reason to abandon, or at least greatly to
restrict; and he has himself stated (p. 675), that it is only ap-
plicable when certain conditions are fulfilled, and when moreover
the nature of the medium is such that normal and transversal
vibrations are propagated with equal velocity. This latter con-
dition, as Green has shewn, is incompatible with the stability of
the medium. In the second memoir M. Cauchy has explained the
principles of a new solution of the problem which he had obtained,
without giving any of the analysis. The principal result, it would
appear, at which he has arrived is, that light incident on an aper-
ture in a screen is capable of being reflected, so to speak, by the
aperture itself (p. 675); and he proposes seeking, by the use of
very black screens, for these new rays which are reflected and
diSracted. But it follows from reasoning similar to that of Art.
34, or even from the general formula (45) or (46), that such rays
would be wholly insensible in all ordinary cases of difiraction, even
were the screen to reflect absolutely no light. The only way
apparently of rendering them sensible would be, to construct a
grating of actual threads, so fine as to allow of observations at
a large angle of diffraction. Such a grating I believe has never
been made ; and even if it could be made it would apparently
be very diflicult, if not impossible, to separate the effect to be
investigated from the effect of reflection at the threads of the
grating.

[A few years after the appeamnce of the above Paper, the
question was re-examined experimentally by M. Holtzmann*, who
at first employed glass gratings, but without getting consistent
results (though there seemed some indication of a conclusion the
same as that which I had obtained), and afterwards had recourse

♦ Poggendorff's Annalen, Vol. 99 (1866) p. 446, or Philosophical Magazine,
Vol. 13, p. 135.

328 ON THE DYNAMICAL THEORY OF DIFFRACTION.

to a Schwerd's lampblack grating. With the latter consistent
results were obtained. But the crowding of the planes of polari-
zation was towards the plane of diffraction ; and when instead of
measuring the azimuths of the planes of polarization of the
incident and di0racted hght, the incident light was polarized
at an azimuth of 45^ to the lines of the grating, and the diffracted
light was divided by a double-image prism into two beams
polarized in and perpendicularly to the plane of diffraction, it was
the latter that was the brighter. From these experiments the
conclusion seemed to follow that in polarized light the vibrations
are in the plane of polarization. The amount of rotation did not
very well agree with theory. The subject was afterwards more
elaborately investigated by M. Lorenz*. He commences by an
analytical investigation which he substitutes for that which I
had given, which latter he regards as incomplete, apparently
jfrom not having seized the spirit of my method. He then gives
the results of his experiments, which were made with gratings of
various kinds, especially smoke gratings. His results with these do
not confirm those of Holtzmann, and he points out an easily over-
looked source of error, which he himself had not for some time
perceived, which he thinks may probably have affected Holtz-
mann's observations. Lorenz's results like mine were decisively
in favour of the supposition that in polarized light the vibrations
are perpendicular to the plane of polarization. He found as I had
done that the results of observation as to the azimuth of the plane
of polarization of the diffracted light agreed very approximately
with the theoretical result, provided we imagine the diffraction
to take place before the light reaches the ruled lines.]

* Poggendorff'B Annalen^ Vol. Ill (1860) p. 815, or Philosophical ilagaxine^
Vol. 21, p. 821.

[From the Transactioiis of the Cambridge Philosophical Society, Vol ix.

Part I.]

On the numerical Calculation of a Class of Definite

Integrals and Infinite Series.

[Read March 11, 1850.]

In a paper " On the Intensity of Light in the neighbourhood
of a Caustic*/' Mr Airy the Astronomer Royal has shewn that the
undulatory theory leads to an expression for the illumination in-

volving the square of the definite integral I cos ^ (i^/* — mw) dw,

where m is proportional to the perpendicular distance of the point
considered from the caustic, and is reckoned positive towards the
illuminated side. Mr Airy has also given a table of the numerical
values of the above integral extending from m = — 4tom = + 4, at
intervals of 0*2, which was calculated by the method of quadratures.
In a Supplement to the same paperf the table has been re-calcu-
lated by means of a series according to ascending powers of m, and
extended to m = ± 5*6. The series is convergent for all values of
m, however great, but when m is at all large the calculation be-
comes exceedingly laborious. Thus, for the latter part of the
table Mr Airy was obliged to employ 10-figure logarithms, and
even these were not sufficient for carrying the table further. Yet
this table gives only the first two roots of the equation W= 0, W
denoting the definite integral, which answer to the theoretical
places of the first two dark bands in a system of spiirious rainbows,
whereas Professor Miller was able to observe 30 of these bands.
To attempt the computation of 30 roots of the equation W=Ohy

• Camb. Phil, Trans, Vol. vi. p. 870. t VoL vui. p. 695.

330 ON THE NUMERICAL CALCULATION OF A CLASS OF

means of the ascending series would be quite out of the question,
on account of the enormous length to which the numerical calcula-
tion would run.

After many trials I at last succeeded in putting Mr Airy's
integral under a form from which its numerical value can be calcu-
lated with extreme facility when m is large, whether positive or
negative, or even moderately large. Moreover the form of the
expression points out, without any numerical calculation, the law
of the progress of the fuuction when m is large. It is very easy to
deduce from this expression a formula which gives the ^ root of
the equation W=0 with hardly any numerical calculation, except
what arises from merely passing from (rn/3)i, the quantity given
immediately, to m itself.

The ascending series in which TFmay be developed belongs to
a class of series which are of constant occurrence in physical ques-
tions. These series, like the expansions of c"^, sin x, cos x, are
convergent for all values of the variable x, however great, and are
easily calculated numerically when x is small, but are extremely
inconvenient for calculation when x is large, give no indication of
the law of progress of the function, and do not even make known
what the function becomes when a;=isoo. These series present
themselves, sometimes as developments of definite integrals to
which we are led in the first instance in the solution of physical
problems, sometimes as the integrals of linear differential equations
which do not admit of integration in finite terms. Now the method
which I have employed in the case of the integral W appears to
be of very general application to series of this class. I shall at-
tempt here to give some sort of idea of it, but it does not well
admit of being described in general terms, and it will be best
understood from examples.

Suppose then that wo have got a series of this class, and let
the series be denoted by y or f (x), the variable according to as-
cending powers of which it proceeds being denoted by x. It will
generally be easy to eliminate the transcendental function / (x)
between the equation y=f{x) and its derivatives, and so form a
linear differential equation in y, the coefficients in which involve
powers of x. This step is of course unnecessary if the differential
equation is what presented itself in the first instance, the series

DEFINITE INTEGRALS AND INFINITE SERIES. 331

being only an integral of it. Now by taking the terms of this
differential equation in pairs, much as in Lagrange's method of
expanding implicit functions which is given by Lacroix*, we shall
easUy find what terms are of most importance when a; is lai^e: but
this step will be best understood from examples. In this way we
shall be led to assume for the integral a circular or exponential
function multiplied by a series according to descending powers of x,
in which the coeflBcients and indices are both arbitrary. The
differential equation will determine the indices, and likewise the
coefficients in terms of the first, which remains arbitrary. We
shall thus have the complete integral of the differential equation,
expressed in a form which admits of ready computation when x is
large, but containing a certain number of arbitrary constants,
according to the order of the equation, which have yet to be deter-
mined.

For this purpose it appears to be generally requisite to put the
infinite series under the form of a definite integral, if the series be
not itself the developement of such an integral which presented
itself in the first instance. We must now endeavour to determine
by means of this integral the leading term mf{x) for indefinitely
large values of x, a process which will be rendered more easy by
our previous knowledge of the form of the term in question, which
is given by the integral of the differential equation. The arbitrary
constants will then be determined by comparing the integral just
mentioned with the leading term mf(x).

There are two steps of the process in which the mode of pro-
ceeding must depend on the particular example to which the
method is applied. These are, first, the expression of the ascending
series by means of a defiuite integral, and secondly, the determina-
tion thereby of the leading term in / (x) for indefinitely large
values of x. Should either of these steps be found impracticable,
the method does not on that account fall to the ground. The arbi-
trary constants may still be determined, though with more trouble
and far less elegance, by calculating the numerical value of / (x)
for one or more values of x, according to the number of arbitrary
constants to be determined, from the ascending and descending
series separately, and equating the results.

• TraiU du Calcul, &c, Tom. i. p. 104.

332 ON THE NUMERICAL CALCULATION OF A CLASS OF

In this paper I have given three examples of the method just
described. The first relates to the integral TT, the second to an
infinite series which occurs in a great many physical investigations,
the third to the ii^tegral which occurs in the case of diffraction
with a circular aperture in front of a lens. The first example
is a good deal the most difficult. Should the reader wish to see
an application of the method without involving himself in the
difficulties of the first example, he is requested to turn to the
second and third examples.

FIRST EXAMPLE.

1. Let it be required to calculate the integral

JF= I cos -^ {v^ — mw) dw (1),

for different values of m, especially for large values, whether posi-
tive or negative, and in particular to calculate the roots of the
equation W==0.

2. Consider the intesrral

u

^p^^iccsZBW^l^nm^-n.)^^ ^2^^

where d ia supposed to lie between — ir/6 and + 7r/6, in order that
the integral may be convergent.

Putting X = (cos — V— 1 sin 0) z,

we get dx = (cos — V— 1 sin 0) dz, and the limits of z are and oo ;
whence, writing for shortness

|) = (cos 2(? + V^ sin 2^) n (3),

we get w = (co8(?-V^8in^)r 6-<«*-i«>d«* (4).

Jo

* The legitimacy of this transformation rests on the theorem that if /(x) be a
continnoas fanotion of x, which does not become infinite for any real or imaginaiy,
but finite, value of x, we shall obtain the same result for the integral of f{x)dje
between two given real or imaginary limits through whatever series of real or
imaginary values we make x pass from the inferior to the superior limit. ^ It is
unnecessary here to enunciate the theorem which applies to the case in which f{x)
becomes infinite for one or more real or imaginary values of x. In the present case

DEFINITE INTEGRALS AND INFINITE SERIES. 333

3. Let DOW 0^ which hitherto has heen supposed less than 7r/6,
become equal to 7r/6. The integral obtained from (2) by putting
=z 7r/6 under the integral sign may readily be proved to be con-
vergent. But this is not suj£cient in order that we may be at
liberty to assert the equality of the results obtained from (2), (4) by
putting « tt/G before integration. It is moreover necessary that
the convergency of the integral (2) should not become infinitely
slow when approaches indefinitely to 7r/6, in other words, that if
X be the superior limit to which we must integrate in order to
render the remainder, or rather its modulus, less than a given
quantity which may be as small as we please, X should not become
infinite when becomes equal to tt/G*. This may be readily
proved in the present case, since the integi*al (2) is even more
convergent than the integral

r

J

-N/-l8inW(«'->ut) T

which may be readily proved to be convergent.
Putting then = tt/G in (2) and (4), we get

tt=j C03{a?-nx)dx-^'-l I sin (a?' - tut) da? (5),

w= ^cos J - V-1 sin ^j J* €-<*'-!") dz (G),

where p= (cos^ + V— 1 sin ^j n (7).

Let u^U-^f^U,

and in the expression for Z7got from (5) put

x = i^ w, ^=(1) 'tn (8);

thenweget ^""(l)" ^ (^)-

the liniits of x are and real infinity, and accordingly we may first integrate with
respect to z from to a large real quantity 2^ , tf (which is supposed to be written
for in the expression for x) being constant, then leave z equal to 2, , make 0^ vary,
and integrate from to 0, and lastly make z^ infinite. But it may be proved
without difficulty (and the proof may be put in a formal shape as in Art. 8), that
the second integral vanishes when Zi becomes infinite, and consequently we have
only to integrate with respect to z from to real infinity.

* See Section in. of a paper **0n the Critical Values of the sums of Periodic
Series." Camh, Phil. Trans, Vol. viii. p. 561. [Ante, Vol i. p. 279.]

334 ON THE NUMERICAL CALCULATION OF A CLASS OF

4. By the transformation of u from the form (5) to the form
(6), w€ are enabled to differentiate it as often as we please with
respect to n by merely differentiating under the integral sign. By
expanding the exponential e*" in (6) we should obtain u, and there-
fore U, in a series according to ascending powers of n. This series
is already given in Mr Airy's Supplement. It is always conver-
gent, but is not convenient for numerical calculation when n is
large.

We get from (6)

.l((».|-V-lsm^),

which becomes by (7)

£+^i^~l • • »

^

Equating to zero the real part of the first member of this
equation, we get

^-^-o (")•

5. We might integrate this equation by series according to
ascending powers of n, and w.e should thus get, after determining
the arbitrary constants, the series which have been already
mentioned. What is required at present is, to obtain for U an
expression which shall be coxivenient when n is large.

The form of the differential equation (11) already indicates
the general form of U for large values of n. For, suppose n large
and positive, and let it receive a small increment Sn. Then the
proportionate increment of the coefficient n/3 will be very small ;
and if we regard this coefficient as constant, and Bn as variable,
we shall get for the integral of (11)

tr=i\rco8|yg).8n| + i«^'sin|yQ).Sn} ... (12).

where N, N' are regarded as constants, hn being small, which does
not prevent them from being in the true integral of (11) slowly
varying functions of n. The approximate integral (12) points out

DEFINITE INTEGRALS AND INFINITE SERIES. 335

the existence of circular functions such as cos/(w), sin /(n) in the
true integral ; and since V(^/3) • ^ must be the small increment
of /(n), we get /(w) =f VC^V^)* omitting the constant, which it is
unnecessary to add When n is negative, and equal to — n', the
same reasoning would point to the existence of exponentials with
±SV(w'V3) in ^he index. Of course the exponential with a posi-
tive index will not appear in the particular integral of (11) with
which we are concerned, but both exponentials would occur in the
complete integral. Whether n be positive or negative, we may, if
we please, employ exponentials, which will be real or imaginary
as the case may be.

6. Assume then to satisfy (11)

U = €^^~J{An- + Bnfi + Cny + ...]* (13),

where A, ByC... a, )8, 7... are constants which have to be deter-
mined. DiflFerentiating, and substituting in (11), we get

a (a - 1) An-'^ + y3 (y3 -) 1 Bnfi'^ + . . .

+ ^|^((42 + l)^n«-i + (4^ + l)5n^-i+...} = 0.

As we want a series according to descending powers of n, we
must put

4a + l=0, i8 = a = f, 7 = )8-f...

* The idea of multipljing the circular functions hy a series according to de-
scending powers of n was suggested to me by seeing in Moigno's Repertoire d'optique
modeme, p. 189, the following formulsB which M. Cauohy has given for the calcu-
lation of Freanel's integrals for large, or moderately large, values of the superior
limit:

/V^ If iff jf

COS- z'dz=i-^cos- ?rt*+if sin^m';

/*'** IT X IT

I sin jr;e*dz=J-Jlf cos;5m*-iVsin-m';
J Q I I A

., 1 1.8 1.8.6.7 ^, 1 1.8.5

where lf= g-wH ^-^ ...; ^—~%-i^ 7\d +•••

The demonstration of these formulse will be foxmd in the 15th Volume of the
Comptea Rendus, pp. 554 and 578. They may be readily obtained by putting
wz^='2x, and integrating by parts between the limits imn^ and oo of x.

336

ON THE mJHERICAL CALCULATION OF A CLASS OF

whence

{'--r

V-1 1.5.7.1 1/ V-l
16 V(3n') "^ 1.2

1.5.7.11.13.17 / V-l Y \ .,,.

1.2.3

Vl6V(3i

By changing the sign of iv/(— 1) ^^h in the index of € and in the
series, writing B for -4, and adding together the results, we shall
obtain the complete integral of (11). with its two arbitrary con-
stants. The integral will have ditferent forms according as n is
positive or negative.

First, suppose n positive. Putting the function of n of which
A is the coejBScient at the second side of (14) under the form
P + VC "- 1) Qi 8^<i observing that an expression of the form

where A and B are imaginary arbitrary constants, and which is
supposed to be real, is equivalent to AB + BQ, where A and B are
real arbitrary constants, we get

Z7 = An-i^Ecosfy|" + flfsinjy'|)

4-fin-l(Esinfy|'-flfcosjy|') (15),

where

-, _ 1.5.7.11 1.5.7.11.13.17.19.23

1.2.16".3n"

1.2.3.4.16\3V

flf =

1.5

1.5.7.11.13.17

1.16(3n»)* 1.2.3. 16'(3n")t

+

...(16).

Secondly, suppose n negative, and equal to — n. Then, writing
— n for n in (14), and changing the arbitrary constant, and the
sign of the radical, we get

rr ^'-i-§>/?fl 1-5 . 1.5.7.11 \ ,.^.

I 1.16(3ii»j4 1.2.1b.3n» j

It is needless to write down the part of the complete in-
tegral of (11) which involves an exponential with a positive

DEFINITE INTEGRALS AND INFINITE SERIES. 337

index, because, as has been already remarked, it does not appear
in the particular integral with which we are concerned.

7. When n or n is at all large, the series (16) or (17) are at
first rapidly convergent, but they are ultimately in all cases hyper-
geometrically divergent. Notwithstanding this divergence, we
may employ the series in numerical' calculation, provided we do
not take in the divergent terms. The employment of the series
may be justified by the following considerations.

Suppose that we stop after taking a finite number of terms of
the series (16) or (17), the terms about where we stop being so
small that we may regard them as insensible ; and let U^ be the
result so obtained. From the mode in which the constants A, B,
(7,... a, ^, 7... in (13) were determined, it is evident that if we
form the expression

dn«+3^- ^^ dn^-3^'-

accoi*ding as n is positive or negative, the terms will destroy each
other, except one or two at the end, which remain undestroyed.
These terms will be of the same order of magnitude as the terms
at the part of the series (16) or (17) where we stopped, and there-
fore will be insensible for the value of n or n for which we are
calculating the series numerically, and, much more, for all superior
values. Suppose the arbitrary constants ^, -B in (16) determined
by means of the ultimate form of Z7 for ?j = 00 , and C in (17) by
means of the ultimate form of U for n' = 00 . Then U^ satisfies
exactly a diflferential equation which dififers from (11) by having
the zero at the second side replaced by a quantity which is in-
sensible for the value of n or n with which we are at work, and
which is still smaller for values comprised between that and the
particular value, (namely 00 ), by means of which the arbitrary
constants were determined so as to make U^ and U agree. Hence
f/j will be a near approximation to U. But if we went too far
in the series (16) or (17), so as, after having gone through the
insensible terms, to take in some terms which were not insensible,
the diflferential equation which U^ would satisfy exactly would
diflfer sensibly from (11), and the value of U^ obtained would be
faulty.

s. 11. 22

338 ON THE NUMERICAL CALCUIJLTION OF A CLASS OF

8. It remains to determiQe the arbitrary constants A, B, C.
For this purpose consider the integral

•'0

e'^'^^'dx (18),

where q is any imaginary quantity whose amplitude does not
lie beyond the limits — 7r/6 and + 7r/6. Since the quantity under
the integral sign is finite and continuous for all finite values of x,
we may, without affecting the result, make x pass from its initial
value to its final value oo through a series of imaginary values.
Let then «? = 9 + y, and we get

Q^^r €'^'^^dy.

where the values through which y passes in the integration are
not restricted to be such as to render x real. Putting y = (89)"* t,
where that value of the radical is supposed to be taken which has
the smallest amplitude, we get

Q = (3j)-*62^/€-<«9)-*^-^de (19).

The limits of t are —3*}* and an imaginary quantity with an
infinite modulus and an amplitude equal to \ol, where a denotes
the amplitude of q. But we may if we please integrate up to
a real quantity /o, and then, putting t = p€^'^^~^\ and leaving p
constant, integrate with respect to d from to J2, and lastly put
p = 00 . The first part of the integral will be evidently convergent
at the limit 00, since the amplitude of the coefficient of t* in the
index does not lie beyond the limits — Jtt and + Jtt ; and calling
the two parts of the integral with respect to < in (19) T, jT^, we
get

T^r^^€-^^^'''''^dt (20),

7; = limit<p„)p V~l {^^-^^y'^^^^^'P^^^>f-^^^>i'^i d0...{21).

We shall evidently obtain a superior limit to either the real or
the imaginary part of T^ by reducing the expression under the
integral sign to its modulus. The modulus is €"* where

e = (3c) -* p' cos (30 - |a) + p" cos 20,
c being the modulus of q. The first term in this expression is

DEFINITE INTEGRALS AND INFINITE SERIES. 339

never negative, being only reduced to zero in the particular case
in which ^ = and a = ± tt/G. The second term is never less than
p' cos Jtt or ^/o', and is in general greater. Hence both the real
and the imaginary parts of the expression of which T^ is the limit
are numerically less than Ja/o€~*^', which vanishes when p = oo,
and therefore T^ = 0. Hence we have rigorously

Q = mY^^'T (22).

Let us now seek the limit to which T tends when c becomes
infinite. For this purpose divide the integral T into three parts
7",, T^y Tj, where 2\ is the integral taken from — 3*g^ to a real
negative quantity — a, T, from — a to a real positive quantity + 6,
and r, from 6 to oo ; and suppose c first to become infinite, a and h
remaining constant, and lastly make a and b infinite.

Changing the sign of < in T^y and the order of the limi^, we get

Put ^ = /o€*^^~^l Then we may integrate first from p = a to
p = 3M while remains equal to 0, and afterwards from ^ =
to = a while f p remains equal to 3*c*. Let the two parts of the
integral be denoted by T\ T\ We shall evidently obtain a
superior limit to T by making the following changes in the
integral : first, replacing the quantity under the integral sign by
its modulus ; secondly, replacing ^ in the index by the product

of ^ and the greatest value (namely 3*0^) which t receives in the
integration ; thirdly, replacing a by the smallest quantity (namely
0) to which it can be equal, and, fourthly, extending the superior
limit to 00 . Hence the real and imaginary parts of T' are both

numerically less than 1 c"'^* dt, a quantity which vanishes in the

limit, when a becomes infinite.

We shall obtain a superior limit to the real or imaginary part
of T' by reducing the quantity under the integral sign to its
modulus, and omitting V(~" 1) ^ii the coefficient. Hence L will be
such a limit if

L = 3M \^-<^mde, where /(^) = 3 cos IB - cos (3d - fa).

We may evidently suppose a to be positive, if not equal to zero,
since the case to which it is negative may be reduced to the case

22—2

S40 ON THE NUMERICAL CALCULATION OF A CLASS OF

in which it is positive by changing the signs of a and 0. When
d = 7r/6, the first term in f{d) is equal to f, which, being greater
than 1, determines the sign of the whole, and therefore /(ff) is
positive; and f{0) is evidently positive from ^ = to = ir/6,
since for such values cos 2d > J. Also in general f{0) = — 6 sin 2^
+ 3 sin (3^ — fa), which is evidently positive from = ^/6 to
^ = 7r/4, and the latter is the largest value we need consider, being
the extreme value of fi when a has its extreme value tt/G. When
has its extreme value f2,/(^) = 2 cos 3a, which is positive when
a < 7r/6, and vanishes when a = 7r/6. Hence /(d) is positive when
d < f a ; for it has been shewn to be positive when < 7r/6, which
meets the case in which a < ir/d or — 7r/9, and to be constantly
decreasing from = 7r/6 to d = |a, which meets the case in which
> 7r/9. Hence when a < ir/G the limit of L for c = x is zero,
inasmuch as the coefficient of (^ in the index of € is negative and
finite ; and when a = tt/G the same is true, for the same reason,
if it be not for a range of integration lying as near as we please to
the superior limit. In this case put for shortness /(d) = S, regard
fa - d as a function of S, F{B), and integrate from S = to S = )8,
where )3 is a constant which may be as small as we please. By
what precedes, -P'(8) will be finite in the integration, and may
be made as nearly as we please equal to the constant F* (0) by
diminishing fi. Hence the integral ultimately becomes

J A

which vanishes when c becomes infinite. Hence the limit of T
is zero.

We have evidently ^a < I e^^dt,

which vanishes when b becomes infinite. Hence the limit of T
is equal to that of T,. Now making c first infinite and afterwards
a and b, we get

limit of T, = limit of [ e"^ dt=j "e'^dt^ V^r,

J —a J —00

and therefore we have ultimately, for very large values of c,

k

^ = (^)^^ »

DEFINITE INTEGRALS AND INFINITE SERIES. 341

In order to apply this expression to the integral u given by (6),
we must put

3}* = ri€* , whence ?=(k) ^ " >
whence we get ultimately

Comparing with (15) we get

9. We cannot make n pass from positive to negative through
a series of real values, so long as we employ the series according
to descending powers, because these series become illusory when
n is small. When n is imaginary we cannot speak of the integrals
which appear at the right-hand side of (-5), because the exponential
with a positive index which would appear under the integral signs
would render each of these integrals divergent. If however we
take equation (6) as the definition of u, and suppose U always
derived from u by changing the sign of V(— 1) in the coefficient
of the integral and in the value of p, but not in the expression
for w, and taking half the sum of the results, we may regard u
and U as certain functions of n whether n be real or imaginary.
According to this dei&nition, the series involving ascending integral
powers of n, which is convergent for all values of n, real or imagi-
nary, however great be the modulus, will continue to represent u

* This result might also have been obtained from the integral U in its original

shape, namely, | cos {x^ - nx) doe, by a method similar to that employed in Art. 21.

If a^ be the positive value of x which renders a^-nx& minimum, we have a^=8'~^n*.
Let the integi-al U be divided into three parts, by integrating separately from x=0
to x=Xi-at from x^Xi-a tox=sXi+&, and from x=Xi-^b toz=ao; then make n
infinite while a and b remain finite, and lastly, let a and b vanish. In this
manner the second of equations (23) will be obtained, by the assistance of the
known formula)

fan /•»

I COS x-dx =1 sin x^dx = 2~ * ir* .

i

342 ON THE NUMERICAL CALCULATION OF A CLASS OF

when n is imaginary. The difiFerential equation (11), and conse-
quently the descending series derived from it, will also hold good
when n is imaginary ; hut since this series contains radicals, while
U is itself a rational function of n, we might expect heforehand
that in passing from one imaginary value of n to another it should
sometimes be necessary to change the sign of a radical, or make
some equivalent change in the coefficients A, B. Let n = njC""^^""^*

where n^ is positive. Since both values of 2 (n/S)^ are employed
in the series, with different arbitrary constants, we may without

loss of getierality suppose that value of n* which has ^v for its
amplitude to be employed in the circular functions or exponentials,
as well as in the expression for 8. In the multiplier we may

always take — 1;/4 for the amplitude of n~i by including in the
constant coefficients the factor by which one fourth root of n diffiers
from another; but then we must expect to find the arbitrary
constants discontinuous. In fact, if we observe the forms of iJ
and 8, and suppose the circular functions in (15) expanded in
ascending series, it is evident that the expression for U will be
of the form

An'^N+Bn^N' (25),

where N and N' are rational functions of n. At least, this will be
the case if we regard as a rational function a series involving de-
scending integral powers of n, and which is at first rapidly con-
vergent, though ultimately divergent, or rather, if we regard as
such the function to which the convergent part of the series is a
very close approximation when the modulus of w is at all large.
Now, if A and B retained the same values throughout, the above
expression would not recur till v was increased by Sir, whereas U
recurs when v is increased by 27r. If we write i; + 27r for v, and
observe that N and N' recur, the expression (25) will become

- J'^An'^i N + J^lBni N' ;

and since Z7 recurs it appears that -4, J? become V(—l) ^y — V(— 1)-B,
respectively, when v is increased by 27r. Also the imaginary part
of the expression (25) changes sign with v, as it ought; so that, in
order to know what A and B are generally, it would be sufficient
to know what they are from i; = to i; = tt.

If we put nj€*^^"^^ for n in the second member of equation (15),

and write ^ for 2 . 3"*n^^, and R^, 8^ for what jR, 8 become when

DEFINITE INTEGRALS AND INFINITE SERIES. 343

Tij is put for n in the second members of equations (16) and all
the terms are taken positively, we shall get as our result

Now the part of this expression which contains (R^ + S^e^ ought
to disappear, as appears from (17). If we omit the first part of
the expression, and in the second part put for A and B their values
given by (24), we shall obtain an expression which will be identi-
cal with the second member of (17) provided

C=-— , (26).

2.8* ^

This mode of determining the constant C is anything but satis-
factory. I have endeavoured in vain to deduce the leading term
in Z/for n negative from the integral itself, whether in the original
form in which it appears in (5), or in the altered form in which it
is obtained from (6) *. The correctness of the above value of G
will however be verified further on.

10. Expressing n, U in terms of m, W by means of (8) and
(9), putting for shortness

. *=Ki)'='(i)' <"'•

where the numerical values of m and n are supposed to be taken
when these quantities are negative, observing that 16 \/(3^') == 72^,
and reducing, we get when m is positive

W= 2* (3m)"* Ircos (* - 5) + Ssin U - ^)| ...(28),

where
7?_i 1^5_:7.11. 1.5.7.11.13.17.19.23

...(29).

1.2(72*^)"^ 1.2.3.4(72*^)^

Q= _LA_ 1.5.7.11.13.17
1.72(f> 1.2.3(72^)' "*■•••

When m is negative, so that W is the integral expressed by writ-
ing — m for m in (1), we get

[* The difficulty was overcome in a later paper entitled **0n the discontinaity
of arbitrary constants which appear in divergent developments." {Transactions
of the Cambridge Philosophical Society^ Vol. x. p. 106.)]

344 ON THE NUMERICAL CALCULATION OF A CLASS OF

11. Reducing the coefficients of ^"^ (pT^... in the series (29)
for numerical calculation, we have, not regarding the signs,

order (i) (ii) (iii)

logarithm 2-841638; 2-569766; 2-579704;
coefficient -0694444; -0371335; -0379930;

(iv) _ (v) (vi)

2-760793; 1064829; 1464775;
-0576490; 116099; -291592.

Thus, for m = 3, in which case (f>=^'rr, we get for the successive
terms after the first, which is 1,

-022105, -003762, -001225, -000592, -000379, -000303.

We thus get for the value of the series in (30), by taking half the
last term but one and a quarter of its first difference, -980816;
whence for wi = 3, W= 6"* x •9808166-''= 0173038, of which the
last figure cannot be trusted. Now the number given by Mr Airy
to 5 decimal places, and calculated from the ascending series and
by quadratures separately, is -01730, so that the correctness of the
value of C given by (26) is verified.

For wi = + 3 we have from (28)

Tr=-3-i(i?-5) = -3-*C9965-0213) = --5632,

which agrees with Mr Airy's result — '56322 or — '56323. As m
increases, the convergency of the series (29) or (30) increases
rapidly.

12. The expression (28) will be rendered more easy of numeri-
cal calculation by assuming jR = Jlfcos'^, iS=J/sin^, and ex-
panding M and tan '^ in series to a few terms. These series will
evidently proceed, the first according to even, and the second
according to odd inverse powers of (f>. Putting the several terms,
taken positively, under the form 1, a<fr\ ab<f>'*, abc^'^, abccUfr*, &c.,
and proceeding to three terms in each series, we get

3f=l-a(6--|)<^-* + a|6c(d--a) + |'(6-|)|<^-^..(31),

tan '^ = a<^"* '-ab{c-a)^'^ + ab [cd (e — a) — ab (c - a)] <^"*...(32).

The roots of the equation W= are required for the physical
problem to which the integral IV relates. Now equations (28),

DEFINITE INTEGRALS AND INFINITE SERIES. 345

(29) shew that when m is at all large the roots of this equation are
given very nearly by the formula ^ = (i — \) tt, where i is an inte-
ger. From the definition of -^ it follows that the root satisfies
exactly the equation

<^ = (*-i)^ + t (33).

By means of this equation we may expand ^ in a series according
to descending powers of 4>, where O = (i — ^) tt. For this purpose
it will be convenient first to expand '^ in a series according to
descending powers of ^, by means of the expansion of tan~^ x and
the equation (32), and having substituted the result in (33) to
expand by Lagrange*s theorem. The result of the expansion
carried as far as to 4>"* is

«^= <> + a<>"' - {oi (c - a) + K + a'} ^'^

+ [ah [cd {e-a)'^ab{c- a)] + a'6 {c-a)-h ^a*

+ 4a[a6(c-a) + Ja'] + 2a'}<>-* (34).

13. To facilitate the numerical calculation of the coefficients
let

and let the coefficients of 4>^y <^"* in (31) be put under the forms

A A

- i-^ij^y 123' 41^ ' ^^^ similarly with respect to (32), (34).

Then to calculate W for a given value of m, we have

Tr=2i(3m)-*ifcosf<^-.^-i/r) (35),

where ^^= ^ " OZ>« *"' + 1727^^^ ^ (^«)'

tan^ = ^-^r-rAz>3*-' + lT2T3-:4T5^
and for calculating the roots of the equation Tr= 0, we have

346 ON THE NUHEBICAL CALCULATION OF A CLASS OF

The coefficients in these formulae are given by the equations

A, = a' {b' - oO; A^ = o' [b'c' {d' - 4o') + Sa" (26' - a')] •)
0, = a'; C, = o'6'(c' - 3a'); G, = a'6' {c'd' (e' - 5a') - 10C7,}

E^ = a'; E, = C, + 2a" (SD + aO
E^=C^ + 20a' (42) + a') (7, + 24a'» + 80o'» D {W + 2a') J

...(39).

14. Putting in these formulae

a' = 1.5; 6' = 7.11;c' = 13.l7;rf'=19.23;e =25.29; D = 72;
we get

il,= 5.72; ^^ = 3.5.72*.457; C, = 5; (7, = 2.5.7.11.103;
C, = 4». 5'. r.l 1.23861; ^. = 5; ^, = 72.1255; ^,=4.5».72M0883;
whence we obtain, on substituting in (36), (37), (38),

^-^"144* +41472* '

, , 5 .., 39655 ^_,^ 321526975 .^
tan t = 72 * - jj^g^^^ p + 2902376448 * '

^ = ^ + l^-._ 1255^*- + _272071
'^ ^72 31104 ^ 2239488 '

Reducing to decimals, having previously divided the last equation
by IT, and put for O its value (t — \) ir, we get

if = 1 -034722^-*+ -055097^-* (40),

tan ^ = 069444^-' - 035414^-' + 110781 ^-» (41),

^_. 028145 -0265 10 . -129402

--1---0+ ^-.— J- -(4i_i). + (4t-_i). (*-i;-

15. Supposing 1 = 1 in (42), we get

^ = -75 + 0094 - -0010 + 0005 = -7589;

whence m = 3 (<^/7r)' = 2*496. The descending series obtained in
this paper fail for small values of m; but it appears from Mr Airy's
table that for such values the function W is positive, the first
change of sign occurring between m = 2'4 and m = 26. Hence the
integer i in (42) is that which marks the order of the root. A
more exact value of the first root, obtained by interpolation from
Mr Airy's tabic, is 2*4955. For i = 1 the series (42) is not conver-

DEFINITE INTEGRALS AND INFINITE SERIES. 347

gent enough to give the root to more than three places of decimals,
but the succeeding roots are given by this series with great
accuracy. Thus, even in the case of the second root the value of
the last term in (42) is only '000007698. It appears then tliat
this term might have been left out altogether.

16. To determine when W is a maximum or minimum we
must put dW/dm=0. We might get dW/dm by direct differen-
tiation, but the law of the series will be more easily obtained from
the differential equation. Resuming equation (11), and putting
V for dU/dn, we get by dividing by n and then differentiating

dn* n dn S

This equation may be integrated by descending series just as
before, and the arbitrary constants will be determined at once by
comparing the residt with the derivative of the second member
of (15), in which A, B are given by (24). As the process cannot
fail to be understood from what precedes, it will be suflScient to
give the result, which is

F= 3- W {-R' cos (f + '^ + 5" sin (* + J)l (*3),

where

p' = l _-l«7 .5.13 -1.7.5 .1 3.11.19,17.25
1.2(72<^)» "^ 1.2.3.4 (72<^)*

_ -1.7 :il.7.5.13. ll.J^
^ "1.72^ 1.2.3(72<^)» '^""

(44).

17. The expression within brackets in (43) may be reduced
to the form ilf cos(^ + Jtt- '^) just as before, and the formulae
of Art. 13 will apply to this case if we put

a =-1.7; 6' = 5.13; c' = 11.19; &c., i) = 72.

The roots of the equation d W/dm = are evidently the same as
those of ^^^=0. They are given approximately by the formula
<^ = (i — f)7r, and satisfy exactly the equation ^ = (t— })7r + '^.
The root corresponding to any integer % may be expanded in a
series according to the inverse odd powers of 4i — 3 by the formula;

348 ON THE NUMERICAL CALCULATION OF A CLASS OF

of Art 13. Putting (i — |)7r for ^, and taking the series to three
terms only, we get

E^ = -7; ^3 = -84168;

whence ^ = ^ - /g ^"^ + ^^^ O'* ;

or, reducing as before,

4>_. .. -039403 ^ 024693 ....

^-^-'^^-^ii^gi +(4^:3)^ ^ ^^•

This series will give only a rough approximation to the first
root, but will answer very well for the others.

For i = 1 the series gives tt"* ^ = 25 - '039 + '025, which
becomes on taking half the second term and a quarter of its first
difference -25 - -019 - 004 = -227, whence m = 112. The value
of the first root got by interpolation from Mr Airy's table is 1*0845.
For the second and third roots we get from (45)

for i = 2, Tr"' <^ = 1-25 - '00788 + -00020 = 1-24232 ;

for i = 3, IT-' 4> = 2-25 - 00438 + '00003 = 224565.

For higher values of i the last term in (45) may be left out
altogether.

18. The following table contains the first fifty roots of the
equation W=^0, and the first ten roots of the derived equation.
The first root in each case was obtained by interpolation from
Mr Airy's table; the series (42) and (45) were sufficiently con-
vergent for the other roots. In calculating the second root of
the derived equation, a rough value of the first term left out in
(45) was calculated, and its half taken since the next term would
be of opposite sign. The result was only — '000025, so that the
series (45) may be used even when % is as small as 2. By far
the greater part of the calculation consisted in passing from the
values of tt"*^ to the corresponding values of m. In this part
of the calculation 7-figure logarithms were used in obtaining the
value of \m, and the result was then multiplied by 3.

A table of differences is added, for the sake of exhibiting the
decrease indicated by theory in the interval between the con-
secutive dark bands seen in artificial rainbows. This decrease
will be readily perceived in the tables which contain the results

DETtNITE INTEGRALS AND INFINITE SERIES.

349

of Professor Miller's observations*. The table of the roots of the
derived equation, which gives the maxima of TF*, is calculated for
the sake of meeting any observations which may be made on the
supernumerary bows accompanying a natural rainbow, since in
that case the maximum of the red appears to be what best admits
of observation.

i

VI diU.

i

m am.

1
i
3
4
5
6
7
8
9
10
U
12
13
U
15
IG

18
19
20
21
S2
23
24
25

2-4955 ,.„-„
4-3631 J.™'^
5-8922 \_^l\

12-7395 ^%

13-6924 Z^
14-6132 Z^

15-5059 .™:'

l-.2;^ -^^^

iH-ili^v '8250
18-8502 :?^
19-8399 .i^
20-4139 .f.*l

21-1-36 .:^g

21'9199 '™
22-G536 .itV,
2.3-3757 ::?*
24-0868 .:i^'
24-7876 lana
2.V4785 .°°^

26
27
28
29
30
31
32
33
34
35
36
37
38

40
41
42
43
44
45
46
47
48
49
50

26-1602 .„-„

28-1546 S';
28-8037 ."»;
29-4456 .J^»
30^)805 ?^
30-7089 .J:^^

sis :"i™

32-5567 ."'"
33-1610 !^
33-7599 .™™
34-3535 .™?»

35-5256 ™*
36-104.1 .5;35

3''»«' -5*6

SSS :-|

38-9323 ™;|
39-4855 .gJJ
40-0349 ,?T''*
40-5805 *"'*'

I

a

3

4
5

3-4669 ; l°l:
%\*tl 1-4M8
' "^'* 1-1914

6
7
8
9
10

S-0599

12-2475 '«' J
13-2185 ' "

• Cambridge Philoiophieal Tmmaeliotu, Vol. vii. p. 377.

350 ON THE NUMERICAL CALCULATION OF A CLASS OP

SECOND EXAMPLE.

19. Let us take the integral

which occurs in a great many physical investigations. If we
perform the operation x.djdx twice in succession on the series
we get the original series multiplied by — sfy whence

d^u 1 du ^ ,^^,

cte« + idi + " = « (*')•

20. The . form of this equation shews that when x Ls very
large, and receives an increment Bx^ which, though not necessarily
a very small fraction itself, is very small compared with w, u is
expressed by A cos hx-\-B sin hx, where under the restrictions
specified A and B are sensibly constant'!'. Assume then, according
to the plan of Art. 5,

w = 6*^^ {ila:* + Ba^ + Cxy-\-..,] (48).

On substituting in (47) we get

J~^ {(2a + 1) -4a;*-! + (2/9 + 1) Bcfi'^ + . . .}

+ a'^a:*-« + /8'J5'»-«+ ... =0.

Since we want a descending series, we must put
2a+l=0; )8=a-l; 7 = )8-l...;

(2i8 + l)5 = N/^aM; (27 + 1)(7 = V^/8'jB...;

* This integral has heen tabulated by Mr Airy from a;=0 to x— 10, at intervals
of 0-2. The table wiU be foond in the 18th Volume of the Philosophical Magaz(nr,
pagel.

t That the Ist and 8rd terms in (47) are ultimately the important terms, may
readily be seen by trying the terms two and two in the way mentioned in the intro-
duction. Thus, if we suppose the first two to be the important terms, we get
ultimately U=A or U=B log x, either of which would render the last term more
important than the 1st or 2nd, and if we suppose the 2nd and 3rd to be the

important terms, we get ultimately u = Ae~' '\ which would render the first term
more important than either of the others.

DEFINITE INTEGRALS AND INFINITE SERIES. 351

whence « = — i; )8= — fj 7 = "-§«'«;

1*.3'.5' /

Substituting in (48), reducing the result to the form

^(P+V-lQ),

adding another solution of the form B (P — ^/— IQ), and changing
the arbitraiy constants, we get

u = -4a?"* (R cos x + Ssinx) + Bx'^ (R sin a? — iScos x) . ...(49),

, 7^ ^ 1.0 1.0.5.7

where ""-^~l.2(8a;)«'*' 1.2.3.4 (8a;)**"

1' 1'.3«.5' I

1.8x 1.2.3 (8a:)' '^•" J

(50).

21. It remains to determine the arbitrary constants A, B. In
equation (46) let cos 0=1 — fi, whence

where M = {2fi - fi^^ - {2fiy *,

a quantity which does not become infinite between the limits of /t.
Substituting in (46) we get

u

a/2 f^ 2 f*

= -^ I cos{(l — fi) x] fjL"^ dfi + - I cos{(l — /Lt)a?}ird/4...(51).

By considering the series whose n^ term is the part of the
latter integral for which the limits of fi are nirx"^ and (n + 1) irx'^
respectively, it would be very easy to prove that the integral has
a superior limit of the form Hx'^, where J7 is a finite constant,
and therefore this integral does not furnish any part of the leading
terms in u. Putting fix = v in the first integral in (61), so that

fi-l dfi^x'^v'^dpf

352 ON THE NUMERICAL CALCULATION OF A CLASS OF

observing that the limits of p are and x, of which the latter
ultimately becomes oo , and that

1 cos V ,v~^dfjL— 2 j cos\^d\= a/ ^

sin \*dX =1 sin 1^ . p-^dv,

Jo

we get ultimately for very large values of x

u = (ttx)"^ (cos X + sin x).

Comparing with (49) we get

whence u=f- -J Rcosfx— j-]-\-( — J 5sin[a: — -rj ...(52).

For example, when a? = 10 we have, retaining 5 decimal places
in the series,

^ = 1 - -00070 + 00001 = -99931 ; 8 = '01250 - '00010 = '01240

TT

Angle a? - ^ = 527" 95780 = 3 x 180' - 12^ 2' 32" ;

whence u = — '24594, which agrees with the number (— '2460)
obtained by Mr Airy by a far more laborious process, namely, by
calculating from the original series.

22. The second member of equation (52) may be reduced to
the same form as that of (28), and a series obtained for calculating
the roots of the equation u = just as before. The formulae of
Art. 13 may be used for this purpose on putting

a'=l*; 6' = 3"; c =5"; &c.; D = S,

and writing x, X for ^, O, where X^(i — J) tt. We obtain
J, = 8; ^, = 3.8'.53; (7^ = 1; C, = 2,'3,\ll;
(7, = 3'.4\5.1139; ^^ = 1; ^, = 8.31; J?, = 4* . 3779 ;

* This expression for u, or rather an expression differing £rom it in nothing but
notation and arrangement, has been already obtained in a different manner by
Sir William B. Hamilton, in a memoir ** On Fluctuating Functions." See Transae-
(torn of the Uoyal Irith Academy, Vol. xix. p. 313.

DEnNITE INTEGRALS AND INFINITE SERIES. 353

whence we get for calculating u for a given value of x

«=(-|yifcos(.-J-^) (33).

For calculating the roots of the equation w = we have

Reducing to decimals as before, we get

Jlf=l - -0023 0?-^ + -103516 a?-* (54),

tan ^=1250?-' -•064453 a;-' +208557 a;-* (55),

X . „. -050661 053041 . -262051 ,.^,

TT 4fi—l(4fi — ly (4t - 1)' ^

As before, the series (56) is not suflSciently convergent when
i = 1 to give a very accurate result. In this case we get

TT-* a: = -75 + -017 - -002 + -001 = 766,

whence a? = 2*41, Mr Airy's table gives ^ = + -0025 for a? = 2*4,
and tt = — -0968 for x = 2-6, whence the value of the root is 2*4050
nearly.

The value of the last term in (56) is -0000156 for i = 2, and
•00000163 for i = 3, so that all the roots after the first may be
calculated very accurately from this series.

THIRD EXAMPLE.

23. Consider the integral

w

tr =r - I I COS (a? COS 0) xdx dd

= 1 uxax ~ "2 "" 2' 4 "^2* 4' 6 "" ••••lo7}»

7? 71^ 2v

* The series 1 - =— j + ^—t^—^ ... or - has been tabulated by Mr Aiiy from x=0

to a; =12 at intervals of 0-2. See Camb. Phil. Trans. Yo\. v. p. 291. The same
function has also been calculated in a different manner and tabulated by M. Schwerd

S. II. 23

354 ON THE NUMERICAL CALCULATION OF A CLASS OF

which occurs in investigating the diffiraction of an object-glass
with a circular aperture.

By performing on the series the operation denoted by
X . didx , x'^ . didx, we get the original series with the sign
changed, whencQ

^^-^5^+"=" <^^>-

We may obtain the integral of this equation in a form similar
to (49). As the process is exactly the same as before, it will be
sufficient to write down the result, which is

v — A'x^ (i2cosa;+ Ssina;) + J5'a;^(i2sina;— iS'cosa:) , (59),

where

-1.3.1.5- 1.8.1.5.3.7.5.9
^""^ 1.2(8a:)» "*■ " 1 . 2.3, 4 (8a;)"
^__ — 1.3 —1.3.1.5.3.7

^ ~ T78^ 1.2.3(8a;)» "■*■•••

...(60),

the last two factors in the numerator of any term being formed by
adding 2 to the last two factors respectively in the numerator of
the term of the preceding order.

The arbitrary constants may be easily determined by means of
the ec^uatioa

dv

dic^-"^ (")•

Writing down the leading terms only in this equation, we have

x^ (- A' sin x-hB'coBx) = tt'M (cos x + sin x),
whence

'=(!)'

hi cos (*• - -^j + ^' sin (a; - ?J)] (62).

k

24. Putting in the formula? of Art. 13,
a=-1.3; 6' = 1.5; c'=3.7; d' = 5.9; e' = 7.11; 2) = 8;

in his work on diflfraction. The argument in the latter table is the angle IQOPIr . jp,
and the table extends from 0« to 1125" at intervals of 16^ «hat is, from «=sO to
X- 19*63 at intervals of 0*262 nearly.

DEFINITE INTEGRALS AND INFINITE SERIES. 355

we get

J, = -3.8; ^,«-8'.8Ml; Oj=-3; C,= -2.8«.5»;

C^ = -3\4*.5M27; E^ = -3; ^3 = -3^8; £', = -3'.4.8M31 ;
whence we get for the formulae answering to those of Art. 22,

X being in this case equal to (i + \) ir.

Reducing to decimals as before, we get for the calculation of v
for a given value of x,

3/= 1+1875 a?"* + -193350 a;-' (63),

tan i|r = - -375 x"' -H '146484 a:"" - -348817 a?'* (64),

v = (^jMcos(x-'~^f^ (65);

and for calculating the roots of the equation i; = 0,

X . . ^. 151982 . -015399 245835 ,^^,

TT 4i + l (4t+l)' (4i+l)*

25. The following table contains the first 12 roots of each of
the equations u = 0, and x'^v = 0. The first root of the former

•

I

1

2
3

4

5

6

7

8

10

11

12

- for i«=0

diflf.

- for v=0

diff.

2-7546 1*^
3-75»* -9993

5-7522 .?^§

7-7516 ^
8-7514 -9999

9-7513 .QOQQ

10-7512 ^^l
11-7511 '^^^

2-2330 };^
0.9000 1*0053

2428 1 ^11

6-2439 trrii

7.0440 1-0009

8I454 ^"^^^

9-^4^9 }^

10*2463 j:^

23—2

356 ON THE NUMERICAL CALCULATION OF A CLASS OF

was got by interpolation from Mr Airy's table, the others were
calculated from the series (56). The roots of the latter equation
were all calculated from the series (66), which is convergent
enough even in the case of the first root. The columns which
contain the roots are followed by columns which contain the
differences between consecutive roots, which are added for the
purpose of shewing how nearly equal these differences are to 1,
which is what they ultimately become when the order of the root
is indefinitely increased.

26. The preceding examples will be sufficient to illustrate
the general method. I will remark in conclusion that the pro-
cess of integration applied to the equations (11), (47), and (58)
leads very readily to the complete integral in finite terms of the
equation

da?

-{^^'^}y-' (67),

where i is an integer, which without loss of generality may be
supposed positive. The form under which the integral imme-
diately comes out is

v^Ac^W ^'(t + l) , (^'-l)^(^fl)(^ + 2) \

1 Jc-fl . M^^ + l) . (^-l)^(^^fl)(^^f2) |

+ ^^ Y^1.2qx^ 1.2i2qxy +-J'

where each series will evidently contain i + l terms. It is well
known that (67) is a general integrable form which includes as a
particular case the equation which occurs in the theory of the
figure of the earth, for q in (67) is any quantity real or imaginary,
and therefore the equation formed from (67) by writing + j'y for
— q^y may be supposed included in the form (67).

It may be remarked that the differential equations discussed
in this paper can all be reduced to particular cases of the equation
obtained by replacing t(i+ 1) in (67) by a general constant. By

taking grj?, where g is any constant, for the independent variable

DEFINITE INTEGRALS AND INFINITE SERIES. 857

in place of n in the differential equations which U, V in the first
example satisfy, these equations are reduced to the form

and (47), (58) are in this form already. Putting now y = x~'e, we
shall reduce the last equation to the form required.

[The four following are from the Beport of the British Association for 1850,

Tart II. p. 19.]

k

Ox THE Mode of Disappearance of Newton's Rings in passing

THE ANGLE OF TOTAL INTERNAL REFLEXION.

When Newton's rings are formed between the under surface of
a prism and the upper surface of a lens, there is no difficulty in
increasing the angle of incidence so as to pass through the angle of
total internal reflexion. When the rings are observed with the
naked eye in the ordinary way, they appear to break in the upper
part on approaching the angle of total internal reflexion, and pass
nearly into semicircles when that angle is reached, the upper edges
of the semicircles, which are in all cases indistinct, being slightly
turned outwards when the curvature of the lens is small.

The cause of the indistinctness will be evident from the follow-
ing considerations. The order of the ring (a term here used to
denote a number not necessarily integral) to which a ray reflected
at a given obliquity from a given point of the thin plate of air
belongs, depends partly on the obliquity and partly on the thick-
ness of the plate at that point. When the angle of incidence is
small, or even moderately large, the rings would not be seen, or at
most would be seen very indistinctly, if the glasses were held near
the eye, and the eye were adapted to distinct vision of distant
objects, because in that case the rays brought to a focus at a given
point of the retina would correspond to a pencil reflected at a
given obliquity from an area of the plate of air, the size of which
would correspond to the pupil of the eye ; and the order of the
rays reflected from this area would vary so much in passing from
the point of contact outwards that the rings would be altogether

PAPERS FROM BRITISH ASSOCIATION REPORT, 1850. 359

confused. When, however, as in the usual mode of observation,
the eye is adapted to distinct vision of an object at the distance of
the plate of air, the rings are seen distinctly, because in this case
the rays proceeding from a given point of the plate of air, and
entering the pupil of the eye, are brought to a focus on the retina^
and the variation in the obliquity of the rays forming this pencil
is so small that it may be neglected.

When, however, the angle of incidence becomes nearly equal to
that of total internal reflexion, a small change of obliquity pro^
duces a great change in the order of the ring to which the reflected
ray belongs, and therefore the rings are indistinct to an eye
adapted to distinct vision of the surfaces of the glass* They are
also indistinct, for the same reason as before, if the eye be adapted
to distinct vision of distant objects.

To see distinctly the rings in the neighbourhood of the angle
of total internal reflexion, the author used a piece of blackened
paper in which a small hole was pierced with the point of a
needle. When the rings were viewed through the needle-hole,
in the light of a spirit-lamp, the appearance was very remarkable.
The first dark band seen within the bright portion of the field of
view where the light suffered total internal reflexion was some-
what bow-shaped towards the point of contact, the next still more
so, and so on, until at last one of the bands made a great bend and
passed under the point of contact and the rings which surrounded
it, the next band passing under it, and so on. As the incidence
was gradually increased, the outermost ring united with the bow-
shaped band next above it, forming for an instant a curve with a
loop and two infinite branches, or at least branches which ran out
of the field of view : then the loop broke, and the curve passed
into a bulging band similar to that which had previously sur-
rounded the rings. In this manner the rings, one after another,
joined the corresponding bands till all had disappeared, and nothing
was left but a system of bands which had passed completely below
the point of contact, and the central black spot which remained
isolated in the bright field where the light suffered total internal
reflexioa Corresponding appearances were seen with daylight or
candlelight, but in these cases the bands were of course coloured,
and not near so many could be seen at a time.

360 PAPERS FROM THE REPORT

On Metallic Reflexion.

The effect which is produced on plane-polarized light by re-
flexion at the surface of a metal, shews that if the incident light
be supposed to be decomposed into two streams, polarized in and
perpendicularly to the plane of reflexion respectively, the phases as
well as the intensities of the two streams are diflFerently affected
by the reflexion. It remains a question whether the phase of
vibration of the stream polarized in the plane of reflexion is acce-
lerated or retarded relatively to that of the stream polarized per-
pendicularly to the plane of reflexion. This question was first
decided by the Astronomer Royal, by means of a phsenomenon
relating to Newton's rings when formed between a speculum and
a glass plate. Mr Airy's paper is published in the Cambridge
Philosophical Transactions. M. Jamin has since been led to the
same result, apparently by a method similar in principle to that of
Mr Airy. In repeating Mr Airy's experiment, the author expe-
rienced considerable diflSculty in observing the phaenomenon. The
object of the present communication was to point out an extremely
easy mode of deciding the question experimentally. Light polar-
ized at an azimuth of about 45^ to the plane of reflexion at the
surface of the metal was transmitted, after reflexion, through a
plate of Iceland spar, cut perpendicular to the axis, and analysed
by a Nicors prism. When the angle of incidence was the smallest
with which the observation was practicable, on turning the Nicol's
prism properly the dark cross was formed almost perfectly ; but on
increasing the angle of incidence it passed into a pair of hyperbolic
brushes. This modification of the ring is very well known, having
been described and figured by Sir D. Brewster in the Philosophical
Transactions for 1830. Now the question at issue may be imme-
diately decided by observing in which pair of opposite quadrants
it is that the brushes are formed, an observation which does not
present the slightest difficulty. In this way the author was led
to Mr Airy's result, namely, that as the angle of incidence increases
from zero, the phase of vibration of light polarized in the plane of
incidence is accelerated relatively to that of light polarized in a
plane perpendicular to the plane of incidence.

OF THE BBITISH ASSOCUTION, 1850. 361

On a Fictitious Displacement of Fringes of

Interference.

The author remarked that the mode of determining the refrac-
tive index of a plate by means of the displacement of a system of
interference fringes, is subject to a theoretical error depending
upon the dispersive power of the plate. It is an extremely simple
consequence (as the author shewed) of the circumstance that the
bands are broader for the less refrangible colours, that the point of
symmetry, or nearest approach to symmetry, in the system of
displaced fringes, is situated in advance of the position calculated
in the ordinary way for rays of mean refrangibility. Since an
observer has no other guide than the symmetry of the bands in
fixing on the centre of the system, he would thus be led to attri*
bute to the plate a refractive index which is slightly too great.

The author has illustrated this subject by the following experi-
ment. A set of fringes, produced in the ordinary way by a flat
prism, were viewed through an eye-piece, and bisected by its cross
wires. On viewing the whole through a prism of moderate angle,
held in front of the eye-piece with its edge parallel to the fringes,
an indistinct prismatic image of the wires was seen, together with
a distinct set of fringes which lay quite at one side of the cross
wires, the dispersion produced by the prism having thus occasioned
an apparent displacement of the fringes in the direction of the
general deviation.

In conclusion, the author suggested that it might have been
the fictitious displacement due to the dispersion accompanying
eccentrical refraction, which caused some philosophers to assert
that the central band was black, whereas, according to theory,
it ought to be white. A fictitious displacement of half an
order, which might readily be produced by eccentrical refraction
through the lens or eye-piece with which the fringes were viewed,
would suffice to cause one of the two black bands of the first
order to be the band with respect to which the system was sym-
metrical

362 PAPERS FROM THE REPORT

k

On Haidinger's Brushes.

It is now several years since these brushes were discovered, and
they have since been observed by various philosophers, but the
author has not met with any observations made with a view of
investigatiug the action of different colours in producing them.
The author s attention was first called to the subject, by observing
that a green tourmaline, which polarized light very imperfectly,
enabled him to see the brushes very distinctly, while he was un-
able to make them out with a brown tourmaline which trans-
mitted a much smaller quantity of unpolarized light. He then
tried the effect of combining various coloured glasses with a Nicol's
prism. A red glass gave no trace of brushes. A brownish yellow
glass, which absorbed only a small quantity of light, rendered the
brushes very indistinct. A green glass enabled the author to see
the brushes rather more distinctly than they were seen in the
light of the clouds viewed without a coloured glass. A deep blue
glass gave brushes of remarkable intensity, notwithstanding the
large quantity of light absorbed. With the green and blue glasses,
the brushes were not coloured, but simply darker than the rest of
the field.

To examine still further the office of the difierent colours in
producing the brushes seen with ordinary daylight, the author
used a telescope and prism mounted for shewing tbe fixed lines of
the spectrum. The sun's light having been introduced into a
darkened room througli a narrow slit, it was easy, by throwing the
eye-piece a little out of focus, to form a pure spectrum on a screen
of white paper, placed a foot or two in front of the eye-piece. On
examining this spectrum with a NicoPs prism, which was suddenly
turned round from time to time through about a right angle, the
author found that the red and yellow did not present the least
trace of brushes. The brushes began to be visible in tbe green,
about the fixed line E of Fraunhofer. They became more distinct
on passing into the blue, and were particularly strong about the
line i^. The author was able to trace them about as far as the
line ; and when they were no longer visible, the cause appeared
to be merely the feebleness of the light, not the incapacity of the
greater part of the violet to produce them. With homogeneous

OF THE BRITISH ASSOCIATION, 1850. 3C3

light, the brushes, when Ihey were fonned at all, were simply
darker than the rest of the field, and, as might have been ex-
pected, did not appear of a different tint. In the blue^ where the
brushes were most distinct, it appeared to the author that they
were somewhat shorter than usual. The contrast between the
more and less refrangible portions of the spectrum, in regard to
their capability of producing brushes, was most striking. The
most brilliant part of the spectrum gave no brushes ; and the in-
tensity of the orange and more refrangible portion of the red,
where not the slightest trace of brushes was discoverable, was
much greater than that of the more refrangible portion of the blue,
where the brushes were formed with great distinctness, although
ccbteris paribus a considerable degree of intensity is favourable to
the exhibition of the brushes.

These observations account at once for the colour of the brushes
seen with ordinary daylight. Inasmuch as no brushes are seen
with the less refrangible colours, and the brushes seen with the
more refrangible colours consist in the removal of a certain
quantity of light, the tint of the brushes ought to be made up of
red, yellow, and perhaps a little green, the yellow predominating,
on accotmt of its greater brightness in the solar spectrum. The
mixture would give an impure yellow, which is the colour ob-
served. The blueness of the side patches may be merely the effect
of contrast, or the cause may be more deeply seated. If the total
illumination perceived be independent of the brushes, the light
withdrawn from the brushes must be found at their sides, which
would account, independently of contrast, both for the comparative
brightness and for the blue tint of the side patches.

The observations with homogeneous light account likewise for
a circumstance with which the author had been struck, namely,
that the brushes were not visible by candle-light, which is ex-
plained by the comparative poverty of candle-light in the more
refrangible rays. The brushes ought to be rendered visible by
absorbing a certain quantity of the less refrangible rays, and ac-
cordingly the author found that a blue glass, combined with a
NicoFs prism, enabled him to see the bnishes very distinctly when
looking at the flame of a candle. The specimen of blue glass
which shewed them best, which was of a tolerably deep colour,
gave brushes which were decidedly red, and were only compara-
tively dark, so that the difference of tint between the brushes and

364 PAPERS FROM BRITISH ASSOCIATION REPORT, 1850.

side patches was far more conspicuous than the difference of in-
tensity. This is accounted for by the large quantity of extreme
red rays which such a glass transmits. That the same glass gave
red brushes with candle-light, and dark brushes with daylight, is
accounted for by the circumstance, that the ratio which the in-
tensity of the transmitted red rays bears to the intensity of the
transmitted blue rays is far larger with candle-light than with
daylight.

INDEX TO VOL. II.

Astigmatio lenses, effect of oomblned,

174
ttstigmatism of a defeetiye eye, mode of

measuring, 172
attractions, propositions respecting, 105,

124

Bessers functions, calculation of, for
large values of the variable, 829

Britannia Bridge, deflection liable to be
produced in, bj a travelling load, 219

Brewster's apparent new polarity of
light, explanation of, 24

Challis, Prof., explanation of difficulties
in the theory of sound discovered by,
51,82

Clairaut's theorem, 112, 142

Decomposition, in a particular way, of
three functions representing com-
ponents of displacement, dc, 256

differential equation, discussion of a,
relating to deflection produced by a
travelling load, 178

diffraction, dynamical theory of, 243

diffraction of polarized light, experi-
ments on, 290

disturbance in an isotropic medium, due
to a force continually acting, 276

disturbances, small, of a dynamical sys-
tem due to initial (1) displacements,
(2) velocities, deduced from each other,
261

Earth's original fluidity, review of geo-
detic and gravitational evidence in
favour of, 120

elastic solid, propagation of an arbitrary

disturbance in an, 257
ether, constitution of the luminiferous, 8
eye (see astigmatism)

Fresners formulas for the intensities of
reflected and refracted light, experi-
mental evidence of the correctness of,
97

Gravity, variation of, at the surface of
the earth, 131

Haidinger*s brushes, 362

Helmholtz's propositions respecting vor-
tex motion, 47

Holtzmann's experiments on diffraction
of polarized light, 327

hydrodynamics, notes on, 1, 36, 221

Instability of motion of sphere in a per-
fect fluid, 8

interference bands seen in the spectrum,
under peculiar conditions, explained, 14

interference fringes, fictitious displaoe-
ment of, 361

irregularities of earth's surface, effect of,
on local gravity, 149

IiOrenz*B researches on diffraction of
polarized light, 828

Moon, effect of earth's oblateness on,
independent of hypothesis of earth's
oiiginal fluidity, 118, 132, 142

Newton's rings, formation of the central
spot of, beyond the critical angle, 56 ;

366

INDEX TO VOL. II.

sudden disappearauce of central spot
of, on inoreasing the angle of internal
incidence, 80 ; explanation of the per-
fect blackness of the central spot of,
89 ; mode of disappearance of, in pass-
ing the critical angle, 858
Nomerioal calculation of a class of
definite integrals and infinite series,
329

Pendulum results applicable to the de-
termination of the earth's figure,
irrespective of hypotheses respecting
internal distribution of matter, 140

polarized light, direction of vibrations in,
determined by experiments on diffrac-
tion, 317

precession and nutation, moment of
force causing, is independent of hy-
potheses respecting distribution of
matter in the earth, 118, 132, 142

Railway Bridges, differential equation
relating to the breaking of, discussed,
178

rarefaction a necessary accompaniment
of condensation in a sound-wave pro-
pagated from a centre, 83

reflexion, metallic, 860

refraction of light beyond the critical
angle, 57

reversion, application of the principle
of, to the demonstration of two laws
relating to the reflection of light, 90

ring, single bright, surrounding a dark
centre, in connexion with Newton*s
rings, 75

Secondary wave of light, law of disturb-
ance in, determined, 280

semi-convergent series, establishment
and application of, 329

sound, on a difficulty in the theory o(
51 ; on some points in the received
theory of, 82

sound-wave, alteration of the type of,
when the motion is not small, 52

sphere, steady motion of, in a visoous
fluid, 10

Stanley, Capt., comparison of observa-
tions made by, on waves in open sea,
with theory, 239

Vortex motion, Helmholtz'a propositions
respecting, deduced from Caaohy*8 in-
tegrals, 47

Wave length, determination of, corre-
sponding with any point of the spec-
trum, 176

waves, 221

Willis, Prof., discussion of an equation
relating to experiments by, 178

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SELECT DISCOURSES, by John Smith, late Fellow of

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with the richest lights of meditative genius...
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AESCHYLI FABULAE.— IKETIAE2 XOHa)OPOI IN

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1—5

10 PUBLICATIONS OF

PINDAR. OLYMPIAN AND PYTHIAN ODES. With

Notes Explanatory and Critical, Introductions and Introductory
Essays. Edited by C. A. M. Fennell, M.A., late Fellow of Jesus
College. Crown 8vo. 9^.

" Mr FenncII deserves the thanks of all clas- scholarship, and we are glad to see that it is to

sical studenu for his careful and scholarly cdi- be continued." — Saturday Rexriew.
tion of the Olympian and Pythian odes. He *'Mr C A. M. Fennell's 'Pindar* displays

brings to his task the necessary enthusiasm for that union of laborious research and unassum>

his author, great industry, a sound judgment, ing directness of style which characterizes the

and, in particular, copious and minute learning best modern scholarship. The introduction

in comparative philology. To his qualifica • contains an essay on the Pentathlon, a table of

tions in this last respect every page bears wit- Metrical Schemes, and an Excursus on the

ness." — Athenaum. Causative Middle. The notes, which are in

" Considered simply as a contribution to the English, amd at the foot of each page, are

study and criticism of Pindar, Mr Fennell's clear and to the point. There is an introduc-

edition is a work of great merit. . . Altogether, tion to each Ode. There are Greek and Eng-

this edition is a welcome and wholesome sign lish Indices, and an Index of Quotations." —

of the vitality and development of Cambridge Weitminster Revirw,

THE ISTHMIAN AND NEMEAN ODES. By the same Editor.
Crown 8vo. 9^.

ARISTOTLE. THE RHETORIC. With a Commentary
by the late E. M. Cope, Fellow of Trinity College, Cambridge, re-
vised and edited by J. E. Sandys, M.A., Fellow and Tutor of St John's
College, Cambridge, and Public Orator. With a biographical Memoir
by H. A. J. MUNRO, M.A. Three Volumes, Demy 8vo. £\. lu. 6^/.

"This work is in many ways creditable to the "Mr Sandys has performed his arduous
University of Cambridge. If an English student duties with marked ability and admirable tacL
wishes to have a full conception of what is con- ... In every part of his work — revising, sup-
tained in the Rhttoric of Aristotle, to Mr Cope's plemcnting, and completing — he has done ex-
edition he must go*'— Academy. ceedingly well." — Exatfttfur.

PRIVATE ORATIONS OF DEMOSTHENES, with In-

troductions and English Notes, by F. A. Paley, M.A. Editor of
Aeschylus, etc. and J. E. Sandys, M.A. Fellow and Tutor of St John's
College, and Public Orator in the University of Cambridge.

Part I. Contra Phormionem, Lacritum, Pantaenetum, Boeotum
de Nomine, Boeotum de Dote, Dionysodorum. Crown 8vo. 6s.

"Mr Paley's scholarship is sound and ac> bears upon his author, and the elucidation of

curate, his experience of editing wide, and if matters of daily life, in the delineation of which

he is content to devote his learning and abili- Demosthenes is so rich, obtains full justice at

ties to the production of such manuals as these, his hands. . . . We hope this edition may lead

they will be received with gratitude throughout the way to a more general study of these

the higher schools of the country. Mr Sandys speeches in schools than has hitherto been

is deeply read in the German literature which possible." — Academy.

Part II. Pro Phorraione, Contra Stephanum I. II.; Nicostra-
tum, Cononem, Calliclem. Crown 8vo. ys, 6d.

" To give even a brief sketch of these have the aid of Mr Sandys's excellent running

speeches {Pro Phomiione and Contra Ste^ commentary . . . and no one can say that he is

phanum\ would be incompatible with our ever deficient in the needful help which enables

limits, though we can hardly conceive a task us to form a sound estimate of tne rights of the

more useful to the classical or professional case,* —Saturday Review.

scholar than to make one for himself. ... It is " the edition reflects credit on Ctun-

a great boon to those who set themselves to bridee scholarship, and ought to be extensively

unravel the thread of arguments pro and con to used." — Athenaum.

DEMOSTHENES AGAINST ANDROTION AND

AGAINST TIMOCRATES, with Introductions and English Com-
mentary, by William Wayte, M.A., late Professor of Greek, Uni-
versity College, London, Formerly Fellow of King's College, Cam-
bridge, and Assistant Master at Eton. Crown 8vo. 7J. dd,

** The editor has devoted special attention "The present edition may therefore be used

to the principles and practice of Attic law, of by students more advanced than school-boys,

which these speeches afford many rich illustra- and to their purposes it is admirably suited,

tions. In the notes, which include serviceable There is an excellent introduction to and ana>

abstracts of the sjpeeches, his object, he tells us, lysis of each speech, and at the beginning of

has been " to afford full help without unduly each paragraph of the text there is a summary

encouraging 'the less industrious sort;'" and of its subject-matter. . . The notes are luu-

they certainly afford as much help, and of the formly good, whether they deal with-<}uestions

right kind, as any reasonable student is likely of schoUrship or with pomts of Athenian law."

to desire."— TA^ Scotsman. — Saturday Review.

London : Cambridge University Press Warehouse ^ 17 Paternoster Row.

THE CAMBRIDGE UNIVERSITY PRESS.

II

THE TYPES OF GREEK COINS. By Percy Gardner,

M.A., F.S.A., Disney Professor of Archaeology. With i6 Autotype
plates, containing photographs of Coins of all parts of the Greek World.
Impl. 4to. Cloth extra, £i. iis.Stf,; Morocco backs, £2, 2s.

"Professor Gardner's book is written with which he supplies) or casts of the original

such lucidity and in a manner so straightfor*
ward that it may well win converts, and it may
be distinctly recommended to that omnivorous
class of readers — ' men in the schools.' The his-
tory of ancient coins is so interwoven with and
so vividly illustrates the history of ancient States,
that students of Thucydides and Herodotus can-

:ya
Pi

not afford to neglect Professor Gardner's intro-
duction to Hellenic numismatics . . . The later
part of Mr Gardner's useful and interesting
volume is devoted to the artistic and archaeo-
logical aspect of coins, and can scarcely be
studied apart from photographs (like diose

medals." — Saturday Review.

* The Types of Greek Coins ' is a work which
is less purely and drylv scientific. Neverthe*
less, it takes high rank as proceeding upon a
truly scientific basis at the same time tnat it
treats the subiect of numismatics in an attrac-
tive style and is elegant enough to justify its
appearance in the drawing-room .... Six-
teen autotype plates reproduce with marvellous
reality more than six hundred types of picked
specimens of coins in every style, from the
cabinets of the British Museum and other col-
ections. — A tkefueum.

THE BACCHAE OF EURIPIDES. With Introduction,
Critical Notes, and Archaeological Illustrations, by J. E. Sandys,
M.A., Fellow and Tutor of St John's College, Cambridge, and Public
Orator. Crown 8vo. loj. dd,

able advance in freedom and lightness of style.
. . . Under such circumstances it is superfluous
to say that for the purposes of teachers and ad-
vanced students this handsome edition far sur.
passes all its predecessors." — Atkemtum.

'I It has not, like so many such books^ been
hastily produced to meet the momentary need
of some particular examination ; but it has em-
ployed for some years the labour and thought
of a highly finished scholar, whose aim seems
to have been that his book should go forth toiu»
teres, atgue rotuHdus^ armed at all points with
all that may throw light upon its subject 'thit
result is a work which will not only assist the
schoolboy or undergraduate in his tasks, but
will adorn the library of the scholar." — The
Guardian.

" Of the present edition of the Baccha by Mr
Sandys we may safely say that never before has
a Greek play, in England at least, had fuller
jastice done to its criticism, interpretation,
and archasological illustration, whether for the
young student or the more advanced scholar.
The Cambridge Public Orator may be said to
have taken the lead in issuing a complete edi*
tion of a Greek play, which is destined perhaps
to gain redoubled favour now that the study of
ancient monuments has been applied to its il-
lustration." — Saturday Review.

"The volume is interspersed with well-
executed woodcuts, and its general attractive-
ness of form reflects great credit on the Uni-
versity Press. In the notes Mr Sandys has more
than sustained his well-earned reputation as a
careful and learned editor, and shows consider-

ESSAYS ON THE ART OF PHEIDIAS. By C. Wald-

STEIN, M.A., Phil. D., Reader in Classical Archaeology in the Uni-
versity of Cambridge. Royal 8vo. With Illustrations.

[In the Press,

M. TULLI CICERONIS DE FINIBUS BONORUM

ET MALORUM LIBRI QUINQUE. The text revised and ex-
plained; With a Translation by James S. Reid, M.L., Fellow and
Assistant Tutor of Gonville and Cnius College. In three Volumes.

[/« the Press,
Vol. III. Containing the Translation. [Nearly Ready,

M. T. CICERONIS DE OFFICIIS LIBRI TRES,

with Marginal Analysis, an English Commentary, and copious
Indices, by H. A. Holden, LL.D., late Fellow of Trinity College,
Cambridge. Fourth EditioiL Revised and considerably enlarged.
Crown 8vo. 91.

*'Dr Holden has issued an edition of what assumed after two most thorough revisions,
is perhaps the easiest and most popular of leaves little or nothing to be desired in the full-
Cicero's philosophical works, the de Officiis^ ness and accuracy of its treatment alike of the
which, especially in the form which it has now matter and the language." — Academy,

M. TVLLI CICERONIS PRO C RABIRIO [PERDVEL-

LIONIS REO] ORATIO AD QVIRITES With Notes Introduc-
tion and Appendices by W E Heitland MA, Fellow and Lecturer
of St John's College, Cambridge. Demy 8vo. ^s, 6d,

London : Cambridge University Press Warehouse^ 17 Paternoster Row,

12

PUBLICATIONS OF

M. TULLII CICERONIS DE NATURA DEORUM

Libri Tres, with Introduction and Commentary by JOSEPH B.
Mayor, M.A., late Professor of Moral Philosophy at King's Col-
lege, London, together with a new collation of several of the English
MSS. by J. H. SWAINSON, M.A., formerly Fellow of Trinity College,
Cambridge. Vol. I. Demy 8vo. los, 6a, [Vol. II. In the Press,

'* Such editions as that of which Prof. Mayor
has given us the first instalment will doubtless
do much to remedy this undeserved neglect. It
is one on which great pains and much learning
have evidently been expended, and is in every

P. VERGILI MARONIS OPERA cum Prolegomenis
et Commentario Critico pro Syndicis Preli Academic! edidit Ben-
jamin Hall Kennedy, S.T.P., Graecae Linguae Professor Regius.
Extra Fcap. 8vo. 5J.

way admirably suited to meet the needs of the
student . . . The notes of the editor are all that
could be expected from his well-known learn-
ing and scholarship."— ^£-a</riw^.

MATHEMATICS, PHTSIGAL SCIENCE, &c.

MATHEMATICAL AND PHYSICAL PAPERS. By
Sir W. Thomson, LL.D., D.C.L., F.R.S., Professor of Natural Phi-
losophy in the University of Glasgow. Collected from different
Scientific Periodicals from May 184I1 to the present time. VoL I.

Demy 8vo. i8j.

"Wherever exact science has found a fol-
lower Sir William Thumsuu's name X's, known as
a leader and a master. For a space of 40 years
each of his successive contributions to know-
ledge in the domain of experimental and mathe-
matical physics has been recognized as marking
a sta^e m the progress of the subject. But, un-
happily for the mere learner, he is no writer of
text-books. His eager fertility overflows into
the nearest available journal . . . The papers in
this volume deal largelv with the subject of the
dynamics of heat. They begin with two or
three articles which were in part written at the
age of 17, before the author had commenced
residence as an undergraduate in Cambridge
. . . No student of mechanical engineering,
who aims at the higher levels of his profession,
can afford to be ignorant of the principles and
methods set forth in those great memoirs . . .
The article on the absolute measurement of
electric and galvanic quantities (1851) has
borne rich and abundant fruit. Twenty years
after its date the International Conference of
Electricians at Paris, assisted by the author
himself, elaborated and promulgated a series of

MATHEMATICAL AND PHYSICAL PAPERS, by
George Gabriel Stokes, M.A., D.C.L., LL.D., F.R.S., Fellow of
Pembroke College, and Lucasian Professor of Mathematics in the
University of Cambridge. Reprinted from the Original Journals and
Transactions, with Additional Notes by the Author. Vol. I. Demy

[Vol. II. In the Press,

rules and units which are but the detailed out-
come of the principles laid down in these
papers." — The Times.

"We are convinced that nothing has had a
greater effect on the progress of the theories of
electricity and magnetism during the last ten
years than the publication of Sir W. Thomson's
reprint of i>aners on electrostatics and magnet*
ism, and we believe that the present volume is
destined in no less degree to further the ad«
vancement of physical science. We owe the
modern dynamical theory of heat almost wholly
to Joule and Thomson, and Clatisius and Ran*
kine. and we have here collected together the
whole of Thomson's investigations on this sub-
iect, together with the papers published jointly
Dy himself and Joule. We would fain linger
over this fascinating subject, but space does not
permit ; and we can only ask those who really
wish to study thermo-dynamics to know some-
thing of the great theory of the dissipation of
energy originated by the author to give his days
and nights to the volume before us." — Glasgow
HeraU.

8vo. 1 5 J.

'* ITie volume of Professor Stokes's papers
contains much more than his hydrodynamical
fxapers. The undulatory theory of light is
treated, and the difficulties connected with its
application to certain phenomena, such as aber-
ration, arc carefully examined and resolved.
Such difficultici* arc commonly passed over with
scant notice in the text -books . . . Those to
whom difficulties like these are real stumbling-
blocks will still turn for enlightenment to Pro-
fessor Stokes's old, but still fresh and still

[Vol. II. Nearly ready.

necessary, dissertations. There nothing is
slurred over, nothing extenuated. We learn
exactly the weaknesses of the theory, and
the direction in which the completer theory of
the future must be sought for. The same spirit
pervades the papers on pure mathematics which
are included in the volume. They have a severe
accuracy of style which well befits the subtle
nature of the subjects, and inspires the oom-
pletest confidencem their author.**— 7!4r Timut.

Volume III. In the Press,

Lottdon: Cambridge University Press Warehouse^ it Paternoster Raw,

THE CAMBRIDGE UNIVERSITY PRESS.

13

THE SCIENTIFIC PAPERS OF THE LATE PROF.

J. CLERK MAXWELL. Edited by W. D. Niven, M.A. In 2 vols.
Royal 4to. [In the Press.

A TREATISE ON NATURAL PHILOSOPHY. By
Sir W. Thomson, LL.D., D.C.L., F.R.S., Professor of Natural
Philosophy in the University of Glasgow, and P. G. Tait, M.A.,
Professor of Natural Philosophy in the University of Edinburgh.
Vol. L Part L Demy 8vo. 16s.

** In this, the second edition, we notice a form within the time at our disposal would be
lar^e amount of new matter^ the importance of utterly inadequate." — Mature.
which is such that any opimon whicn we could

Part n. Demy 8vo. iSs,

ELEMENTS OF NATURAL PHILOSOPHY. By Pro-
fessors Sir W. Thomson and P. G. Tait. Part I. Demy 8vo.
Second Edition, gs.

A TREATISE ON THE THEORY OF DETERMI-

NANTS AND THEIR APPLICATIONS IN ANALYSIS AND
GEOMETRY, by Robert Forsyth Scott, M.A., of St John's
College, Cambridge. Demy 8vo. I2J.

" This able and comprehensive treatise will searches on this subject which have hitherto
be welcomed by the student as bringing within been for the most part inaccessible to him." —
his reach the results of many important re- AtMenaum.

HYDRODYNAMICS, a Treatise on the Mathematical
Theory of the Motion of Fluids, by Horace Lamb, M.A., formerly
Fellow of Trinity College, Cambridge ; Professor of Mathematics in
the University of Adelaide. Demy 8vo. 12s.

THE ANALYTICAL THEORY OF HEAT, by Joseph

Fourier. Translated, with Notes, by A. Freeman, M.A., Fellow
of St John's College, Cambridge. Demy 8vo. i6j.

process employed by the author." — Contempo-
rary Review^ October, 1878.

There cannot be two opinions as to the
value and importance of the Thiorie de la Cha-
Uur ... It is still tfu text-book of Heat Con-
duction, and there seems little present prospect
of its beinz superseded, though it is already
more than half a century q\^,*'— Nature,

" It is time that Fourier's masterpiece, The
Analytical Theory of Heaty translated by Mr
Alex. Freeman, should be introduced to those
English students of Mathematics who do not
follow with freedom a treatise in any language
but their own. It b a model of mathematical
reasoning applied to physical phenomena, and
is remarkiable for the ingenuity of the analytical

THE ELECTRICAL RESEARCHES OF THE Honour-
able Henry Cavendish, F.R.S. Written between 1771 and 1781.
Edited from the original manuscripts in the possession of the Duke
of Devonshire, K.G., by the late J. Clerk Maxwell, F.R.S.
Demy 8vo. i8j.

"Every department of editorial duty ap- faction to Prof. Maxwell to see this goodly
pears to have been most conscientiously pcr-
tormed ; and it must have been no small satis-

volume completed before his life's work was
done. " — yj theneeum.

An elementary TREATISE on QUATERNIONS.

By P. G. Tait, M.A., Professor of Natural Philosophy in the U ni-
versity of Edinburgh. Second Edition, Demy 8vo. I4.r.

THE MATHEMATICAL WORKS OF ISAAC BAR-
ROW, D.D. Edited by W. Whewell, D.D. Demy 8vo. 7J. 6//.

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r4 PUBLICATIONS OF

NOTES ON QUALITATIVE ANALYSIS. Concise and

Explanatory. By H. J. H. Fenton, M.A., F.I.C., F.C.S., Demon-
strator of Chemistry in the University of Cambridge. Late Scholar
of Christ's College. Crown 4to. Ts. dd,

A TREATISE ON THE PHYSIOLOGY OF PLANTS,

by S. H. Vines, M.A., Fellow of Christ's College. [In the Press.

THE FOSSILS AND PAL^EONTOLOGICAL AFFIN-
ITIES OF THE NEOCOMIAN DEPOSITS OF UPWARE
AND BRICKHILL with Plates, being the Sedgwick Prize Essay
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loif. 6^.

COUNTERPOINT. A Practical Course of Study, by Pro-
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revised. Demy 4to. 75. dd,

ASTRONOMICAL OBSERVATIONS made at the Obser-

vatory of Cambridge by the Rev. James Challis, M.A., F.R.S.,
F.R.A.S., late Plumian Professor of Astronomy and Experimental
Philosophy in the University of Cambridge. For various Years, from
1846 to i860.

ASTRONOMICAL OBSERVATIONS from 1861 to 1865.

Vol. XXI. Royal 4to. 15J. From 1866 to 1869. Vol. XXII.
Royal 4to. [Nearly ready,

A CATALOGUE OF THE COLLECTION OF BIRDS

formed by the late H. E. Strickland, now in the possession of the
University of Cambridge. By Osbert Salvin, M.A., F.R.S., &c.
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"The discriminating^ notes which Mr Salvin ^ "The author has formed a definite and, as

has here and there introduced make the book it seems to us, a righteous idea of what the

indispensable to every worker on what the catalogue of a collection should be, and, allow-

Americans call "the higher plane" of the ing for some occasional slips, has effectively

science of birds. "—A cademy. carried it out, " — Notes and Qturies.

A CATALOGUE OF AUSTRALIAN FOSSILS (in-
cluding Tasmania and the Island of Timor), Stratigraphically and
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ontologist, H.M. Geol. Survey of Scotland. Demy 8vo. loj. 6d.

"The work is arranged with great clearness, consulted by the author, and an index to th«
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ILLUSTRATIONS OF COMPARATIVE ANATOMY,

VERTEBRATE AND INVERTEBRATE, for the Use of Stu-
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Plates, £1. I J.

A CATALOGUE OF THE COLLECTION OF CAM-

BRIAN AND SILURIAN FOSSILS contained in the Geological
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CATALOGUE OF OSTEOLOGICAL SPECIMENS con-

tained in the Anatomical Museum of the University of Cambridge.
Demy 8vo. 2s, 6d,

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THE CAMBRIDGE UNIVERSITY PRESS. 15

LAW.

AN ANALYSIS OF CRIMINAL LIABILITY. By E. C.
Clark, LL.D., Regius Professor of Civil Law in the University of Cam-
bridge, also of Lincoln's Inn, Barrister-at-Law. Crown 8vo. ys, 6d,

"Prof. Clark's little book is the substance Students of jurisprudence will find much to

of lectures delivered by him upon those por- interest and instruct them in the work of Prof,

tions of Austin's work on jurisprudence which Clark.** — Athtmeum,
deal with the "operation of sanctions" ...

PRACTICAL JURISPRUDENCE, a Comment on Austin.

By E. C. Clark, LL.D. Regius Professor of Civil Law. Crown
8vo. 9J.

A SELECTION OF THE STATE TRIALS. By J. W.
Willis-Bund, M.A., LL.B., Barrister-at-Law, Professor of Con-
stitutional Law and History, University College, London. Vol. I.
Trials for Treason (1327 — 1660). Crown 8vo, i8j.

as it may be gathered from trials before the
ordinary courts. The author has very wisely
distinguished these cases from those of im-
peachment for treason before Parliament, which
ne proposes to treat in a future volume under
the general head 'Proceedings in Parliament'"
— Tne Academy.

" This is a work of such obvious utility that
the only wonder is that no one should have un-
dertaken it before ... In many respects ther^
fore, although the trials are more or less
abridged, this is for the ordinary student's pur-
pose not only a paore handy, but a more useful
work than Howell's." — Saturday Review.

"Mr Willis-Bund has edited 'A Selection of
Cases from the State Trials' which is likelv to
form a very valuable addition to the standard
literature . . . There can be no doubt, therefore,
of the interest that can be found in the State
trials. But they are large and unwieldy, and it
is impossible for the seneral reader to come
across them. Mr Willis-Bund has therefore
done good service in making a selection that
is in the first volume reduced to a commodious
form." — The Examiner.

"This work is a very useful contribution to
that important branch of the constitutional his-
tory of England which is concerned with the
growth and development of the law of treason.

Vol. II. In two parts. Price 145. each.

" But, although the book is most interesting
to the historian of constitutional law, it is also
not without considerable value to those who
seek information with regard to procedure and
the growth of the law of evidence. We should
add that Mr Willis-Bund has given short pre-
faces and appendices to the trials, so as to form
a connectea narrative of the events in history
to which they relate. We can thoroughly re-
commend the book. " — Lava Times.

**To a large class of readers Mr Willis-
Bund's compilation will thus be of great as-
sistance, for he presents in a convenient form a

judicious selection of the principal statutes and
the leading cases bearing on the crime of trea-
son . . . For all classes of readers these volumes
possess an indirect interest, arising from the
nature of the cases themselves, from the men
who were actors in them, and from the numerous
points of social life which are incidentally illus-
trated in the course of the trials. On these
features we have not dwelt, but have preferred
to show that the book is a valuable contribution
to the study of the subiect with which it pro-
fesses to deal, namely, the history of the law of
treason." — A thenteum.

Vol. III. In the Press.

THE FRAGMENTS OF THE PERPETUAL EDICT

OF SALVIUS JULIANUS, collected, arranged, and annotated by
Bryan Walker, M.A., LL.D., Law Lecturer of St John's College, and
late Fellow of Corpus Christi College, Cambridge. Crown 8vo. dr.

II

In the present book we have the fruits of
the same kind of thorough and well-ordered
study which was brought to bear upon the notes
to the Commentaries and the Institutes . . .
Hidierto the Edict has been almost inac-
cessible to the ordinary English student, and

such a student will be interested as well as per-
haps surprised to find how abundantly the ex-
tant fragments illustrate and clear up points
which have attracted his attention in the Com-
mentaries, or the Institutes, or the Digest"—
Law Times,

London : Cambridge University Press Warehouse, 17 Paternoster Row,

i6

PUBLICATIONS OF

DIGEST. Book VII. Title I. De Usufructu. With

Introduction and full Explanatory Notes, intended as an Introduction
to the study of the Digest. By Henry John Rory, M.A., formerly
Fellow of St John's College. [/« the Press,

THE COMMENTARIES OF GAIUS AND RULES OF
ULPIAN. (New Edition, revised and enlarged.) With a Trans-
lation and Notes, by J. T. Abdy, LL.D., Judge of County Courts,
late Regius Professor of Laws in the University of Cambridge, and
Bryan Walker, M.A., LL.D., Law Lecturer of St John's College,
Cambridge, formerly Law Student of Trinity Hall and Chancellor's
Medallist for Legal Studies. Crown 8vo. i6j.

"As scholars and as editors Messrs Abdy way of reference or necessary explanation,

and Walker have done their work well . . . For Thus the Roman jurist is allowed to speak for

one thing the editors deserve special commen- himself, and the reader feels that he is really

dation. They have presented Gains to the studying Roman law in the original, And not a

reader with few notes and those merely by fanciful representation of it. ''—^/A^Mrmw.

THE INSTITUTES OF JUSTINIAN, translated with
Notes by J. T. Abdy, LL.D., Judge of County Courts, late Regius
Professor of Laws in the University of Cambridge, and formerly
Fellow of Trinity Hall ; and Bryan Walker, M.A., LL.D., Law
Lecturer of St John's College, Cambridge ; late Fellow and Lecturer
of Corpus Christi College ; and formerly Law Student of Trinity
Hall. Crown 8vo. idr.

"We welcome here a valuable contribution
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Institutes is occasionally oerplexing, even to
practised scholars, whose Knowledge of clas-
sical models does not always avail them in
dealing with the technicalities of legal phrase-
ology. Nor can the ordinary dictionaries be
expected to furnish all the help that is wanted.
This translation will then be of great use. To

the ordinary student, whose attention is dis-
tracted from the subject-matter by the dif-
ficulty of struggling through the language in
which it is contained, it will be almost indis-
pensable. " — S^ctator.

" The notes are learned and carefully com-
piled, and this edition will be found useful to
students." — Law Times.

SELECTED TITLES FROM THE DIGEST, annotated
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M. T. CICERONIS PRO CN. PLANCIO ORATIO.

Edited by H. A. Holden, LL.D., late Head Master of Ipswich School.

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commentary most likely to be generally useful ; and he has carried out his views with admirable
thoroughness." — Academy.

" Dr Holden has given us here an excellent edition. The commentary is even unusuallv full
and complete ; and after going through it carefully, we find little or nothine to criticize. There
is an excellent introduction, lucidly explaining the circumstances under which the speech was
delivered, a table of events in the life of Cicero and a useful index." Sftctatar, Oct 29, z88z.

London : Cambridge University Press Warehouse^ 17 PaiemosUr Row,

i6

PUBLICATIONS OF

DIGEST. Book VII. Title I. De Usufructu. With

Introduction and full Explanatory Notes, intended as an Introduction
to the study of the Digest. By Henry John Roby, M.A., formerly
Fellow of St John's College. \In the Press,

THE COMMENTARIES OF GAIUS AND RULES OF
ULPIAN. (New Edition, revised and enlarged.) With a Trans-
lation and Notes, by J. T. Abdy, LL.D., Judge of County Courts,
late Regius Professor of Laws in the University of Cambridge, and
Bryan Walker, M.A., LL.D., Law Lecturer of St John's College,
Cambridge, formerly Law Student of Trinity Hall and Chancellor's
Medallist for Legal Studies. Crown 8vo. i6j.

** As scholars and as editors Messrs Abdy
and Walker have done their work well . . . For
one thing the editors deserve special commen-
dation. They have presented Gains to the
resider with few notes and those merely by

way of reference or necessary explanation.
Thus the Roman jurist is allowed to speak for
himself, and the reader feels that he is really
studying Roman law in the original, and not a
fanciful represcnution oiil,"—Aihgtutum.

THE INSTITUTES OF JUSTINIAN, translated with
Notes by J. T. Abdy, LL.D., Judge of County Courts, late Regius
Professor of Laws in the University of Cambridge, and formerly
Fellow of Trinity Hall ; and Bryan Walker, M.A., LL.D., Law
Lecturer of St John's College, Cambridge ; late Fellow and Lecturer
of Corpus Christi College ; and formerly Law Student of Trinity
Hall. Crown 8vo. idr.

"We welcome here a valuable contribution
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InstitHtes is occasionally perplexing, even to
practised scholars, whose Knowledge of clas-
sical models does not always avau them in
dealing with the technicalities of legal phrase-
ology. Nor can the ordinary dictionaries be
expected to furnish all the help that is wanted.
This translation will then be of great use. To

the ordinary student, whose attention is dis-
tracted from the subject-matter by the dif-
ficulty of struggling through the language in
which it is contained, it will be almost indis-
pensable." — spectator.

" I'he notes are learned and carefully com-
piled, and this edition will be found useful to
students." — Law Times.

SELECTED TITLES FROM THE DIGEST, annotated
by B. Walker, M.A., LL.D. Part L Mandati vel Contra. Digest
XVII. I. Crown 8vo. 5^.

"This small volume is published as an ex-
periment The author proposes to publish an
annotated edition and translation of several
books of the Digest if this one is received with
favour. We arc pleased to be able to say that

Mr Walker deserves credit for the wav in which
he has performed the task undertaken. The
translation, as might be expected, is scholarly.**
— Law Time*,

Part IL De Adquirendo rerum dominio and De Adquirenda vel
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Part IIL De Condictionibus. Digest xii. i and 4 — 7 and Digest
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GROTIUS DE JURE BELLI ET PACIS. with the Notes

of Barbeyrac and others ; accompanied by an abridged Translation
of the Text, by W. Whewell, D.D. late Master of Trinity College.
3 Vols. Demy 8vo. 12s, The translation separate, dr.

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17

HISTORY.

THE GROWTH OF ENGLISH INDUSTRY AND
COMMERCE. By W. Cunningham, M.A., late Deputy to the
Knightbridge Professor in the University of Cambridge. With
Maps and Charts. Crown 8vo. 12s,

*'He is, however, undoubtedly sound in the
main, and bis work deserves recognition as the
result of immense industry and research in a
fidd in which the labourers have hitherto been
comparatively h^.**— Scotsman.

"Mr Cunninf^ham is not likely to disap-
pmnt any readers except such as besin by mis-
taking the character of his book. He does not
promise, and does not give, an account of the

dimensions to which English industry and com>
merce have grown. It is with the process of
growth that he is concerned ; and this process
he traces with the philosophical insight which
distinguishes between what is important and
what is trivial. He thus follows with care,
skill, and deliberation a single thread through
the maze of general English history." — Guar-
dian.

LIFE AND TIMES OF STEIN, OR GERMANY AND
PRUSSIA IN THE NAPOLEONIC AGE, by J. R. Seeley,
M.A., Regius Professor of Modem History in the University of
Cambridge, with Portraits and Maps. 3 Vols. Demy 8vo. 48J.

*' If we could conceive anything similar to
a protective system in the intellectual depart-
ment, we might perhaps look forward to a time
when our hbtorians would raise the cry of pro-
tection for native industry. Of the unquestion*
ably greatest German men of modern history —
I speak of Frederick the Great, Goethe and
Stem— the first two found long since in Carlyle
and Lewes biographers who have undoubtedly
driven their German competitors out of the
field. And now in the year just past Professor
Seeley of Cambridge has presented us with a
biography of Stein which, though it modestly
declines competition with German works and
disowns the presumption of teaching us Ger-
mans our own history, yet casts into the shade
by its brilliant superiority all that we have our-
selves hitherto written about Stein." — Deutsche
Rundschau.

" In a notice of this kind scant justice can
be done to a work like the one before us : no
short risumi can give even the most meagre
notion of the contents of these volumes, which
contain no page that is superfluous, and none
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Germany of to-day one must study the Ger-
many of^ many yesterdays, and now that study
has been made easy by this work, to which no
one can hesitate to assign a very high place
among those recent histories which have aimed
at original research." — Athemrum.

"The book before us fills an important gap
in English — nay, European — historical litera-
ture, and bridges over the history of Prussia
from the time of Frederick the Great to the
days of Kaiser Wilhelm. It thus gives the
reader standing ground whence he may regard
contemporary events in Germany in their pro-
per historic light . . . We congratulate Cam-
oridge and her Professor of History on the
appearance of such a noteworthy production.
And we may add that it is something upon
which we may congratulate England that on
the especial field of the Germans history, on
the history of their own country, by the use of
their own literary weapons, an Englishman has

f)roduced a history of^ Germany in the Napo-
eonic age far superior to any that exists in
German. "-—Examiner.

ii<

THE UNIVERSITY OF CAMBRIDGE FROM THE

EARLIEST TIMES TO THE ROYAL INJUNCTIONS OF
1535* t>y Ja.mes Bass Mullinger, M.A. Demy 8vo. (734 pp.)i i2j.

tinue his labours, and give us a history of the
University during the troublous times of the
Reformation and the Civil War. " — A themtum.
** Mr Mullinger's work is one of great learn-
ing and research, which can hardly fail to
become a standard book of reference ol the
subject . . . We can most strongly recommend
this book to our readers." — Spectator,

'We trust Mr Mullinger will yet continue
his history and bring it down to our own day. "
— Academy,

"He has brought together a mass of in-
structive details respecting the rise and pro-
gress, not only of his own University, but of
all the principal Universities of the Middle
Ages . . . We hope some day that he may con-

VOL. 11. In the Press,

London: Cambridge University Press Warehouse^ 17 Paternoster Row,

i8 PUBLICATIONS OF

CHRONOLOGICAL TABLES OF GREEK HISTORY.

Accompanied by a short narrative of events, with references to the
sources of information and extracts from the ancient authorities, by
Carl Peter. Translated from the German by G. Chawner,
M.A., Fellow and Lecturer of King's College, Cambridge. Demy
4to. icxr.

"As a handy book of reference for genuine ticular point as quickly as possible, the Tables
students, or even for learned men who want to are M^axX.*'— -Academy.
lay their hands on an authority for some par-

CHRONOLOGICAL TABLES OF ROMAN HISTORY.

By the same. {Preparing.

HISTORY OF THE COLLEGE OF ST JOHN THE
EVANGELIST, by Thomas Baker, B.D., Ejected Fellow. Edited
by John E. B. Mayor, MJV., Fellow of St John's. Two Vols.
Demy 8vo. 24J.

"To antiquaries the book will be a source **The work displays very wide reading, and

of almost inexhaustible amusement, by his- it will be of great use to members of the col

torians it will be found a work of considerable lege and of the university, and, perhaps, of

service on questions respecting our social pro- stul greater use to students of English his-

gress in past times ; and the care and thorough- tory, ecclesiastical, political, social, literary

ness with which Mr May or has discharged his and academical, who have hitherto had to be

editorial functions are creditable to his learning content with ' Dyer.'" — Academy.
and industry." — Atkeneeum.

HISTORY OF NEPAL, translated by MUNSHI Shew
Shunker Singh and Pandit ShrI Gunanand; edited with an
Introductory Sketch of the Country and People by Dr D. Wright,
late Residency Surgeon at Kathmanda, and with facsimiles of native
drawings, and portraits of Sir JUNG Bahadur, the King of Nepal,
&c. Super- royal 8vo. 7.\s,

"The Cambridge University Press have interesting." — Nature.

done well m publishing this work. Such trans- ** The history has appeared at a very op-

lations are valuable not only to the historian portune moment... The volume.. .is beautifully

but also to the ethnologist ; . . . Dr Wright's printed, and supplied with portraits of Sir Jung

Introduction is based on personal inquiry and Bahadoor and others, and with excellent

observation, is written intelligently and can- coloured sketches illustrating Nepaulese archi-

didly, and adds much to the value of the tecture and religion." — Examiner.
volume. The coloured lithographic plates are

SCHOLAE ACADEMIC AE : some Account of the Studies
at the English Universities in the Eighteenth Century. By Chris-
topher Wordsworth, M.A., Fellow of Peterhouse; Author of
" Social Life at the English Universities in the Eighteenth Century."
Demy 8vo. 15J.

"The general object of Mr Wordsworth's "Only those who have engaged in like la-
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has collected a great quantity of minute and sustained industry and conscientious accuracy
curious information about the working of Cam- discernible in every page ... Of the whole
bridge institutions in the last century, with an volume it may be said that it is a ^nuine
occasional comparison of the corresponding service rendered to the study of University
state of things at Oxford ... To a great extent history, and that the habits of thought of any
it is purely a book of reference, and as such it writer educated at either seat of learning in
will be of permanent value for the historical the last century will, in many cases, be far
knowledge of English education and learning. " better understood after a consideration of the
— Satuirlay Review. materials here collected." — Academy.

THE ARCHITECTURAL HISTORY OF THE UNI-

VERSITY AND COLLEGES OF CAMBRIDGE, by the late
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tions. Continued to the present time, and edited by John Willis
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[In the Press.

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THE CAMBRIDGE UNIVERSITY PRESS.

19

MISCELLANEOUS.

A CATALOGUE OF ANCIENT MARBLES IN GREAT

BRITAIN, by Prof. Adolf Michaelis. Translated by C. A. M.
Fennell, M.A., late Fellow of Jesus College. Royal 8vo. £2, 2s,

"The object of the present work of Mich-
aelis is to describe and make known the vast
treasures of smcicnt sculpture now accumulated
in the galleries of Great Britain, the extent and
value of which are scarcely appreciated, and
chiefly so because there has hitherto been little
accessible information about them. To the
loving labours of a learned German the owners
of art tieasures in England are for the second
time indebted for a full description of their rich
possessions. Waagen gave to the private col>
lections, of picttu^s the advantage of his in-
spection ana cultivated acquaintance with art,
and now Michaelis performs the same office
for the still less known private hoards of an-
tique sculptures for which our country is so
remarkable. The book is beautifully executed,
and with its few handsome plates, and excel-
lent indexes, does much credit to the Cam-
bridge Press. It has not been printed in

German, but appears for the first time in the
English translation. All lovers of true art and
of good work should be grateful to the Syndics
of the University Pre.ss for the liberal facilities
afibrded by them towards the production of
this important volume by Professor Michaelis."
— Saturday Kevinv.

'"Ancient Marbles' here mean relics of
Greek and Roman origin which have been
imported into Great Britain from cla.ssicai
soil. How rich this island is in respect to
the.se remains of ancient art, every one knows,
but it is coually well known that these trea-
sures h.id been most inadeqtmtely described
before the author of this work undertook the
labour of description. Professor Michaelis has
achieved so high a fame as an authority in
classical arclueology that it seems unneces-
sary to say how good a book this is."— Tfif
Antiquary.

JLECTURES ON TEACHING, delivered in the University

of Cambridge in the Lent Term, 1880. By J. G. FiTCH, M.A., Her
Majesty's Inspector of Schools. Crown 8vo. New Edition. 5J.

"The lectures will be found most interest-
ing, and deserve to be carefully studied, not
only by per.^ns directly concerned with in-
struction, but by parents who wish to be able
to exercise an intelligent judgment in the
choice of schools and teachers for their chil-
dren. For ourselves, we could almost wish to
be of school age again, to learn history and
geography from some one who could teach
them alter the pattern set by Mr Fitch to his
audience . . . But perhaps Mr Fitch's observa-
tions on the general conditions of school- work
are even more important than what he says on
this or that branch of study.'* — Saturday Re-
Xfirw.

" It comprises fifteen lectiu«s, dealing with
such subjects as organisation, discipline, ex-
amining, language, fact knowledge, science,
and methods of instruction; ana though the
lectures make no pretention to systematic or
exhaastive treatment, they yet leave very little
of the g[Tound uncovered; and they combine in
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ciples with practical suggestions and illustra-
tions which are evidently derived from wide
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cxxmxnmg.*'— Scotsman.

"As principal of a training college and as a
Government inspector of schools, Mr Fitch has
got at his fingers' ends the workmg of j)rimary
education, while as assistant commissioner to
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schools . . . Mr Fitch's book covers so wide a
field and touches on so m.any burning cjuestions
that we must be content to recommend it as

the best existing vad* mecum for the teacher.
. . • He is always sensible, always judicious,
never wanting in tact ... Mr Fitch Is a scholar ;
he pretends to no knowledge that he does not
possess ; he brings to his work the ripe expe-
rience of a well- stored mind, and he possesses
in a remarkable degree the art of exposition."
^Pall Mall Gazette.

'* In no other work in the English language,
so far as we know, are the principles and
methods which most conduce to successful
teaching laid down and illustrated with such

Etecision and fulness of detail as they are
^n.'*— Leeds Mercury.

"Therefore, without reviewing the book for
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of the opportunity of c.-illing attention to the
rc-issue of the volume in the five-shilling form,
bringing it within the reach of the rank and
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casion pass without making special reference to
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lecture on * Discipline.' "—School Board Chron-
tele.

"From beginning to end Mr Fitch's book
is eminently practical. ^ No teacher can read it
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conception of his duties, or without obtaining
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helpful to him in a profession which is not
always so generously considered as it ought to
be. Members of School Boards who wish to
understand the work they are engaged in su-
perintending, will also find it of use."— i'f<?/.
tish Review.

THEORY AND PRACTICE OF TEACHING. By the
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late Fellow of King's College, Cambridge. Crown 8vo. 6jr.

London: Cambridge University Press Warehcusey ij Patertioster Row^

30 PUBLICATIONS OF

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HAUFF. DAS WIRTHSHAUS IM SPESSART. Edited

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DER OBERHOF. A Tale of Westphalian Life, by Karl

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Wagner, Ph.D., late Professor at the Johanneum, Hamburg, y.

A BOOK OF GERMAN DACTYLIC POETRY. Ar-

ranged and Annotated by the same Editor. 31.

3)er crpe Stxmmi (THE FIRST CRUSADE), by Fried-

RICH VON Raumer. Condensed from the Author's 'History of the Hohen-
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A BOOK OF BALLADS ON GERMAN HISTORY.

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DER STAAT FRIEDRICHS DES GROSSEN. By G.

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** Prussia under Frederick the Great, and France under the Directory, bring us face to face
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GOETHE'S HERMANN AND DOROTHEA. With

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5)a« 3a^r 1813 (The Year 1813), by F. Kohlrausch.

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i

V. ENGLISH.

OUTLINES OF THE PHILOSOPHY OF ARISTOTLE.

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THREE LECTURES ON THE PRACTICE OF EDU-
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GENERAL AIMS OF THE TEACHER, AND FORM

MANAGEMENT. Two lectures delivered in the University of Cambridge
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MILTON'S TRACTATE ON EDUCATION. A fac-

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LOCKE ON EDUCATION. With Introduction and Notes

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THE TWO NOBLE KINSMEN, edited with Intro-

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BACON'S HISTORY OF THE REIGN OF KING

HENRY VII. With Notes by the Rev. J. Rawson Lumby, D.D., Nor-
risian Professor of Divinity ; late Fellow of St Catharine's College. 31.

SIR THOMAS MORE'S UTOPIA. With Notes by the

Rev. J. Rawson Lumby, D.D., Norrisian Professor of Divinity ; late Fellow

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Lumby's edition of the ' Utopia.' We are afraid to say more lest we should be thought ex-
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'* It was originally written in Latin and does not find a place on ordinary bookshelves. Avery
great boon has therefore been conferred on the general English reader by the managers of the
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linguistic interest to the intrinsic merit of the work. . . . AH this has been edited in a most com-
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alone is a sufHcient warrant for its accuracy. It is a real addition to the modem stock of classical
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MORE'S HISTORY OF KING RICHARD III. Edited

with Notes, Glossary and Index of Names. By J. Rawson Lumby, D.D.
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A SKETCH OF ANCIENT PHILOSOPHY FROM

THALES TO CICERO, by Joseph B. Mayor, M.A., late Professor of

Moral Philosophy at King's College, London. 3J. 6d.

"In writing this scholarly and attractive sketch, Professor Mayor has had chieflv in view
* undergraduates at the University or others who are commendns the study of the philoso{^cal
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found interesting and useful by educated readers generally, not merely as an introduction to the
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