(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Mathematical and physical papers"



LIBRARY 

UNIVERSITY OF 
CALIFORNIA 



MATH* 
STAT. 



s* 



MATHEMATICAL 



AND 



PHYSICAL PAPERS. 



Sonfcon : C. J. CLAY, M.A. & SON, 

CAMBEIDGE UNIVEESITY PEESS WAEEHOUSE, 
17, PATEBNOSTEK Eow. 




CAMBRIDGE: DEIGHTON, BELL, AND CO. 
LEIPZIG : F. A. BROCKHAUS. 



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. 



Hep rinted from the Original Journals and Transactions, 
with Additional Notes by the Author. 



VOL. II. 



CAMBRIDGE : 
AT THE UNIVERSITY PRESS. 

1883 
[The rights of translation and reproduction arc reserved.] 



v, X. 





PRINTED BY C. J. CLAY, M.A. AND SON, 
AT THE UNIVERSITY PRESS. 



CONTENTS. 



PAGE 

Notes on Hydrodynamics. III. On the Dynamical Equations . . . 1 
On the constitution of the Luminiferous Ether ...... 8 

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

SECTION I. Explanation of the Formation of the Bands on the imper 
fect Theory of Interferences. Mode of Calculating the Number 
of Bands seen in a given part of the Spectrum .... 15 

SECTION II. Investigation of the Intensity of the Light on the com 
plete Theory 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 Theory of Sound ....... 51 

On the Formation of the Central Spot of Newton s Kings beyond the 

Critical Angle ........... 56 

On some points in the Received Theory of Sound ..... 82 

On the perfect Blackness of the Central Spot in Newton s Rings, and on 
the Verification of Fresnel s Formula 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 

PART I. Theoretical Investigation. 

SECTION I. Preliminary Analysis ...... 250 

SECTION II. Propagation of an Arbitrary Disturbance in an 

Elastic Medium ......... 257 

SECTION III. Determination of the Law of the Disturbance in a 

Secondary Wave of Light ....... 280 



8140C5 



viii CONTENTS. 

PAGE 

PART II. Experiments on the Eotatiou of the Plane of Polarization 

of Diffracted Light. 

SECTION I. Description of the Experiments .... 290 
SECTION II. Discussion 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 Brushes 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. in. 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 surfaces 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. n., in. 
and rv. of the Cambridge and Dublin Mathematical Journal, were written by agree 
ment between Sir William Thomson and myself mainly for the use of Students. As 
far as my own share in the series is concerned, there is little contained hi the 
"notes" which may not be found elsewhere. Acting however upon 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. n. p. 282. 

II. On the Equation of the Bounding Surface (Thomson), Vol. in. p. 89. 
IIL (Stokes) as above. 

IV. Demonstration of a Fundamental Theorem (Stokes), Vol. in. p. 209. 

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

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

s. n. 1 



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 necessary 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 the 
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 the 
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. 



OX 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 the 
theory of the friction of fluids in motion*, and accordingly the 
hypothesis 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, 2, and x -f dx, y + dy, z + dz. Let X, Y, 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 pX dx dy dz. The 
difference between the pressures, referred to a unit of surface, at 
opposite points of the faces dy dz is ultimately dp/dx . dx, acting in 
the direction of x negative, and therefore the difference 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 effective moving force parallel to x is ultimately 
p . D*x/Df . dx dy dz, where, in order to prevent confusion, D is 
used to denote differentiation 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, 

V dD 



* The reader who feels an interest in the subject may consult a memoir by 
Navier, Memoires de VAcademie, torn. vi. p. 389 ; another by Poisson, Journal de 
I Ecole Poll/technique, Cahier xx. p. 139 ; an abstract of a memoir by M. de Saint- 
Venant, Comptes Eendus, torn. xvn. (Nov. 1843) p. 1240; and a paper in the Cam 
bridge Philosophical Transactions, Vol. vui. p. 287. [Ante, Vol. i. p. 75.] 

12 



4 NOTES ON HYDRODYNAMICS. 

with similar equations foi> y and z. Dividing by p dx dy dz, trans 
posing, and taking the limit, we get 



ldp_ = = 

pdx~ D?> pdy ~ L>f> pdz ~ W 

These are the dynamical equations which must be satisfied at 
every point in the interior of the fluid mass ; but they are 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 y z, t are the 
independent variables. Now Dx/Dt = u, and on the second suppo 
sition u is a function of t, x, y, z, each of which is a function of t 
on the first supposition. We have, therefore, by Differential Cal 
culus, 

Du D*x du duDx duDu duDz 

__ s\Y _ , _ I . __ l ___ 7 i __ _ __ . 

Dt Dt 2 dt "*" dx Dt^ dy ~Dt "*" dz Dt 
or, since by the definitions of u, v, w, 

Dx Dy Dz 

M = U > to= v > Dt =W > 

, D*x du du du du 

we have -n72 = ~T,+ U -J- + v :r + w T~ 

JJt at ax dy dz 

with similar equations for y and z. 

Substituting in (1), we have 

1 dr> ^ du du du du ~\ 
--f X -- . -- u-j -- v-, -- w -j- 
pdx dt dx dy dz 

1 dp ^ T dv dv dv dv 

~=Y -j- rr u -^ v-j ^-7- 

p dy dt dx dy dz 

1 dp dw dw dw dw 

- -- Z ^ u --. -- v -j -- w -=- 
p dz dt dx dy 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 



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 8 may 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 DpjDt = 0, 



which expresses the fact of the iricompressibility. 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 



(4), 

where k is a given constant, 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 the 
squares of small quantities, we have, putting Pl , Pi for the values 
of p, p in equilibrium, 



f) 



K 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 volume 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 d log p/d log p for p = p l} 
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 K, 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 
differentiation, integration, &c. are sensibly correct, so far as this 
cause 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 
great, &> an element of the surface about P, PN a normal at P, 
directed into the fluid, and let PN = h. Through N draw a plane 
A perpendicular to PN, arid project o> on this plane by a circum 
scribing cylindrical surface. Suppose h greater than the radius r 



OX THE DYNAMICAL EQTTATIOXS. 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 &>. For 
the reason already mentioned r will be neglected, and therefore no 
restriction imposed on k on the first account. Let II be the pres 
sure sustained by the solid, referred to a unit of surface, II 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 
&>, its projection on the plane A, 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 NP, and is ulti 
mately equal to peo ; secondly, the pressure of the solid, which 
ultimately acts along PN and is equal to IIco; 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 effective 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 ah, and therefore, whatever may be 
their directions, vanish in the limit compared with the force pa), 
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, 

^ = 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 x, 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, II 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 Magazine, Vol. xxxn. p. 343, May, 1848.] 



ON THE CONSTITUTION OF THE LUMINIFEROUS ETHER. 

THE phenomenon of aberration may be reconciled with the 
undulatory theory of light, as I have already shown (Phil. Mag., 
Vol. xxvil. 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 -f- vdy + wdz is an exact diffe 
rential. It becomes an interesting question to inquire on what 
physical properties of the ether this sort of motion can be explained. 
Is it sufficient 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. 6"f*). 

In my paper last referred to, I have expressed my belief that 
the motion for which udx + &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 condensations and rarefactions, analogous to the 
vibrations of the air in the case of sound, since such vibrations 
would be propagated with about 10,000 times the velocity of the earth, 

* Ante, Vol. i. p. 13 - t Ante, Vol. i. p. 153. 



ON THE CONSTITUTION OF THE LUMINIFEROUS ETHER. 

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 velocit} 7 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. vm. p. 297*) 

d du d 2 u d*u d 



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 A ,~. 

-r + ;r + :r =0 ........................ ( 2 )> 

dx dy dz 

and the conditions, (1) that the fluid at the surface of the sphere 
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+...is an exact 

O 

differential, which would be obtained from the common equations, 
were stable, the motion which would be obtained from equations 
(1) would approach indefinitely, as p, vanished, to one for which 
udx+ ... was an exact differential, and therefore, for anything 
proved to the contrary, the latter motion might be stable ; but if, 
011 the contrary, the motion obtained from (1) should turn out 
totally different from one for which udx + ... is an exact differen 
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. 93. 



10 ON THE CONSTITUTION OF THE LUMINIFEROUS ETHEE. 

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 sufficient 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 

u = 0, v = 0, w = 0, when r = a (3) ; 

u= lv } v = mv, w = nv, when r = oo (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, 

p = II -f - JJLV -3 (lx + my + nz\ 

u =- f-. ?} t\L :?_ 

with symmetrical expressions for v and w. II 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 //, vanishes, to become one for which udx + ... is an 
exact differential, and therefore the motion which would be 
obtained by supposing udx -\- ... an exact differential, and applying 
to the ether the common equations of hydrodynamics, would be 



ON THE CONSTITUTION OF THE LUMIXIFEROUS 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 on the ether. Hence the 
stars would appear to be displaced in a manner different 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 udx + . . . is an exact differential, 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-}- ... is an exact differential. Accord 
ing to this explanation, any nascent irregularity of motion, any 
nascent deviation from the motion for which udx + ... 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 a 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 we have a medium such as we may conceive the ether to 



OX THE CONSTITUTION OF THE LUMINIFEROUS ETHER. 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 introduction to an admirable memoir by Green, "On the laws of the 
Reflexion and Refraction of Light at the common surface of two nou-crystaUized 
media." Cambridge Philosophical Transactions, 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 
bands. 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 allows 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 Koyal 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 we 
should have bands of the same nature as those described by Sir 
David Brewster, the only difference 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 difference 
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 Powell s experiment 
the fluid fills the double office 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 Powell s 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 diffraction, 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 all theory, it is certain that the image of a point or 
line of homogeneous light seen with a small aperture is diffused. 
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 aperture. With a mode 
rate aperture, however, the diffusion is so slight as not to prevent 
s. II. 2 



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 a 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 M is an odd multiple of J, 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 M is an even multiple of 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 A, ; 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 be the breadths of the interfering streams ; then 
we may take 

_ . o \ 

h sin vt, k sin f vt p j 

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

which varies between the limits (h - k) z and (h + 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 
when 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 

B, G H of the spectrum, or for any of them that may be 

selected, and thence the values of M t by dividing by the known 
values of X. Set down the results with their proper signs opposite 
to the letters B y C ... denoting the rays to which they respectively 
refer, and then form a table of differences by subtracting the 
value of J/ for B from the value for 0, the value for G 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 
F to G ; then the numerical value of N, that is to say, N or 3", 
according as N is positive or negative, gives the number of bands 
seen between F and G. 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 of 



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 
lengths. 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, //- the refractive 
index of the fluid, fjf that of the plate ; then 

R = W-tiT. (2). 

If the plate had been placed so as to intercept light from the 
thick end of the prism, we should have had R = (/* //,) 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 formulae will 
apply to both of the arrangements mentioned in Art. 2. 



BANDS SEEN IN THE SPECTRUM. 21 

If we put /i = 1, 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 
effect 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 effected 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 (j>, $ be the angles of incidence and refraction at the first 
surface of the fluid, i, i those at the surface of the plate, 2e the 
angle of the prism. Since the deviation of F is a minimum, the 
angle of refraction <p f for F is equal to 6, and the angle of inci 
dence (/> is given by sin <j> = /JL^ sin <j> F , and cf> 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 (/> = I/fj, . sin $ = fj, f /fi . sin e ; then i = 2e </> , and i is 
known from the equation 

p sin i = /JL sin i ........................ (3). 

The retardation is given by either of the formula? 



= 



Bnt 

R = T^ cosi -pcosi) ............... (5). 



22 ON THE THEORY OF CERTAIN 

These formulae might be deduced from that given in Airy 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 

R = T sec i . (jjtf fj,) Tp sec i versin (ii } ......... (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 i at all. 

8. The formulse (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 /ji , fjb e be the principal indices of refraction referring to the 
ordinary and extraordinary rays respectively, JJL in the case last 
supposed must be replaced by fJL for the bands polarized in a plane 
perpendicular to the plane of incidence, and by ^ e 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 formula? 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, 



(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), 

P e 

the formula (7) becomes 

R=T(p cosj-tj,cosi) (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 //, e , fjb 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 tne 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 //, . 

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 difference 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 
should 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 polarity of the bands. 

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 difference 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 light, 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 IX THE SPECTRUM. 25 

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 iuvestigation. 

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 I., 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, f 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 C be 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 M, 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 # 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 ; 
x, 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 



2G OX 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 

c 1 ^TT 

(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 I/A,. 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" 1 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 assumption that in the 
immediate neighbourhood of the normal the intensity may be re 
garded as sensibly constant. 

In the expression (a) we have 
PM = *~ 2 + x -* + - 



> nearly, 

if we write / for V(/ 2 +P* + <f}- It will be sufficient to replace 
l/PM outside the circular function by l/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 M 
due to the unretarded stream, 



or on performing the integrations and reducing, 

2chl A/ . 2-rrql \f . irpli . 2?r / pq ph\ .,. 
- - sin -T^T. ^ sm -f? .sin - Ivt-^ - . ...(6). 
\f Trph X/ \ \ f tfj ^ 



For the retarded stream, the only difference is that we must 
subtract H 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, 



\f . 27rql \f . wpk . 2?r / . n pq pk 

. _ + 7 sm -^-.-^y sm -^- . sm -, (vt- R+^- + ^ 

2-Trql X/ Trpk \f X \ / -/ 



BANDS SEEN IN THE SPECTRUM. 27 

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

u sinr + vsin (T a), 
so that 

1= u 2 + v 2 + -iiv cos or, 

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



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 ft at the distance 
of the object-glass varies as ft, and may be represented by Aft. 
Let the former origin be referred to a new origin situated in 
the plane xy t and in the image of the line ; and let 77, q be the 
ordinates of 0, M referred to , so that q = q 77. In 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 2 by Ad ft or Af~ l d7) ; 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 77, the largeness being esti 
mated by comparison with \f/l. Now the angle ^irql/\f changes 
by TT when q changes by \f/2l, which is therefore the breadth, in 
the direction of y, of one of the diffraction 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 ON THE THEORY OF CERTAIN 

sensible error suppose the limits of ij to be oo and + o . 
have then 



V ^(q -r)}}* \ff /sin ^ . 

by taking the quantity under the circular function in place of 77 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, 



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 Bft-fy be the illumination on the 
object-glass due to a length of the line which subtends the small 
angle /3, 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 0", and let f, p be the abscissae of , 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 Bdty, or Bf~ l 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 a?, 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 f ; and we may 
moreover suppose the limits of f to be GO and + oo . Let p be 
the value of p } and w that of dpfdg, when f =p, so that we may 



BANDS SEEN IN THE SPECTRUM. 29 

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



)=,,, ...... (13), 

V A / 

we have for the intensity, 
/= 2 ,- 1 (sin 2 A^+sin 2 &,p + 2 sin ^_p . sin k t p . cos(p -ffjal)} -^ . 

Now I sin 2 &,p -2 = /H sin 2 -^ = TT^, . 

J - oo _p J - oo 

Similarly, sin 2 &,p . -^ = TT&,. 

J - P 

Moreover, if we replace 

cos (p f g t p) by cos p . cos </,) + sin p . sin ^r^, 
the integral containing sin p will disappear, because the positive 
and negative elements will destroy each other, and we have only to 
find w, where 

r t 7 ^ 

w = I sm hjp . sin ^^p . cos g t p . ~ . 

Now we get by differentiating under the integral sign, 

dw r i -7 dp 

-j = I sm h ( p . sin k t p .smg t p. 

1 f 00 
= - (sin (g t + h t + k t ) p + sin (g t -h t - k)p 

* J -00 

- sin (g t + h,- k) p - sin (g t + ^ - ^) ^} -^ . 

But it is well known that 

/"* sin sp j 

dp = 7r ) or = TT, 

J -00 P 

according as 5 is positive or negative. If then we use F (s) to de 
note a discontinuous function of 5 which is equal to + 1 or 1 
according as 5 is positive or negative, we get 



This equation gives 

- = 0, from g t = - oo to g, = - (h t + &,) 



30 ON THE THEORY OF CERTAIN 

= | , from g t = - (A, + k) to g t = - (A, ~ & ) 
= 0, from g t = - (A, - k,) to g t = + (A, ~ & ) 
= - ^ , from g t = A, ~ k, to g, = h t + k, 
= 0, from g t = A, + Jc t to g t = GO . 

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

w 0, from g t GO to g t (]i / + h,) \ 

w = ^ (A, + k, + g), from g t = - (h, + k) to g t = - (A, ~ Jc) ; 

w = irk i or = >irh iy (according as h, > k t or h / < k /} ) 

from g = - (h t - k) to g t = + (^ - A; ,) ; 

^ = -- (A^ -f ^ <7 y ), from g t = ti ~ k t to g t = A / 4- k t ; 

w = 0, from ^ = A 7 + ^ to g t = co . 

Substituting in the expression for the intensity, and putting 
in (13) g t = irtffKf, so that 

g = ^--*g-h-k ..................... (14), 

we get 

07?7 

/-^(A + i) ........................... (15), 

when the numerical value of g exceeds h + k; 

9 7^7 
I=~~{h + k+(h + k-Jg *)cosp} ............... (16), 

when the numerical value of g lies between A -f k and li~k\ 



- ) ...(17), 

according as A or k is the smaller of the two, when the numerical 
value of g is less than A - k. 

The discontinuity of the law of intensity is very remarkable. 



BANDS SEEX IX THE SPECTRUM. 31 

By supposing g t = 0, Jc / = h / in the expression for w, and observ 
ing that these suppositions reduce w to 



we get 



f. 



- 



f 00 

J - 



p 

a result already employed. This result would of course have been 
obtained more readily by differentiating with respect to h r 

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 ty by the angular extent of the portion of the spectrum 
considered, diminished in the ratio of m to 1, m 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 i/r must be taken to refer to the magnified, and 
not the unmagnified spectrum. 

Let the axis of x 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 N 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 OT is negative, and therefore 
g <- (h+k), 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 cr is positive ; and since CT varies as the thickness of 
the plate, g may be made to assume any value from (4sg + h+ k) 
to + oo 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 h, k of the interfering streams are 
equal, and that the streams are contiguous, so that g = 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 g = - (h + k), T= T Q to g = 0, and T= < 2T tog = k + k; 
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 2T , there are dark and bright bands formed, the vividness of 
the bands being so much the greater as T is more nearly equal to 
jP , for which particular value the minima are absolutely black. 

Secondly, suppose the breadths h, k of the two streams to be 
equal as before, but suppose the streams separated by an interval 
2g ; then the only difference is that g = (h + k) corresponds to a 
positive value, T z suppose, of T. If T be less than T 2 , or greater 
than 2T T^ there are no bands; but if T lie between T 2 and 
2T T z bands are formed, which are most vivid when T=T , in 
\vhich case the minima are perfectly black. 



BANDS SEEN IN THE SPECTRUM. 33 

Thirdly, suppose the breadths h, fc of the interfering streams 
unequal, and suppose, as before, that the streams are separated by 
an interval 2g ; then g = (h + k) corresponds to a positive value, 
T 2 suppose, of T : g = (h ~ k) corresponds to another positive 
value, T l suppose, of T, 2\ lying between T 2 and T , T being, as 
before,, the value of T which gives g = 0. As T increases from T Q , 
fj becomes positive and increases from 0, and becomes equal to 
h ~ k when T=2T Q -T lt and to h + k when T=2T - T 2 . When 
T < T z there are no bands. As T increases to T l bands become 
visible, and increase in vividness till T T lt when the ratio of the 
minimum intensity to the maximum becomes that of h k to 
h + 3/i , or of k h to k + 3/z, according as h or k is the greater of 
the two, h, k. As T increases to 27J, T lt the vividness of the 
bands remains unchanged ; and as T increases from 2T T l to 
2T T Z) the vividness decreases by the same steps as it before in 
creased. When T = 2T 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 
woe T, 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 Q 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 l 
to 2T T x 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 f by e, if e be the linear breadth of a band; and for this small 

s. ii. 3 



34 ON THE THEORY OF CEKTAIN 

change of f we may suppose the changes of p and f proportional, 
or put dp/di; = %7T/e. Hence the best aperture for a given thick 
ness is that for which 

4# -|- h + k = ^ . 

If g = and k = h, this equation becomes h = \f/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 S be the change of f when this difference changes by X, 

40 + h + k = -^ 

Hence, when the thickness of the plate is equal to the best thick 
ness, e = 8, 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 

1 Philosophical Transactions for 1841, Part i. p, 6. 



BANDS SEEN IN THE SPECTRUM. 35 

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. 






3-2 



[From the Cambridge and Dublin Mat/wmatical Journal, Vol. in. p. 209 
(November, 1848)]. 



NOTES ON HYDRODYNAMICS. 

IV. Demonstration of a Fundamental Theorem. 

THEOREM. Let the accelerating forces X, Y, Z, acting on the 
fluid, be such that Xdx -f Ydy 4- Zdz is the exact differential d V 
of a function of the coordinates. The function V may also contain 
the time t explicitly, hut 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 
one instant such that udx + 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, " Me- 
moire sur la The orie des Ondes, &c." (Mem. des Savans Etran- 
gers, 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 1). 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 dP dx dP dy dP dz 

. I & _j 

da dx da dy da dz da y 

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



(1). 



In these equations d^xjdf, dx/da, &c. have been written for D*x/Df, 
Dx/Da, &c., since the context will sufficiently explain the sense in 
which the differential coefficients are taken. By differentiating 
the first of equations (1) with respect to b, the second with respect 

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

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



dV dP_d*xdx tfydy d?z dz^ 
da da ~ df da + ~di? da + df da 

dV dP tfxdx d?ydy d*z dz 




db db dt* db + df db + dt 2 db 

dV dP tfxdx tfydy d*z dz 
dc dc ~Wdc + d? dc + ~df dc . 





38 NOTES ON HYDRODYNAMICS. 

to a, and subtracting, we get, after putting for dx/dt, dy/dt, dzjdt 
their values u y v, w, 



d?u dx d 2 u dx d*v dy d?v dy d*w dz 
dtdb da dtda db dtdb da dtda db dtdb da 



^ 

* w 



dtdadb 

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 dx/dt = u, &c., that the first side of (2) is the dif 
ferential coefficient with respect to t of 

du dx ^ du dx dv dy _ dv dy dw dz ^ dw dz . 

~ + ~~ + ~~ ...... 



the differential coefficient in question being of course of the kind 
denoted by D in No. in. of these Notes. Hence the expression 
(3) is constant for the same particle. Let w , v , w be the initial 
velocities of the particle which at the time t is situated at the 
point (x, y, z); then if we observe that x = a, y = b, z = c, when 
t = 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 n 

db da da db db da da db db da da db db da 

du dx du dx dv dy dv dy dw dz dw dz dv n dw n 

dc db db dc do db db dc dc db db dc ~ dc db 

du dx du dx dv dy dv dy dw dz dw dz dw n du 

da dc dc da da dc 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 u, v and w, taken with respect to a, b and c, by differential 
coefficients of the same quantities taken with respect to x, y 
and z, and differential coefficients of x, y and z taken with respect 
to a, b and c, the first sides of equations (4) become 



DEMONSTRATION OF A FUNDAMENTAL THEOREM. 



39 



du dv\ fdy dx dy dx\ fdv dw\ fdz dy dz dy\ 
dy dx) \db da da db) \dz dy ) \db da da db) 

fdw du\ fdx dz dx dz\ 

f Us "" dz) (db da ~ da db) 

fdu dv\ fdy dx dy dx\ fdv dw\ fdz dy dz dy\ 
\dy dx) \dc db db dc) \dz dy) \dcdb db dc) 

fdw du\ fdx dz dx dz\ 
^ \dx~fa) \dcdb~ dbdc) 

fdu dv\ fdy dx dy dx\ fdv dw\ fdz dy dz dy 
\dy dx) \da dc dc da) \dz dy ) \da dc dc da 

fdw du 

f \dx ~ dz 



dx dz 
da dc 



dx dz 
dc da 



...(5). 



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

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

as the unknown quantities. For this purpose multiply equations 
(4) by dzjdc, dzjda, 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 dyjdc, di//da, dy/db, and add ; then the first 
and second will disappear. Putting for shortness 

dx dy dz dx dy dz dx dy dz dx dy dz 
da db dc da dc db db dc da db da dc 

dx dy dz dx dy dz _ ^ ,, 
dc da db dc db da 

we thus get 

dy ~~ Tx = ~R (dc (~db ~ ~da) + da \dc ~ ~db ) + db \da ~ ~dc . 

dv dw _ 1 (dx fdu Q __ dv\ dx fdv dw \ dx fdv\ ^ du^ 
dz~dy~R\fc\^~fa) + fa(^~~db) + db \da ~~ ~db 

du, 

etc 



dx dz~Rc\ab da 



- Q 4- 

da\dc db)^cib\da 



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 y 
b + db, 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 x, y, z at the 
time t ; and the coordinates of the extremities of the three edges 
of the oblique-angled parallelepiped which meet in the point 
(x, y, z) are 

dx j dy 7 dz , 

x + -=- da, y + -/-da, z + -y- da ; 
da da da 

dx , 7 dy ,, dz , T 

x +db db y + db db - z+ M db 

dx 1 dy , dz , 

xj--j-dc t y + / dc, z -f dc. 

dc dc dc 

Consequently, by a formula in analytical geometry, the volume 
of the element which at first was da db dc is R dadbdc at the 
time t. Hence if p be the initial density, 

R = p ^ (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, b, 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 Q da + v Q db + w dc is an exact differential, and in 
particular if the motion begin from rest, we shall have 

^ _ o = ^o _ ffo> _ n dw du Q _ 
db da dc db ~ da dc " 

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

time t 

du dv _ . dv dw _ dw du _ n 
dy dx dz dy ~ dx dz ~ 

or u dx + v dy + w dz is an exact differential. 



DEMONSTRATION OF A FUNDAMENTAL THEOREM. 



41 



Since any instant may be 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 + wdz 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 vr du 

- -f =X - -TT 
p ax at 

1 dp Tr dv 
-~ = Y -j- 
p dy at 

I dp ~ dw 

- -f = Z -IT - 
p dz at 



du 


du 


du - 




dx~ V 


Ty~ 


Tz 




dv 


dv 


dv 




dx 


dy 


Tz 




dw 


dw 


dw 




-j V 

dx 


%-* 


dz\ 





(9). 



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



(d d d dWcZw cfoA du du dv du 
\cfa dx dy dz) \dy dx) dy dx dy dy 

dw du du dv _ dv dv dw dv _ . 
dy dz dx dx dx dy dx dz 

According to the notation before employed, 

d d d d 

-T. + U-J- + V-J-+W-T- 
dt dx dy dz 

means the same as D/Dt. Let 

dw dv _ 9 , du dw _ _ dv du _ _ , 
dy dz ~ dz dx ~ dx dy ~ 



(10). 



(11); 
v 



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

du , dv (du dv 

2 -y-ft) -f 2-j-O) -2-T-+^- 

dz dz \dx dy 



_ dw dw 

-T- -3- would have done as well 
ax ay 



42 



NOTES ON HYDRODYNAMICS. 



Da> " 


du , 


dv 


n (du 


dv\ , 




"Dt 


~T~ W 

dz 


+ -j-e 

dz 


~(dx~ 


dyr 




Deo 


dv n 


dw 


, (dv 


dw\ , 
j eo 




dt 


dx 


dx 


\dy 


dW 




Day" 


dw ,, 


, du 


, (dw 


du\ 




Dt " 


dy 


dy 


\dz 


dx) 





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



(12). 



Now the motion at any instant varying continuously from one 
point of the fluid to another, the coefficients of &> , o>", to" on 
the second sides of equations (12) cannot become infinite. Sup 
pose that when t = 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 a/, a>", a/" vanish when t = 0. Let 
L be a superior limit to the numerical values of the coefficients 
of &) , a/ , a" on the second sides of equations (12) from the time 
to the time t : then evidently a) , co", w" cannot increase faster 
than if they satisfied the equations 



T\ 






L (a/ 4- to" + to" ) 



ft) 



(13), 



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

&) + &)" + ft) " = fl, 

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



DEMONSTRATION OF A FUNDAMENTAL THEOREM. 43 

The integral of this equation is H= (7e"; and since = when 
= 0, (7=0; therefore the general value of H is zero. But H 
is the sum of the three quantities &/, ", a/", which evidently 
cannot be negative, and therefore the general values of , a)", w" 
are each zero. Since, then, o> , &>", &/" w r ould 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 
Dco /Dt, &c. and not dw /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 Q , V Q , 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 

1 da . x 1 dq . 1 dq . . 

pl--< w -^ -pTy ^-** par 

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 w r hich 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 - <) dx + (v- v ) dy + (w- w ) dz 

is an exact differential d(qjp}\ and therefore if u ot v , W be 
zero, or if they be such that u Q dx + v Q dy + w dz is an exact dif 
ferential d(f) Q , udx + vdy + wdz will also be an exact differential 



When udx + vdy + wdz is an exact differential cZ<, 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 <f>. 

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 Traite de Mecanique. 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]:, 
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 u, 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 coefficients of the expanded 
functions cannot in general be obtained by direct differentiation, 
that is by differentiating under the sign of summation, but must 

* Traite de Mecanique, torn. u. p. 688 (2nd edition). 

t See Cambridge Philosophical Transactions, Vol. viu. p. 305. [Ante, Vol. i.p. 110.] 

Traite de Mecanique, torn. u. p. 690. 



DEMONSTRATION OF A FUNDAMENTAL THEOREM. 45 

be got from formulas 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 u, v, and w vanish with t, unless it be proved that 
if u l} v lt w^ be the results obtained by dividing u, v, w by the 
lowest power of t occurring as a factor in u, v, w, and then putting 
t= 0, Ujdx + Vjdy + w^dz is an exact differential. Whether we call 
udx -f 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 exact differential 
at all. The theorem might have been enunciated as follows. 
If the three quantities dujdy 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, 
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 w r hen 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 Values of the sums of Periodic Series," Cambridge 
Philosophical Transactions, Vol. vin. Part 5. [Ante* Vol. i, p. 236.] 



4G NOTES ON HYDRODYNAMICS. 

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 
udoo + 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 difference ; 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 effect 
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 R the expression given by (8), and 
introducing the notation of angular velocities, as in (11), equa 
tions (7) become 

, p fdx t dx dx 



I db w ^dc" 

/// _ _/>_ fdz , dz , r dz , fl \ 
~~p^(da (0 db 5c W / 
* Crelle s Journal, Vol. LV. p. 25. 



(a). 



48 NOTES ON HYDRODYNAMICS. 

We see at once from these equations that if 

da db dc 



(b), 



., dx dy dz , . 

then = -4, = -,- ........................ ( c ), 

CO CO (i) 

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

dx dy _ dz 
~P = ~Q = ~R> 

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, R, as 
exhibited in (c), we see that (c) are also the differential equations 
of the system of vortex lines at the time t. Therefore the same 
loci of particles which at one moment are vortex lines remain 
vortex lines throughout the motion. 

Let I1 be the resultant angular velocity at the time of a 
particle P which at the time t is at P, and has fl for its angular 
velocity ; let d* Q 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 (b) is equal 
to c?s /f! , and each member of equations (c) equal to cfo/H. Hence 
we get from any one of equations (a) 



Let A 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 A 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. 40 

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

pAds = p Q A ds , 
whence 



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 t 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) 

da) da" dw" f 

~7 -- 1 -- ~J -- f~ / - == Oj 

dx ay dz 
and consequently 

day day" da> 



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

ffu dyds + jj <*"dzdx + ff** d*dy = 0, 

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



where dS is an element of the surface of the space, and 6 the 
s. ii. 4 



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 6 90, 
and at the two ends 6 = 180 and = 0, respectively, and therefore 
if fl denotes the angular velocity at the second extremity of the 
portion of the vortex thread considered 



which proves the theorem.] 



[From the Philosophical Magazine, Vol. xxxm., p. 349 (November, 1848.)] 



OX A DIFFICULTY IN THE TflEORY 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 notation. 

* [The following quotation will suffice to put the reader in possession of the 
apparent contradiction discovered by Professor Challis. It should be stated that 
the investigation relates to plane waves, propagated in the direction of *, 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, 

w m sin - {z - (a + w) t], 

A 

This equation shows that at any time fj we shall have ic = at points on the axis 
of 2, for which 



or 

At the same time tr will have the value =tm at points of the axis for which 



or = -_ + W jf 1 - - 



4-2 



52 ON A DIFFICULTY IN THE THEORY OF SOUND. 

Without entering into the consideration of the mode in which 
Poisson obtained the particular integral 

(1), 



it may easily be shown, by actual differentiation and substitution, 
that the integral does satisfy our equations. The function / being 
arbitrary, we may assign to it any form we please, as representing 
a particular possible motion, and may employ the result, so long as 
no step tacitly assumed in the course of our reasoning fails. The 
interpretation of the integral (1) will be rendered more easy by 
the consideration of a curve. In Fig. 1 let oz be the axis of z> 
and let the ordinate of the curve represent the values of w for 
t = 0. The equation (1) merely asserts that whatever value the 



Fig. 2. 



velocity w may have at any particular point when t 0, the same 
value will it have at the time t at a point in advance of the former 
by the space (a + w) t. Take any point P in the curve of Fig. 1, 
and from it draw, in the positive direction, the right line PP 
parallel to the axis of z, and equal to (a + w) t. The locus of all the 
points P will be the velocity-curve for the time t. This curve is 
represented in Fig. 2, except that the displacement at common 
to all points of the original curve is omitted, in order that the 
modification in the form of the curve may be more easily perceived. 
This comes to the same thing as drawing PP equal to wt instead 
of (a + w) t. Of course in this way P will lie on the positive or 
negative side of P, according as P lies above or below the axis of z. 
It is evident that in the neighbourhood of the points a, c the curve 
becomes more and more steep as t increases, while in the neigh- 

Here it is observable that no relation exists between the points of no velocity 
and the points of maximum velocity. As m, t lt and X are arbitrary constants, we 
may even have 



in which case the points of no velocity are also points of maximum velocity,"] 



ON A DIFFICULTY IN THE THEORY OF SOUND. 53 

bourhood of the points o, b, z its inclination becomes more and 
more gentle. 

The same result may easily be obtained analytically. In 
Fig. 1, take two points, infinitely close to each other, whose 
abscissas are z and z + dz ; the ordinates will be iy and 

dw j 

10+-T- dz. 
dz 

After the time t these same ordinates will belong to points whose 
abscissas will have become (in Fig. 2) z + wt and 

dw 



(** 



Hence the horizontal distance between the points, which was dz, 
will have become 



and therefore the tangent of the inclination, which was dwjdz, will 
have become 

dw 

..(A). 




At those points of the original curve] at which the tangent is 
horizontal, dwjdz = 0, and therefore the tangent will constantly 
remain horizontal at the corresponding points of the altered curve. 
For the points for which dwjdz is positive, the denominator of the 
expression (A) increases with t, and therefore the inclination of 
the curve continually decreases. But when dwjdz is negative, 
the denominator of (A) decreases as t increases, so that the curve 
becomes steeper and steeper. At last, for a sufficiently large 
value of t, the denominator of (A) becomes infinite for some value 
of z. Now the very formation of the differential equations of 
motion with which we start, tacitly supposes that we have to deal 
with finite and continuous functions ; and therefore in the case 
under consideration we must not, without limitation, push our 
results beyond the least value of t which renders (A) infinite. 
This value is evidently the reciprocal, taken positively, of the 
greatest negative value of dwjdz ; w here, as in the whole of this 
paragraph, denoting the velocity when t = 0. 



54 ON A DIFFICULTY IN THE THEORY OF SOUND. 

By the term continuous function, I here understand a function 
whose value does not alter per saltum, and not (as the term 
is sometimes used) a function which preserves the same alge 
braical expression. Indeed, it seems to me to be of the utmost 
importance, in considering the application of partial differential 
equations to physical, and even to geometrical problems, to con 
template functions apart from all idea of algebraical expression. 

In the example considered by Professor Challis, 

2?r 

w = m sin [z (a + w) t], 
A 

where m may be supposed positive ; and we get by differentiating 
and putting t 0, 

dw 2 



T- = COS - , 

dz A A 

the greatest negative value of which is 2?rm/\ ; so that the 
greatest value of t for which we are at liberty to use our results 
without limitation is X/2?rm, whereas the contradiction arrived at 
by Professor Challis is obtained by extending the result to a larger 
value of t, namely X/4m. 

Of course, after the instant at which the expression (A) be 
comes infinite, some motion or other will go on, and we might 
wish to know what the nature of that motion was. Perhaps the 
most natural supposition to make for trial is, that a surface of 
discontinuity is formed, in passing across which there is an abrupt 
change of density and velocity. The existence of such a surface 
will presently be shown to be possible*, on the two suppositions 
that the pressure is equal in all directions about the same point, 
and that it varies as the density. I have however convinced 
myself, by a train of reasoning which I do not think it worth while 
to give, inasmuch as the result is merely negative, that even on 
the supposition of the existence of a surface of discontinuity, it is 
not possible to satisfy all the conditions of the problem by means 
of a single function of the form f{z-(a + w)t}. Apparently, 
something like reflexion must take place. Be that as it may, it 
is evident that the change which now takes place in the nature 
of the motion, beginning with the particle (or rather plane of 
particles) for which (A) first becomes infinite, cannot influence a 

* [Not so: see the substituted paragraph at the end.] 



ON A DIFFICULTY IN THE THEORY OF SOUND. 55 

particle at a finite distance from the former until after the expi 
ration of a finite time. Consequently even after the change in 
the nature of the motion, our original expressions are applicable, 
at least for a certain time, to a certain portion of the fluid. It 
was for this reason that I inserted the words " without limitation," 
in saying that we are not at liberty to use our original results 
without limitation beyond a certain value of t The full discussion 
of the motion which would take place after the change above 
alluded to, if possible at all, would probably require more pains 
than the result would be worth. 

[So long as the motion is continuous, and none of the diffe 
rential coefficients involved become infinite, the two principles 
of the conservation of mass and what may be called the conserva 
tion of momentum, applied to each infinitesimal slice of the fluid, 
are not only necessary but also sufficient for the complete determi 
nation of the motion, the functional relation existing between the 
pressure and density being of course supposed known. Hence any 
other principle known to be true, such for example as that of the 
conservation of energy, must be virtually contained in the former. 
It was accordingly a not unnatural mistake to make to suppose 
that in the limit, when we imagine the motion to become dis 
continuous, the same two principles of conservation of mass and 
of momentum applied to each infinitesimal slice of the fluid should 
still be sufficient, even though one such slice might contain a 
surface of discontinuity. It was however pointed out to me by 
Sir William Thomson, and afterwards independently by Lord 
Bayleigh, that the discontinuous motion supposed above involves 
a violation of the principle of the conservation of energy. In fact, 
the equation of energy, applied to the fluid in the immediate 
neighbourhood of the surface of discontinuity, and combined with 
the two equations deduced from the two principles first mentioned, 
leads in the case ofpxp to 



where p, p are the densities at the two sides of the supposed 
surface of discontinuity ; but this equation has no real root except 
P = /> ] 



[From the Transactions of the Cambridge Philosophical Society, 
Vol. VIIL p. 642.] 



ON THE FORMATION OF THE CENTRAL SPOT OF NEWTON S 
RINGS BEYOND THE CRITICAL ANGLE. 

[Read December 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 difficulty 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 light*)*. 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 Kings, 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 
mula 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 allude especially to the phenomena described by Mr Airy in a paper printed 
in the fourth volume of the Cambridge Philosophical Transactions, p. 409. 

t Eeport on the present state of Physical Optics. Reports of the British 
Association, Vol. in. p. 310. 

$ Cambridge Philosophical Transactions, Vol. Tin. p. 20. 



58 ON 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. Kegard the surface as plane, and take it for the 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 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, V t , 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 : 

2jr -} 

V = a cos- (vt Ix nz), 

2 
V = a cos ~ (vt Ix + nz), 5* (A). 



V = a cos ~ (v t I x n z), 
Ai 

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



NEWTON S RINGS BEYOND THE CRITICAL ANGLE. 59 

cosine of the angle of refraction, these quantities being connected 
by the equations 

i r 



v v" v ^ V * 

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 
separation, for which z = 0. These equations lead to certain 
relations between a, a /5 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 X. The equations, being satisfied identically, 
will continue to be satisfied when I becomes greater than 1, and 
consequently n 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 \/( 1), v being 
equal to *J(l 2 ].), z instead of appearing under a circular func 
tion in the third of equations (A), appears in one of the expo 
nentials ** * * , k 1 being equal to 2ir/\ . 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 + X/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 e +k v z , 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 2?r/X, so that 
kv = k v ; 

V 

(B). 



60 ON THE FOKMATION OF THE CENTRAL SPOT OF 

In these expressions, if each exponential of the form e p ^~V be re 
placed by co&P+fj( 1) sinP, the real part of the expressions 
will agree with (A), and therefore will satisfy the equations of the 
problem. The coefficients of /^/( 1) in the imaginary part will be 
derived from the real part by writing t + \/4<v 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 I to become greater than 1, so that ri 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 ri becomes 
z/>v/( 1), let a, a /} a become 

pe^, p,e < vrl , //e e vrl , respectively: 
then we have the symbolical system 

y 6 -0 V^T ^ Jc(vt-lx-nz)^/^l 



of which the real part 

V = p cos {/ (vt Ix nz) 6], 1 

V I = p / cos{k(vt-lx + nz)-e / } ) [ ............ (D) 

V = p e-W* cos {& (v t - I x) - ff], \ 

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 



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 , (\TT i ) to the axes of #, z y i 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 V, consider its resolved 
parts V x , V, in the direction of the axes of x, z. Then when the 
angle of incidence is less than the critical angle, we have 

F; = - ri v = - cos i f . v ; F; = i f w = sin i f . v, 

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 coefficient of V in V x , namely ri, by 
j/gin-V-i^ an( j then, retaining v for the coefficient, add JTT 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, V 
in (D) must be interpreted to mean an expression from which the 
vibrations in the directions of x, z may be obtained by multiplying 
by v, I 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, z, and being proportional to 
i/, I , 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 affect 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 transversal 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, ri remaining real, or that n and n both become imaginary. 

* Cambridge Philosophical Transactions, Vol. vn. p. 2. 



NEWTONS RINGS BEYOND THE CRITICAL ANGLE. Go 

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, n remaining real ; and let *J(l 2 1) = v. Then 
it may be shown as before that we must put v \J( 1), and not 
v V( 1), for n ; and using p, 6 in the same sense as before, we get 
the symbolical system, 




to which corresponds the real system 

V = pe-*> z cos [k(vt-lx)-6} 9 

cos vt-lz -0 . (F). 



When the vibrations take place in the plane of incidence, 
V and V t 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 formulas 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 refracted, becomes superficial, remove 
z from under the circular function, and insert the exponential 



64 ON THE FORMATION OF THE CENTRAL SPOT OF 

-h>z f 6 Jcvz f or -k v z } 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 + v J (- 1) for n or ri, 

retain the modulus p for coefficient, and subtract 9 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 

+ v 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 Kings, 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 



NEWTON S RINGS BEYOND THE CRITICAL ANGLE. 05 

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 show 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 reflected 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, Fresnel s 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 (I 2 v 2 / v 2 ) z, which occurs in the expres 
sion for the refracted transversal supeificial 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 
s. ii. 5 



66 ON THE FORMATION OF THE CENTRAL SPOT OF 

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 coefficient 
of vibration in the incident light be taken for unity; and, accord 
ing to the notation employed in Airy s Tract, let the coefficient be 
multiplied by b 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,f become imaginary, but that will be taken into ac 
count further on. Then the incident vibration will be represented 

symbolically by 

J (vt-lx-ns) tt 

) 

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



ce - Ic v z e k (v t - I x) V -T 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 cq, q being put for shortness in place of e~ Vv D \ 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 

[I + cefq* (1 + e y + eV +...)} e^*-*****)^. 

Summing the geometric series, we get for the coefficient of the 
exponential 



Now it follows from Fresnel s expressions that 
b=-e, cf=I-e z *> 

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

&-*^-*^/=T ..................... (G). 



Let the coefficient, which is imaginary, be put under the form 
p {cosi/r + / v /( l)sim/r}; then the real part of the whole expres 
sion, namely 

p cos [k (vt -lx + nz] + -^r}, 

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

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



T > / o . o . =- 

K v ~ *Jir am i 1. 
X 

* I have proved these equations in a very simple manner, without any reference 
to Fresnel s formulae, in a paper which will appear in the next number of the 
Cambridge and Diillin Mathematical Journal [p. 89 of the present volume]. 

52 



68 OX 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 *J~1, 

b = cos 2(9 + V^ sin 20, 
where 



fj, COS I 

Putting C for the coefficient in the expression (G), we have 



_ 

b~ l - fb (i _ f) C os 2(9 - V - 1 (1 + f) sin 2(9 

_ (i _ g ) {(i _ <f) cog 2(9 + J^l (1 + <f) sin 20} . 

(l-2*) a + V sin 2 2(9 
whence 

1 i 2 

ten /-~ tea 20 ..................... (3), 



where 

2irD . _ 
-- j Vu2siu a i-l 

2 = 6 ..................... (5). 

If we take p positive, as it will be convenient to do, we must 
take ^r so that cos ty and cos 2$ may have the same sign. Hence 
from (3) sin ^ must be positive, since sin 20 is positive, inasmuch 
as 6 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 6 in the equations (3) and (4), 
where 

Li J u? sin 2 i l ,. 

A -^- -- -- ..................... (6). 



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

* Mr Airy speaks of "vibrations perpendicular to the plane of incidence," and 
"vibrations parallel to the plane of incidence," adopting the theory of Fresnel; but 
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. G9 

13. For the transmitted light we have an expression similar 
to (G), 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^l .2? sin 2(9 

" 



- 2 ) cos 20 - V (1 + 2 2 ) sin 2<9 

_ 2g sin 2(9 {(1 + g 2 ) sin 20 - V^T (1 - g 2 ) cos 2(9} . 

(l-g 2 ) 2 + 42 2 sin 2 20 

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

1 tf 2 
tan -f , = ~= 2, cot 20 ........................ (8), 

22 



If we take p, positive, as it will he supposed to be, we must 
take ^ such that cos ^ i 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 + JTT. Hence, since from 
(3) and (8) ^ is of the form ^jr -f- |TT + mr, n being an integer, we 
must take ^ = ^ \ TT. 

For light polarized perpendicularly to the plane of incidence 
we have only to put c/> for 6. 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 D = 0, and therefore from (5) q = 1, and 
therefore p z = 0, or there is absolute darkness. On receding from 
the point of contact q decreases, but slowly at first, inasmuch as D 
varies as ?* 2 , r being the distance from the point of contact. It 
follows from (4) that the intensity p 2 varies ultimately as r 4 , 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 9 and cf> 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 smaller 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 
D oc X, and therefore r oc *J\ 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 X, or in dispersion, 
where they are arranged according to values of X~ 2 nearly. These 
conclusions agree with observation. A faint blueish tint may be 
perceived about the dark spot seen by reflection ; and the fainter 
portions 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 s be the ratio of the 



NEWTON S RINGS BEYOND THE CRITICAL ANGLE. 71 

intensity of the transmitted light to that of the reflected; s lt s 2 , 
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 

4g 2 sin 2 2# 4^ 2 sin 2 2^> 

& + "^ -. o\ o i &~ " ~ 7"-* o\~o 



sin 20V r 2 . 2 . N 

. = ](/A +1) Sin I 11 (!"/ 

1 2<p/ 

Now according as 5 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) s z = fjfs l , 
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 



When i becomes greater than L the order of magnitude is 
reversed ; and the spots become more and more unequal as i 
increases. When i 90 we have s^ /jfs 2 , 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 difference between the spots is 
insensible through a considerable range of incidence. 






17. It is worthy of remark that the angle of incidence L 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 i = i we have 

sin 26 = sin 2< = 8 ; whence cot = tan (j> = p ......... (11) ; 

f^ 4~ 1 

and J- (1 + /0 (1 -9*) 

-(l + / *T(l- 2 7 + 16A 2 

27rZ> m 2 -! 

where = e * ^ +1 .......................... (12). 

If we determine in succession the angles 6, f, T; from the equa 
tions cot 6 p,, tan f = g, tan 97 = sin 20 tan 2f, 

we have pf = 1 p 2 = -| versin 2?;. 

The expression for the intensity may be adapted to numerical 
computation in the same way for any angle of incidence, except 
that 9 or </> 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 VO"- 2 sin 2 { !) = w, we have ultimately 



, . 

A, /A cos i J p? 1 

and we get in the limit 



V A, 

according as the light is polarized in or perpendicularly to the 
plane of incidence. The same formulae may be obtained from the 
expression given at page 304 of Airy s Tract, which gives the 
intensity when i < 7, and which like (4) takes the form 0/0 when i 
becomes equal to 7, in which case e becomes equal to 1. 

19. When i 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 co . The same remark applies to the series at page 



73 

303 of Airy s Tract. If we had included the coefficient in each 
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 

f( x > y} x g i n y + i # 3 s i n 3j/ + i # 5 sin 5^ + . . . = J tan ~ J 



, 

where x is less than 1, and may be supposed positive. When x 
becomes 1 and y vanishes / (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/(x, y} = ; but if we suppose x 
first to become 1, and then y to vanish, f (x, y) becomes + ?r/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 x = 1, or nearly = 1. 
Consequently, in the case of Newton s Rings when i ~ y 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>% and the ordinary 
expression which applies when i < 7, alter continuously as i alters, 
and agree with (13) when i=y, 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 i = 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, Ave 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, 



7* ON THE FORMATION OF THE CENTRAL SPOT OF 

and for three angles of incidence, namely, the critical angle, the 
angle i before mentioned, and a considerable angle, for which I 
have taken 60. I have supposed //, = 1*63, which is about the 
refractive index for the brightest part of the spectrum in the case 
of flint glass. This value of //, gives 7 = 37 51 , i = 42 18 . The 
numerals I., II. refer to light polarized in and perpendicularly to 
the plane of incidence respectively. 



4D 


i = y 


i=L 
I. and II. 


i= 


60 
IL 


\ 


I. II. 


7 





100 


100 


100 


100 


100 


1 


49 


87 


33 16 


6 


2 


20 


63 


5 1 





3 


10 


43 


1 




4 


6 


30 







5 


4 


22 






6 


3 


16 








7 


2 


12 








8 


2 


10 








9 


1 


8 








10 


1 


6 








11 


1 


5 








12 


1 


5 








13 


1 


4 








26 




i 








27 














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 purpose 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 i = i 
or for i 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 RINGS BEYOND THE CRITICAL 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 6 and </> become equal to 90, and 
therefore sin 29 and sin 2< 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 sufficient 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 
(j) for 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 q = 1, 
these two expressions agree in giving -fy= 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 or, 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 ^r t) i/r /y , and 
the coefficients of vibration will be multiplied by p t , /?, if ^ /y , p tl 



70 ON THE FORMATION OF THE CENTRAL SPOT OF 

are what ^r,, p t in Art. (13) become when < is put for 6. Hence 
we may take 



p / cos a . cos -j - (vt 

;27r , . 

p it sin a . cos <- (vt 



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, 



sin a cos a {p cos I (vt px ) + ty ) /o cos 

( \ A- / x . 

which gives for the intensity (/) at any point of the ring 

= i sm22 * {/>; + p, 2 - 2/o /P// cos (f - f ,)}. 

Let P0, QQ be respectively the real part of the expression at the 
second side of (7) and the coefficient of *J(1), and let P^, Q^ be 
what P0, Q0 become when (j> is put for 6. Then we may if we 
please replace (14) by 

H<2* - (W 2 ! (15). 



The ring is brightest, for a given angle of incidence, when 
a = 45. When i = i, 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 <, are comple 
mentary ; so that cot 2(/>, cot 20 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 

tan" 



It follows from (14) that the ring cannot vanish unless 
p / cos ty t p n cos i/r //} and p t sin ^ = p n sin -^r n . This requires 



NEWTON S RINGS BEYOND THE CRITICAL ANGLE. 77 

that pj = p*, which is satisfied only when i = i, 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 i = 7, the expressions for P d , Q e , take the form 0/0, 
and we find, putting for shortness 7rD/\ =p, 



_ 






If we take two subsidiary angles %, co, determined by the 
equations 

Jrf 1 = tan % = /A 2 tan &>, 

A. 

we get 

P e = cos 2 %, P$ = cos 2 G>, 

Q e = sin % cos %, Q<t> sin o> cos w. 
Substituting in (15) and reducing we get, supposing a= 45, 

-2*>) ..................... (16). 



When i = i, cos 20 = -cos 20, sin 20 = sin 20 ; and therefore 
P^ = P 0) Q$ = - Qe, which when a = 45 reduces (15) to 7 = $ fl 2 . 

If we determine the angle r from the equation 

1 - (f = 2q sin 20 tan r, or tan w = cot 2? . cosec 20, 

we get 

7=isin 2 2sr.cos 2 20 ..................... (17). 

In these equations 



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



78 



ON THE FORMATION OF THE CENTRAL SPOT OF 



D 

. X 


/ 

i = y 


/ 
l* 


D 
X 


/ 

i=y 











1 6 


1 4 


1 


1 3 


3 2 


1 7 


1 2 


2 


3 5 


5 1 


1 8 


1 1 


3 


4 8 


3 6 


1 9 


1 


4 


5 1 


1 9 


2 


9 


5 


4 9 


9 


2 1 


9 


6 


4 5 


4 


2 2 


8 


7 


4 


2 


2 3 


7 


8 


3 6 


1 


2 4 


7 


9 


3 1 





2 5 


6 


1 


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 




3 


4 


1 5 


1 5 









The column for i = y may be continued with sufficient 
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 CRITICAL 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 due 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 discussed. 



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 
different 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 sufficient 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 



NEWTON 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 sub3tance to be 
examined is less than that of the prism or plate. 



s. ii. 



[From the Philosophical Magazine, Vol. xxxiv. p. 52, (January, 1849.)] 



ON SOME POINTS IN THE EECEIVED THEOKY 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. xxxn. 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. xxxni. 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 s 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 r 2 s ; and as the constancy 
of mass requires that this excess should be constant, s must vary inversely as 
r 2 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 
so 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 + at, in which case the expressions for v and s will likewise con 
tain functions of r at only. Let 

a, =/>;" ) (1). 

We are at liberty to suppose f (z) = 0, except from z = b to 
z = c, where b and c are supposed positive ; and we may takey (2) 
to denote any arbitrary function for which the portion from z = b 

the time of travelling from the origin to the distance r. Then assuming the 
expression (1), and putting 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 occupy the same space hi the quiescent state of the 
fluid, 

4.TT . a-t-xp If (r - at) dr+a-t 

very nearly, or ^irpAt, putting A for the value of the integral If (r - at) dr taken from 

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

62 



84 ON SOME POINTS IN THE 

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

fc=M^dL + +(r),. ..(2), 



-\Jr (r) being an arbitrary function of r, to determine which we must 
substitute the value of < given by (2) in the equation which < has 
to satisfy, namely 



r~ s u/ ; a * \ / 

df dr 2 

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

^ _/ ( r ~ ot } ^_f( r ~^) _ -^ (4) 

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, 

,^J (5). 

r 

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



.(6). 



The velocities expressed by (5) and (6) are evidently such 
as could take place in an incompressible fluid. Now Professor 
Challis s 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 D = 0, A = ; but if A = 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 e. 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 Traite de Mecanique 
(second edition) do not apply to the whole wave from r = at-e 



RECEIVED THEORY OF SOUND. 85 

to r = at + e, 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 s by a single function 1/ar. % (r ai), where 

*(-*) =/(*). * (*)=*". 

z being positive ; and we shall have 

A = [ X (z) dz =/(e) -/(O) +*()- **(<>). 

J -e 

Now Poisson has proved, and moreover expressly stated at 
page 706, that the functions F, f vanish at the limits of the wave ; 
so that/(e) = 0, .F(e) = 0. Also Poisson s equations (6) give in 
the limiting case for which z = 0, /(O) + ^(0) = 0, so that A = Q 
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 
suppose the condensation and velocity when t = given by the 
equations 

_>, =/>)-/>), 

r r r 2 

from r = b 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 diffi 
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 6 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 A, or f(c) f(b\ 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 c/a 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/rrdr taken throughout the wave, 
or A t 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 b to increase indefinitely, the condensation or 
rarefaction in the wave which travels towards the centre will be a 
small quantity, of the order 6" 1 , compared with that in the shell. 
In the limiting case, in which b = oo , the condensation or rarefac 
tion in the backward travelling wave vanishes. If in the equations 
of paragraph 3 we write b + x for r, bar (x) for/ (r), and then sup 
pose b to become infinite, we shall get as = cr (x), v = <r(x). Con 
sequently a plane wave in which the relation v = as is satisfied 
will be propagated in the positive direction only, no matter 
whether f cr (x) dx taken from the beginning to the end of the 
wave be or be not equal to zero ; and therefore anv 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. 8? 

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 waves, 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 6, 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 r, and then 
ceasing. This problem is evidently just as gocd 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 unwilling 
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 (/> which belongs to the 



88 ON SOME POINTS IN THE KECEIVED THEORY OF SOUND. 

present case, we need not burden our equations with this function, 
but we may assume as the expression for < 



(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 t 0, from r = b to r oo (8) ; 

for <f> 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 ; 

J?= F when r = 6, from = to t = r (9); 

-p = when r = 6, fronU = r to t = oo (10). 

(8) determines /(V) from z = b to z oo ; (9) determines / (V) from 
z = b to z b ar, and (10) determines f(z) from z = b ar to 
z = oo , 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 b, 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 Cambridge and Dublin Mathematical Journal, Vol. iv. 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 formed 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. Report s of the British Association, 
Vol. in. p. 344. 



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 8 may be supposed a plane. Let A be a point in the 
surface S where a ray is incident along IA in the first medium. 
Let AR, AF be the directions of the reflected and refracted rays, 
APi 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 taken for unity, and, 
according to the notation employed in Airy s Tract, let the coeffi 
cient of vibration be multiplied by b 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 KINGS. 91 

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



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 AI or AR , and negatively along AR or AF, the 
reflected ray reversed will give rise to the rays represented by 

6 2 sin X, reflected along A I; 
be sin X } refracted along AR ;* 

and the refracted ray reversed will give rise to 
c/sin X, refracted along A I 
ce sin X t 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 

be + ce = 0, 6 2 + cf= 1 ; 

* It does not at once appear whether on reversing a ray we ought or ought not 
to change the sign of the coefficient ; but the following considerations will shew 
that we must leave the sign unaltered. Let the portion of a wave, in which the 
displacement of the ether is in the direction which is considered positive, be called 
the positive portion, and the remaining part the negative portion; and let the 
points of separation be 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 X 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 versa, we must more 
over add TT to the phase, which comes to the same thing as changing the sign 
back again. Thus we must take I- sin X, as in the text, and not - 6 8 sin X, to 
represent the ray reflected along A I, and so in other cases. 



92 ON THE PERFECT BLACKNESS OF THE 

and therefore, since c is not zero, 



(1), 
(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 EA, 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 indifferently 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 NEWTON 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 b, c, e, or/, according to the previous notation, let the phase of 
vibration be accelerated by the angle /3, 7, e, or c, a retardation 
being reckoned as a negative acceleration. Then, if we still take 
sin X to represent the incident ray, we must take 6sin(X + /8), 
c sin (X+ 7) to represent respectively the reflected and the re 
fracted rays. After reversion we must change the signs of /3 and 
7, 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 /3), csin(X 7) to represent the rays incident along 
RA, FA, which together produce the ray sinX along AL Now 
the ray along RA alone would produce the rays 

6 2 sin X along AT, be sin (X ft + 7) along AR ; 
and the ray along FA alone would produce the rays 

cf sin (X 7 4- <) along AT, ce sin (X 7 -f e) along AR . 
We have therefore in the same way as before 

c/sin (X - 7 + (/>) = (1 - 1-} sin X, 
b sin (X - j3 + 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 f in the first equation, or of b or e in the second. We 
are therefore at liberty to take 

= 7 ( 3 )> 

/3 + e = 2 7 (4); 

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



94 ON THE PERFECT BLACKNESS OF THE 

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 the accelerations of 
phase 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 
fj 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 SvrF/X (according to the notation in Airy s Tract) 
by 2-TrF/A, 2e, or replace D, the distance of the media, by 
D Ae/27r cos j3. 

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 vibra 
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), (8), (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 ON 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 interval 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 //,, fjf for the refractive 
indices of the first and second media, i, i for the angles of in 
cidence and refraction, we easily find, by calculating the retarda 
tions, that 

Q 2-TrA . TrA yu, . .., .. 



from which we get, by interchanging i and i , p and /& , and chang 
ing the signs, since for the first reflection and refraction the true 
surface conies before the supposed, but for the second the supposed 
surface comes before the true, 

2-TrA , ., TrA u! .., , 

= -- - fj, cosz , 6 = -; . sin u t); 
A, X sm i 

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 re version 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 equally 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 A,, 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 we 
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 Fresnel s 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 formulae, which admit 

of being put, and which have been put, to the test of experiment. 

s. ii. 7 



98 ON THE VERIFICATION OF FRESNEL S FORMULA FOR 

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 Avhich 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 formulae, 
chosen so as to satisfy the more remarkable results of experiment. 
On the contrary, they were obtained by him from dynamical con 
siderations and analogies, which, though occasionally somewhat 
vague, are sufficient to lead us to regard the formulas 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 difference 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 different formulae, corre 
sponding to different 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 REFLECTED 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 formulas, 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 Fresnel s formula 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 different particular hypo 
theses, leading to formula differing 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 given-f-for 
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 fju 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 formulas 
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 Eoyal Irish Acadimy, vol. xvii. p. 171. 

t Transactions of the Camlridge Philosophical Society, vol. vn. p. 22. 



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 f, from whence the result 
which would have been observed at an angle of incidence 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 confirmation of 
Fresnel s formula?. 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 
mulae do not apply to metals. 

By the fundamental hypotheses of the theory of transverse 
vibrations, are here meant the suppositions, first, 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- 

* Philosophical Magazine (Netv Series), vol. xxu. (1843) p. 92. 
t Ibid. p. 262. 



THE INTENSITIES OF REFLECTED AND REFRACTED IP.YS. 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 formulae for the three particular 
angles of incidence above mentioned is, as we have seen, not suffi 
cient: the formulae however admit of a very searching comparison 
with experiment in an indirect way, which does not require any 
photometries] processes. When light, polarized in a plane making 
a given angle with the plane of incidence, is incident on the sur 
face 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 formulae 
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 refracted light is 
plane polarized, at least if substances of a very high refractive 
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 formulas for 
reflection are both wrong 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 theory, to 



ON THE VERIFICATION OF FRESNEL s FORMULA FOR 

speak, not of the absolute displacement within a refracting medium, 
but of the equivalent displacement in vacuum, of which all that we 
are concerned to know is, that it is proportional to the absolute 
displacement. By the equivalent displacement in vacuum, is here 
meant the displacement which would exist if the -light were to pass 
perpendicularly, and therefore without refraction, out of the medium 
into vacuum, without losing vis viva by reflection at the surface. 
It is easy to prove that Fresnel s formulae for refraction would be 
adapted to this mode of estimating the vibrations by multiplying 
by /\//z; indeed, the formulas for refraction might be thus proved, 
except as to sign, by means of the principle of vis viva, the 
formulae for reflection being assumed. It will be sufficient to shew 
this in the case of light polarized in the plane of incidence. 

Let i, i be the angles of incidence and refraction, A any area 
taken in the front of an incident wave, I the height of a prism 
having A for its base and situated in the first medium. Let r be 
the coefficient of vibration in the reflected wave, that in the inci 
dent wave being unity, q the coefficient of the vibration in vacuum 
equivalent to the refracted vibration. Then the incident light 
which fills the volume Al will give rise to a quantity of reflected 
light filling an equal volume Al, and to a quantity of refracted 
light which, after passing into vacuum in the way supposed, would 
fill a volume Al cos i /cosi. We have therefore, by the principle 
of vis viva, 

cos % sin 2 (i i} 4 sin i cos i 1 sin i cos i 

f." _ _ __ 1 _ *j* - 1 _ _ J __ / _ _ ; __ 

1 cos i sin 2 (i + i) sin 2 (i -\- i) 

This equation does not determine the sign of q: but it seems 
impossible that the vibrations due to the incident light in the 
ether immediately outside the refracting surface should give rise 
to vibrations in the opposite direction in the ether immediately 
inside the surface, so that we may assume q to be positive. We 
have then 

_ 2 cos i V(sin % sin i) 2 sin i cos i 



sin ( + <) sin(t*+0 



f .. 



as was to be proved. The formula for light polarized perpendicu 
larly to the plane of incidence may be obtained in the same way. 
The formula (5), as might have been foreseen, applies equally well 
to the hypothesis that the diminished velocity of propagation 
within refracting media is due to an increase of density of the 



THE INTENSITIES OF REFLECTED AND REFRACTED RAYS. 103 

ether, which requires us to suppose that the vibrations of plane 
polarized light are perpendicular to the plane of polarization, and 
to the hypothesis that the diminution of the velocity of propaga 
tion is due to a diminution of elasticity, which requires us to sup 
pose the vibrations to be in the plane of polarization. 

If the refraction, instead of taking place out of vacuum into a 
medium, takes place out of one medium into another, it is easy 
to shew that we have only got to multiply by .Jfjf/fjL instead of 
\//x; /-t, fj! being the refractive indices of the first and second media 
respectively. 



[From the Cambridge and Dublin Mathematical Journal, Vol. iv. p. 194 
(May and November, 1849).] 



ON ATTRACTIONS, AND ON CLAIKAUT 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 

* Sec the Mccanique Celeste, Liv. in., or the reference to it in Pratt s Mechanics, 
Chap. Figure of 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 Propositions respecting Attractions. 

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 particle situated at the point 
P, x, y t z the rectangular co-ordinates of P referred to any origin, 
x, y, z those of P ; X, Y, Z the components of the attraction of 
m on m, measured as accelerating forces, and considered positive 
when they tend to increase x, y, z ; then, if PP = r, 

,,- 111 f , .,^711,, \ ry HI f , ^ 



* My object in giving 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 Clairaut 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. "NYith one or two exceptions, these propositions will all be found in 
a paper by Gauss, of which a translation is published in the third volume of 
Taylor s Scientific Memoirs, p. 153. The demonstration here given of Prop. iv. is 
the 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. ix. is given merely for the sake of exemplifying the application of the same 
mode of proof to a theorem of Gauss s. 



106 ON ATTRACTIONS, AND 

Since r 2 = (x - xf + (y f - yf + (z - z}\ 

,dr , , N , v m dr d m 

we have r -y- = (x x} ; whence X. ^ j = -j r ; 

dx r ax dx r 

with similar equations for Y and Z. 

If instead of a single particle m we have any number of 
attracting particles m } m ... situated at the points (x , y, 2 ), 
(#", y", z"}..., and if we put 

r tt r 

"7 + 7 7 + " = r = V ^ 

we get 

v d /m m" \ dF . ., , v dV c^F 
JT=^- I 7 H 77 + ... ="^ ; similarly Y = -j , ^ = -y-. . . (2). 
dx\rr J dx dy dz 

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 

F-W.. -.(3), 



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 M . 
Nevertheless, even when P lies within M r t or at its surface, the 

expressions for F and dV/d%, namely III r and / / 1 (x x) j- , 

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

PROP. II. To express the attraction resolved along any line 
by means of the function V. 

Let s be the length of the given line measured from a fixed 
point up to the point P ; X, ft, r, the direction-cosines of the 
tangent to this line at P, F the attraction resolved along this 
tangent ; then 

dV dV dV 
- r - + v- r -. 
dy dz 



ON CLAIRAUT S THEOREM. 107 

Now if we restrict ourselves 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 z, each of which is a function of s ; and we shall have, by 
Differential Calculus, 

dV = dV dx dV dy dV dz t 
ds ~ dx ds dij ds dz ds 

and since dxjds X, dy/ds = /z, dzjds = v, we get 



PROP. in. 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 F be the attraction 
of M resolved along a tangent to ra s path, reckoned as an accele 
rating force, the moving force of the attraction resolved in the 
same direction will be mF, and therefore the work done by the 
attraction while m describes the elementary arc ds will be ulti 
mately mFds, or by (4) in . dV/ds . ds. Hence the whole work done 
as m moves from P to P is equal to m (V- F ), F being the 



108 ON ATTRACTIONS, AND 

value of V at P . If P be situated at an infinite distance, F 
vanishes, and the expression for the work done becomes simply 
mV. Hence V 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 to x, y, z 
are equal to the components of the accelerating force; a function 
which exists whenever Xdx -f 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 8, dn an element of the 
normal drawn outwards at dS, then 



If: 



the integral being taken throughout the whole surface 8. 

Let ra 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 t , P 2 , P 8 , &c., P l 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 lt A 2 ... the areas which it cuts out from 8 about the points 
P!, P 2 .... Let 1? 2 ... be the angles which the normals drawn 
outwards at P lt P 2 ... make with the line L, taken in the direction 
from Pj to P ; N lt N 2 ... the attractions of ra at P t , P 2 ... resolved 
along the normals; r lt r 2 ... the distances of P x , P 2 ... 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 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.] 



ON CLATRAUT S THEOREM. 109 

is evident that the angles lt 2 ... will be alternately acute and 
obtuse. Then we have 

^ = ^ cos lt N 2 = - z cos (TT - t ) &c. 

7 \ r 2 

We have also in the limit 

A^ = a?\ 2 sec l , A 2 = ar* sec (TT - 2 ), &c. ; 
and therefore N^A^ = am, N 2 A 2 = am, N 3 A 3 = am, &c.; 
and therefore, since the number of points P lt P z ... is even, 
h\A l + N 2 A 2 + N 3 A 3 + N^A^. . . = am - a m + am - am ... - 0. 

Now the whole solid angle contained within a conical surface 
described with P r for vertex so as to circumscribe S 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 S will thus be exhausted. We have therefore 

limit of 2-V4 = 0; 
or, by the definition of an integral, 



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 JfNdS=0. Bat by (4) N=dV/dn, which proves 
the proposition. 

PROP. v. If V be equal to zero at all points of a closed surface 
S, 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 & 

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 r Call the point, or 
the assemblage of connected points, at which V has its maximum 
value F I} T r 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 l at other points within T, 
such points must not be included in T r Then, all round T lf V is 
decreasing, positively or negatively according as V l is positive or 
negative. Circumscribe a closed surface S l around T l9 lying 



110 ON ATTRACTIONS, AND 

wholly within 8, which is evidently possible. Then if S l be drawn 
sufficiently close round T lt V will be increasing in passing out 
wards across $ x *; and therefore, if n^ denote a normal drawn out 
wards at the element dS 1 of S t , dV/dn^ will be negative or positive 

according as F x is positive or negative, and therefore lldS lt 



taken throughout the whole surface $ 1? 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 F be equal to a constant A at all points of the 
surface S, it must be equal to A at all points within S. For it 
may be proved just as before that F cannot be either greater or 
less than A within 8. 

COR. 2. If F be not. constant throughout the surface 8, and if 
A be its greatest, and B its least value in that surface, F cannot 
anywhere within S 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 F at the 
bounding surface of T we speak of the value of F 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 $, and vanish at an infinite 
distance, F must be equal to zero everywhere outside S. If F 
vanish at an infinite distance, and range between the limits A and 
B at the surface S, V cannot anywhere outside S lie beyond the 
limits determined by the two extremes of the three quantities A, 
B, and 0. 

* It might, of course, be possible to prevent this by drawing S t sufficiently 
puckered, but S l is supposed not to be so drawn. Since V is decreasing from T x 
outwards, if we consider the loci of the points where F has the values F 2 , F 3 , F 4 ... 
decreasing by infinitely small steps from F 1; 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 7\ if T l be a point, a line, or 
a surface, or with the surface of T 1 if 2\ be a space. If now we take for S 1 one of 
these "surfaces of equilibrium," or any surface cutting them at acute angles, what 
was asserted in the text respecting Sj_ will be true. 



ox CLAIRAUT S THEOREM. Ill 

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



dx* dif dz* ~ 



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

T/ , m dV m dr m d*V 3m , m 

= r" ~d^- -V*^x=^ (x -*> ~Sf = /* ( X -*) -? * 

with similar expressions for cPV/dy* and d 2 V /dz*; and therefore 
V satisfies (6). This equation will be also satisfied by the poten 
tials V", V"... of particles m", m "... situated at finite distances 
from the point (x, y, z\ and therefore by the potential V of all the 
particles, since F= V + V" + V" + ... 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 V 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) will be proved by means of (5) further on (Prop, vin.), or 
rather an equation of which (6) is a particular case, by means of an equation of 
which (5) is a particular case. Equation (o) may be proved from (6) by a known 
transformation of the equation fff\V dx dij dz = 0, where TjV denotes the first 
member of (6), and the integration is supposed to extend over the space T. For, 
taking the first term in yF, we get 



where ( - ) , ( ) denote the values of -= at the points where S is cut by 
\dx ) a \dx ), dx 

a line drawn parallel to the axis of .T through the point whose co-ordinates are 
0, y, z. Now if X be the angle between the normal drawn outwards at the element 
of surface dS and the axis of 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 within 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 I 7 satisfies the equation (6). It should be 
observed that a quantity like r~ l is not to be regarded as such, if r 
denote the distance of the point (x, y, z] from a point P t which lies 
within T, because r 1 becomes infinite at P t . 

Clairaut s 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 x, and the second integration over the remainder 
of S. If then we extend the integration over the whole of the surface S, we get 



His 



dx dy dz I I cos X . dS. 



Making a similar transformation with respect to the two remaining terms of yF, 
and observing that if ju, v be for y, z what X is for x, 

. dV dV dV dV 

COS X -7- + COS it h COS V -r- = ^r , 

dx dy dz dn 

we obtain equation (5). 

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



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, co the angular velocity, X, Y, Z the components of the 
whole force at the point (x, y, z) ; then 

av dv dv 



Now the general equation to surfaces of equilibrium is 

$(Xdx + Ydy + Zdz] = const., 
and therefore we must have at the earth s surface 

r+K(^+jf) = 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 F t , V^ outside the earth, and 
let Fj F 3 = V. Since F, and F 2 have the same value 



at the surface, V 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, 6 the angle between this radius and the axis 
of z, $ 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-ecos 2 0) ........................ (8); 

and from (7) we must have at the surface 

F+ JG>V sin 2 = c ........................ (9). 

s. ii. 8 



114 ON ATTRACTIONS, AND 

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



= ......... (10). 

4. The form of the equations (8) and (9) suggests the occur 
rence of terms of the form ijr (r) + %(?*) cos 2 in the value of F. 

Assume then 

F-*Vr(r) + x(r)cos f tf+t0 .................. (11). 

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



sin 9 d 
we get from (10) and (11) 

t" W + *V W + p % W + {%" W + ^ % M ~ p % ( r )l cos ^ 

+ V^ = ...... (12). 

If now we determine the functions ty, % from the equations 



x"W+% W-xW=0 ............ (14), 

we shall have yw - 0. 

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



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 



>* (15). 

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

* Cambridge Mathematical Journal, Vol. r. (Old Series), p. 122, or O Brien s 
Tract on the Figure of the Earth, p. 12. 



ON CLAIRAUT S THEOREM. 115 

case with w provided we take B = 0, D = 0, when we get from 
(11) and (15) 



5. It remains to satisfy (9). Now this equation may he 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 C 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 Fin (9), and retaining the first order only of small 
quantities, we get 



w t 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. We get by equating to zero the sum of the constant 
terms, and the coefficient of cos 2 0, 



A __ G_ 
a &? 



These equations combined with (17) give 1^ = 0. Now we 
have seen that w satisfies the equation y^ = 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 t 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 , w" be respectively the greatest and least values of w t , w cannot 
anywhere outside the earth lie beyond the limits determined by 
the two extremes of the three quantities 10 , 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 

82 



116 ON ATTRACTIONS, AND 

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



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) 

IT E " / a/i i \ /i rv\ 

F= ~ ~" ( * .......... ( 



6. If g 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, g cos v will be the resolved part of gravity along the 
radius vector ; and we shall have 

.................. (20), 

where after differentiation r is to be put equal to the radius vector 
of the surface. Now v 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), 

g = ? (1 + 2e cos 2 0) - 3 ( 2 - \<Ja } (cos 2 6 - ) - a>*a (1 - cos 2 0), 

d \ & " / 

or g =(l +e )|_| a +(4 B a- J) cos 2 ............ (21). 

At the equator 6 = 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 o) 2 a = mGr, and 



whence J=(l+fm-e) a 2 ..................... (22); 

and (21) becomes g= G (1 + (fm - e) cos 2 <9} .................. (23). 

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



ox CLAIRAUT S THEOREM. 117 

have been omitted throughout, we may regard 6 as the comple 
ment of the true latitude, and therefore replace cos 6 by the sine 
of the latitude. In the case of the earth, ra is about -^ and e 
about -jj<y, and therefore f >?i 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 lt and mean gravity G lt 
we have only to remark that the mean value of cos 2 6, or 



is J, which gives 

0,=0(1-J), ff,= 0(1 +}*-$), 

which reduces equations (8), (22), and (23) to 
r = a 1 {l-e(cos 0-l)}, 



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

rr 771 z 

*) ................ (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, 6 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), 

dV IdV. 
2 dr V rde 



118 ON ATTRACTIONS, AND 

whence, from (24), 



Ea * \ 

Q = 2 (e - \m] -- sin 6 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) by E -\- M. If p be 
the moment of the couple, we have 

^ = 2 (e-^m)^f 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 by 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 ILL, 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 EO?K ; let M = Ejti ; let b be the annual precession, 



ox CLAIRAUT S THEOREM. 110 

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



where A, B, C denote certain known quantities. Hence the 
observed values of b and f will serve to determine the two unknown 
quantities n, 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 understood 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 E, a, by M, a , where a is the 
moon s radius, take 6 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 t and a about 4 times as great as a , and there 
fore Ea? is about 1200 times as great as J/a 2 . 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 
effect 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 

* !/( + !) will appear in these equations rather than l[n, because, if S be the 
mass, and r, the distance of the sun, the ratio of 37/r 3 to Sjr ^ is equal to l/(n+l) 
multiplied by that of (E + J/)/r 3 to S/r, 3 , and the latter ratio is known by the mean 
motions of the sun and moon. 



120 ON ATTKACTIONS, 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 effects 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 effect 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 expression for V be formed directly, and be expanded according to 
inverse powers of r, the first term will be Jl//r. The terms involving r~ 2 will 
disappear if the centre of gravity of the moon be taken for origin, those involving 
r~ 3 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. 



ox 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 (|??i) 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, affords no 
additional evidence whatsoever in favour of the hypothesis of 
original fluidity, being a direct consequence of the law of universal 
gravitation*f. 

* It may be proved without difficulty that the value of e corresponding to this 
supposition is T ^ 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 ATTRACTIONS, AND 

If the expression for F 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 

dicular to the surface, as well as tlie 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 be 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 whole 
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, follow from the form of the surface. 

It must not here be supposed that these irregularities are actually neglected. 
Such an omission would ill accord with the accuracy of modern 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 2J 
times that of water ; and therefore the ratio 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 = p l when r = 1. 
From the mean density and. the value of K we know the ratios 

of the integrals I prdr and I pr 4 Jr to p^ Now it is probable 

J o * o 

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 l when ? = !, and of giving 
to the two integrals 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 affected by 
giving to the nearly spherical strata of equal density one form or 
another, but the form of the surface will be materially affected. 
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 priori, 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 Ecfic 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 different 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. vn*. If V be the potential of any mass M lt and if M Q 
be the portion of M l contained within a closed surface S, 



!l^ ds *^ -w- 



n and dS having the same meaning as in Prop. IV., and the inte 
gration being extended to the whole surface 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 



is given. In the present paper a different order of investigation is followed ; direct 
geometrical demonstrations of the equations 

I I -j- dS = in one case, and / / dS= - 47rJ/ in another, 

are given in Props, iv. and vn. ; and a new proof of the equation (a) is deduced 
from them in Prop. viu. 

These equations may be obtained as very particular cases of a general theorem 
originally given by Green (Essay on Electricity, p. 12). It will be sufficient to 
suppose U=l in Green s equation, and to observe that dw=-dn, and 5F=0 
or= -47Tp, if V 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 S in 
one point ; but if S be a re-entrant* surface it may be cut by L 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 
lf 8 , will be alternately obtuse and acute, and we shall have 

AT m t A \ m / 

# l = -- i C06 (TT- 0J = -5 cos e v 
1 1 i 

A l = ar, 2 sec (TT #J = ar* sec 6 V , 
and therefore -^1^-1 ~ ~ am/ - 

Should there be more than one point of section, the terms N 2 A Z , 
N Z A^ &c. will destroy each other two and two, as in Prop. IV. 
Now all angular 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 %NA = Sam = m"Zi ; 
or, since 2a = 4?r, JfNdS = 47rm . 

The same formula will apply to any other internal particle, and it 
has been shewn in Prop. iv. that for an external particle f/NdS = 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 ffNdS = 4?rJ/ . But N= d V/dn, which proves the proposi 
tion. 

Prop. vm. At an internal point (x, y, z) about which the 
density is p, the potential F satisfies the equation 



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 I-T- dSis ultimately dV/dx . dy dz, and for the opposite 

fdV d*V \ 

face it is ultimately + ( -^ h -73 dx \dydz\ and therefore for this 

\ctx (IX / 

* This term is here used, and has been already used in the demonstration of 
Prop, iv., to denote a closed surface which can be cut by a tangent plane. 



126 ON ATTRACTIONS, AND 

pair of faces the value of the integral is ultimately d* V/da? . dx dy dz. 
Treating the two other pairs of faces in the same way, we get ulti 
mately for the value of the first member of equation (27), 



J* 

\ dx" dy* dz 

But the density being ultimately constant, the value of J/ , which 
is the mass contained within the parallelepiped, is ultimately 
p dx dy dz, whence by passing to the limit we obtain equation 

(28). " 

The equation which (28) becomes when the polar co-ordinates 
r, 0, <j> are employed in place of rectangular, may readily be 
obtained by applying equation (27) to the elementary volume 
dr . rdO . r sin 0d<j), or else it may be derived from (28) by transfor 
mation of co-ordinates. The first member of the transformed 
equation has already been written down (see equation (10),) ; the 
second remains 



Example of the application of equation (28). In order to give 
an example of the practical application of this equation, let us 
apply it to determine the attraction which a sphere composed of 
concentric spherical strata of uniform density exerts on an internal 
particle. 

Refer the sphere to polar co-ordinates originating at the centre. 
Let p be the density, which by hypothesis is a function of r, R the 
external radius, V the potential of the sphere, which will evidently 
be a function of r only. For a point within the sphere we get 
from (28) 



For a point outside the sphere the equation which V has to satisfy 
is that which would be obtained from (29) by replacing the second 
member by zero ; but we may evidently apply equation (29) to all 
space provided we regard p as equal to zero outside the sphere. 
Since the first member of (29) is the same thing as 1/r . eZV V/dr 2 , 
we get 



ox CLAIRAUT S THEOREM. 127 

Now we get by integration by parts, 

f(jprdr) dr = rfprdr fpr*dr, 

whence V = - Rfprdr + - 7" $pr*dr, 

where the arbitrary constants are supposed to be included in the 
signs of integration. Now F vanishes at an infinite distance, and 
does not become infinite at the centre, and therefore the second 
integral vanishes when r = 0, and the first when r = oc, or, which 
is the same, when r = R, since p.= when r>R. We get there 
fore finally, 

rR 4,^ rr 

V= 4?r I pr dr H -- I pr z dr. 



If F be the required force of attraction, we have F= d V/dr ; and 
observing that the two terms arising from the variation of the 
limits destroy each other, we get 



Now 4-7T I prdr is the mass contained within a sphere de- 

o 

scribed about the centre with a radius r, and therefore the attrac 
tion is the same as if the mass within this sphere were collected at 
its centre, and the mass outside it were removed. 

The attraction of the sphere on an external particle may be 
considered as a particular case of the preceding, since we may first 
suppose the sphere to extend beyond the attracted particle, and 
then make p vanish when r > R. 

Before concluding, one or two more known theorems may be 
noticed, which admit of being readily proved by the method 
employed in Prop. v. 

Prop. ix. If T be a space which contains none of the attract 
ing matter, the potential V cannot be constant throughout any 
finite portion of T without having the same constant value through 
out the whole of the space T and at its surface. For if possible 
let F have the constant value A throughout the space T lt which 
forms a portion of T, and a greater or less value at the portions of 
T adjacent to T r Let R be a region of T adjacent to T^ where F 
is greater than A. By what has been already remarked, Fmust 



128 ON ATTRACTIONS, AND 

increase continuously in passing from T^ into R. Draw a closed 
surface cr lying partly within T l and partly within R, and call the 
portions lying in T^ and R, a 1 , cr 2 respectively. Then if v be a 
normal to cr, drawn outwards, d V/dv will be positive throughout cr, 
if (T I be drawn sufficiently close to the space T t (see Prop. V. and 
note), and dV/dv is equal to zero throughout the surface cr 2 , since 

V is constant throughout the space T l ; and therefore 1 1 -, da-, 

taken throughout the whole surface cr, will be positive, which is 
contrary to Prop. IV. Hence V cannot be greater than A in any 
portion of T adjacent to T lt and similarly it cannot be less, and 
therefore F must have the constant value A throughout T, and 
therefore, on account of the continuity of F, at the surface of T. 

Combining this with Prop. v. Cor. 1, we see that if F be 
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 F be 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 F would be constant 
within it. 

Prop. x. The potential F cannot admit of a maximum or 
minimum value in the space T. 

It appears from the demonstration of Prop. v. that F 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 F 
have the maximum or minimum value V l throughout T^ if T t 
reach up to the surface 8 of T; though the term maximum or 
minimum is not strictly applicable to this case. By Prop. IX. F 
cannot have the value F t throughout a space, and therefore T l can 
only be a surface or a line. 

If possible, let F have the maximum value V l throughout a 
line L which reaches up to S. Consider the loci of the points 
where F has the successive values F 2 , F 3 ..., decreasing by infi 
nitely small steps from F r 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 s be an element of L not adjacent to S, nor reaching 



ON CLAIRAUT S THEOREM. 129 

up to the extremity of L, in case L terminate abruptly. At each 
extremity of 5 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 T 2 , 
and apply equation (5) to this space. Since Fis a maximum at L, 
dV/dn is negative for the tube surface of T 9 , and it vanishes for 
the other surfaces, as readily follows from equation (4). Hence 

dV 

-T- dS, taken throughout the whole surface T 2 , is negative, 

Ollli 

which is contrary to equation (5). Hence F 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 F t throughout a surface 8^ which 
reaches up to S. For this purpose it will be sufficient to draw a 
line of force through a point in S l} and make it travel round an 
elementary area a which forms part of 8 lf 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 8 lt of a surface of 
equilibrium corresponding to a value of V very little different 
from F r 

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 m ; 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, n. 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 V cannot be a maximum or minimum, 
and therefore either 7 must 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. 

* This theorem was first given by Mr Earnshaw in his memoir on Molecular 
Forces read at the Cambridge Philosophical Society, March 18, 1839 (Tram. 
Vol. vii.). See also a paper by Professor Thomson in the first series of this Journal, 
Vol. iv. p. 223. 



[From the Transactions of the Cambridge Philosophical Society, Vol. vin. p. 672.] 



Ox THE VARIATION OF GRAVITY AT THE SURFACE OF THE 

EARTH. 

[Read April 23, 1849.] 

Ox adopting 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 earth, 
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 

92 



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), 
suffices 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 afforded 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 differ from the 
observed value. 

* It has been remarked by Professor O Brien (Mathematical Tracts, 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 SUEFACE 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 OX THE VARIATION 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, Z be its components along 
three rectangular axes, Xdx + Ydy + Zdz is the exact differential 
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 8 to be a surface of equi 
librium. The general equation to such surfaces is 

V+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 S, when some one particular value is assigned to c. For 
any point external to S, 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 S. Now these conditions 
are sufficient for the complete determination of the value of V for 
every point external to S, the quantities U 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 S, on the conditions, 

(1) that the temperature at the surface S shall be equal to c U, 

(2) that it shall vanish at an infinite distance. This problem is 
evidently possible and determinate. The possibility has moreover 
been demonstrated mathematically. 

If U 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 V, or of one of its differential coefficients, at 
one point situated either in the surface S 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 number of forms of the 
surface 8 that the solution has hitherto been effected. The 
most important of these forms is the sphere. When S 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. Although 
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 f of Art. 1 being in this 
case the centrifugal force. The nearly spherical form of the 
surface renders the determination of the variation easy. 



AT THE SURFACE OF THE EAETR. 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 

d\rV 1 d . dV 



f . Q dV\ I (TV 

sm# T +^3 -77^- = ..... (3). 
\ dB J sm 2 d$* 



, 2 ^ 
dr* sm 

Let co be the angular velocity of the earth ; then 

7=io>Vsin 2 0, 
and equation (1) becomes 

F+|a>Vsin 2 6> = 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 

F.+ F,+ F,+...; 

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 n 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 



we get . 

where all integral values of n from to oo are to be taken. 

Now the differential coefficients of V n with respect to r are 
Laplace s functions of the n ih order as well as V n itself; and since 
a series of Laplace s functions cannot be equal to zero unless 



138 ON THE VARIATION OF GRAVITY 

the Laplace s functions of the same order are separately equal 
to zero, we must have 

(P rV 
r-jJ-"-n(n + l)V n = ..................... (6). 

The integral of this equation is 



where Y n and Z n are arbitrary constants so far as r is concerned, 
but contain 6 and (/>. Since these functions are multiplied by 
different powers of r, Y n cannot be a Laplace s function of the n ih 
order unless the same be true of Y n and Z w We have for the 
complete value of V 

Y Y Y 

^ + -.} + -*+... + Z Q + Z 1 r + ...... 

y* ry** tv^ 

Now V vanishes when r = oo , which requires that Z Q = 0, Z 1 = 0, 
&c. ; and therefore 

Y Y Y 



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 47r/or 2 , 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 coefficient 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 -f u) 
be the equation to the earth s surface, where u is a small quantity 
of the first order, a function of 6 and <f>. Let u be expanded in a 
series of Laplace s functions u + 1^+ ... 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 1 + ?/ 2 + ...) ........................ (8). 

If the surface were spherical, and the earth had no motion of 
rotation, V would be independent of 6 and <, and the second 
member of equation (7) would be reduced to its first term. Hence, 
since the centrifugal force is a small quantity of the first order, as 
well as u t 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 sufficient 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 sufficient in that term likewise to put r = a. We 
thus get from (4), (7), and (8), omitting the squares of small 
quantities, 



The most general Laplace s function of the order is a con 
stant ; and we have 

sin 2 = f + (4-cos 2 0), 

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, 



7, = 



140 ON THE VARIATION OF GRAVITY 

The first of these equations merely gives a relation between 
the arbitrary constants F and c; the others determine F I} F 2 , 
&c. ; and we get by substituting in (7) 



6. Let g 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 g = d(V+ U)/dn. Let ^ be the angle between 
the normal and the radius vector ; then g cos ^ is the resolved 
part of gravity along the radius vector, and this resolved part is 
equal to d ( V+ 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 

9 = 5 ( 1 -2t* 1 -2 M ,-2n 8 ...)+ (2^ + 3^ + 4*,...) 

- 1 *a ( J ~ cos 2 6} - o) 2 a (f + i - cos 2 0), 
which gives, on putting 

I -fo. a-e, J = m .................. (11), 

and neglecting squares of small quantities, 

9= {l-fm(i-cos 2 <9)-f-u 2 +2^ 3 + 3 ? 4 ...... } ...... (12). 

In this equation G is the mean value of g taken throughout 

the whole surface, since we know that I I u n sin d6d<k> = 0, if n 

Jo Jo 

be different from zero. The second of equations (11) shews that ra 
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 observed that on the latter 



AT THE SURFACE OF THE EARTH. 141 

supposition there is nothing to determine w 4 . The most general 
form of u t is 

a sin 6 cos (f> -f fB sin 6 sin </> -f 7 cos 0, 

where a, 0, y 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, aft, ay respectively, the line from which 6 is measured 
being taken for the axis of z, and the plane from which <f> is 
measured for the plane of scz. 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 already a small quantity of 
the first order, would not be affected 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 6, (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 E-r. Comparing 
this with (10), we get Y Q = E y and therefore, from the first of 
equations (11), 

E= a* + Ja>V=Ga 8 (l+Jw) (13), 

which determines the mass of the earth from the value of G deter 
mined by pendulum experiments. 

9. If we suppose that the surface of the earth 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 = a{l + 6(4-cos*0)} (14), 



142 ON THE VARIATION OF GRAVITY 

where e is a constant which may be considered equal to the ellip- 
ticity. We have therefore in this case u^ = 0, u ti = ^~ cos 2 0, u n = 
when n> 2; so that (12) becomes 

0={l-(fm-e)(i-cos 8 0)} .................. (15), 

which equation contains Clairaut s Theorem. It appears also from 
this equation that the value of G 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> 2 a 5 = mOa* = mEa 2 , 

V=~ +(e-im)^- (J-cos*0) ............ (1C). 

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, 6 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 



whence we get from (16) 



(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 E by 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 

m0cos0 ............... (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 I 7 refers, we have 



dx d dz 



- y Y+(z - ) 



,J 



the integrals extending throughout the interior of the earth. 
Writing dm for p dx dy dz, putting X, yu,, v for the direction- 
cosines of the radius vector drawn to the point (x, y, z}, so that 
x = \r, y = pr, z = vr, and expanding the radical according to 
inverse powers of r, we get 

V = ~ fffdm + S ^ fffx dm + ~ 2 (3X 2 - 1) fjfx" 2 dm 

yd m + ...... (19), 



2 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 = jffdm = E, as before remarked, and 

//jy dm = o, j/jy dm = o, //jv dm=o ............ (20), 

J 2 (3X 2 - 1) Jffjc" dm + 32Xyu fffx y dm 

= (e-Jm) J E a 2 (J-cos 2 6>) ......... (21); 



ON THE VARIATION OF GRAVITY 

together with, other equations, not written down, obtained by 
equating to zero the coefficients of 1/r 4 , 1/r 5 &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 X, //>, v are not independent, 
but connected by the equation X 2 + ^ + v 2 1. If now we insert 
X 2 + /u, 2 + z; 2 as a coefficient in each term of (21) which does not 
contain X, //,, or v, the equation will become homogeneous with 
respect to X, ^, 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, /it, v. By the transformation just men 
tioned, equation (21) becomes, since cos 6 v, 

2 (X 2 - i/u, 2 - |z/) /J> /2 dm + 3I,\pffJa/y dm 



and we get 

Sffx y dm = 0, jjjy z dm = 0, fffz x dm = ......... (22), 



= fffy *dm - J //jyw - 1- fffaTdm (23) 



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 

JJfaf>dm - fjfz 2 dm = f (e - J m) Ea\ 

If we denote the principal moments of inertia by A, A, C, this 
equation becomes 

C-A = %(e-m)Ea* .................. (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 V given by (10) and (19). In 
the first place it may be observed that the term ^ can always 
be got rid of by taking for origin the centre of gravity of the 
volume. Equations (20) shew that in the generai 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 u, and let u = u + u", where 
u" = Jm Q- - cos 2 0). Then u r may be expanded in a series of 
Laplace s functions u\ + u\ + ... ; and since Y = E, equation (10) 
will be reduced to 

(25). 



If the mass were collected at the centre of gravity, the second 
member of this equation would be reduced to its first term, which 
requires that u t = 0, u s = 0, &c. Hence (8) would be reduced to 
r a(\ + u f ), 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 2 of u. From the general form of Laplace s functions 
it follows that au 2 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 \ = e (s - sin2 cos2 <W + " (i ~ sin 2 # sin2 <) + e " (i - cos 2 (9), 
e , e", e" being three arbitrary constants, and 0, < denoting angles 
related to the new axes of x, y, z in the same way that the 
angles before denoted by 0, $ 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 



ON THE VARIATION OF GRAVITY 

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 - le Ea? = B- y Ea* = C- %e" f Ea\ 
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 e" (-J cos 2 0) will unite 
with the term u" due to the centrifugal force. Thus the most 
general value of u 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 + w a ), which was established by Laplace 
on the supposition that the earth consists of nearly spherical strata 
of equal density, holds good whatever be 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 SUKFACE 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- 

102 



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 are 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 y which is given by (10). 
In this equation, the term containing r~ z will disappear if r be 
measured from the centre of gravity ; the term containing r~ 4 , 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~ z . Now the 
most general value of u z contains five terms, multiplied by as many 
arbitrary constants, and of these terms one is ^ cos 2 6, and the 
others contain 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. 1 2 ; but even if they did not disappear their effect 
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< or cos 20 may be neglected ; so that nothing remains but a 
term which unites with the last term in equation (10). Let e be 
the coefficient of the term -J- cos 12 6 in the expansion of n : then e 
is the constant which determines the effect 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, 
e is the ellipticity. Hence the constant e seems of sufficient 
dignity to deserve a name, and it may be called in any case the 
ellipticity. 

Let r be the radius vector of the earth s surface, regarded as 



AT THE SURFACE OF THE EARTH. 140 

coincident with the level of the sea; and take for shortness 
m {/(#, <)} to denote the mean value of the function / (0, (/>) 
throughout all angular space, or 



Then it follows from the theory of Laplace s functions that 

e^mKj-sin QrJ ..................... (26), 

I being the latitude, or the complement of 9. To obtain this 
equation it is sufficient to multiply both sides of (8) by l/4?r x 
(J cos 2 0) sin 0dOd(j), and to integrate from 6 = to # = TT, and 
from $ = to </> = 2?r. Since J cos 2 6 is a Laplace s function of 
the second order, none of the terms at the second side of (8) will 
furnish any result except u^ and even in the case of u z the terms 
involving the sine or cosine of <f> or of 2(/> will disappear. 

15. Let g be 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 u 2 
its value we get by means of the properties of Laplace s functions 

# = m (ff), G (f m - e) = - V tn (tt - sin I) g} ...... (27). 

If we were possessed of the values of g 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 G 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 G and e are to be found. 

Let g=g i + g , where g is the attraction just mentioned, so 
that a, is the result obtained by reducing the observed value of 

U 1 i/O 

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, cr 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 = %7ro-h. 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 = ^Trcrh will evidently apply to the surface of the sea, provided 
we regard h as a negative quantity, equal to the depth of the sea, 
and replace a by cr 1, the density of water being taken for the 
unit of density ; or we may retain <j as the coefficient, and diminish 
the depth in the ratio of cr to a 1. 

Let p be the mean density of the earth, then 



^2ir<rAGj ~**G~. 

4 "^ 2.QLI 



If we suppose <r = 2J, p = 5J, a = 4000 miles, and suppose h 
expressed in miles, with the understanding that in the case of the 
sea h is a negative quantity equal to f ths of the actual depth, we 
have g = 00017 Gh nearly. 



* Phil. Trans, for 1819. Dr Young 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/t/a which gives the correction for elevation alone must 
be reduced in the ratio of 1 to l-3<r/4/), or 1 to 66 nearly, if <r = 2i, p = 5^. Mr 
Airy, observing that the value <r = 2^ is a little too small, and p = 5^ a little too 
great, has employed the factor -G, instead of GO. 



AT THE SURFACE OF THE EARTH. 151 

17. Consider first the value of G. We have by the preceding 
formula, and the first of equations (27), 

G = m (g}> + G x -00017 m (k). 

According to Professor Rigaud 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 Jth of a mile, and 
the mean depth of the sea 3 J miles, we shall have 

-. $x3J x276-ixlOO 
(*)--* - = -1-49 nearly; 



so that the value of G determined by g l 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 coefficient 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 e. Let e l be the value which 
would be determined by substituting g l for g in (27), and let 



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 e = e l q, 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 (-J- sin 2 ) cos I, or 
3 cos 31 + 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 Philosophical Transactions, 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 2 ^ = 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 e, 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 q = 000302. The difference between these values of 
q is about one fifth of the whole ellipticity. 

19. The attraction g 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 the 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 off. 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 of g v 

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 AS be the mass contained in a small 
element A of this surface. Then S = o-h in the case of the land, 
and 8 = (a 1) h in the case of the sea, li being in that case the 
depth of the sea. Let V c be the potential of the coating, V, V" 
the values of V c outside and inside the surface respectively. Con 
ceive 8 expanded in a series of Laplace s functions S Q + 8 X -f . . ., then 
it is easily proved that 



AT THE SURFACE OF THE EARTH. 153 



(28), 



r being the distance of the point considered from the centre. 
These equations give 



(29)< 



i v * 

= 4-TrZ 



7 " *" * O ~t 

dr 2i -f 1 \u,/ j 

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 s, and let s 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 #, 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 2-TrS*. The mean of the values of dV /dr and dV"/dr 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 
lying adjacent to the points where the attraction is considered. 
Denoting this mean by dVJdr, we get, on putting r = a, 
dV S; 



This equation becomes by virtue of either of the equations (28) 

<^ = -f ............................ (30), 

dr 2a 

This result readily follows from equations (29), which give, on putting r=a, 



dr dr 

This difference 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 be hi each case 
2?r5, and will act outwards in 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 g" . Then gravity will be increased by g" , in 
consequence of the attraction of the general coating. 

20. But besides its direct effect, the attraction of the coating 
will produce an indirect effect by altering the sea-level. Since the 
potential at any place is increased by V c 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 c , that is, we 
must ascend through a space VJg, or VJG nearly. In consequence 
of this ascent, gravity will be diminished by the quantity corre 
sponding to the height VJG, or h suppose. If we take account 
only of the alteration of the distance from the centre of the earth, 
this diminution will be equal to G . Zti/a, or 2 VJa, or 4$r" , and 
therefore the combined direct and indirect effects of the general 
coating will be to diminish gravity by 3</ . 

But the attraction of that portion of the stratum whose thick 
ness is h , which lies immediately about the station considered, 
will be a quantity which involves li as a factor, and to include this 
attraction we must correct for the change of distance h 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 3#", but only 4 (1 - 3o-/4p) 
g" g" , or kg" suppose. If cr : p :: 5 : 11, we have &=16 4 
nearly. 

If we cared to leave the mean value of gravity unaltered, we 
should have to use, instead of 8, its -excess over its mean value S . 
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 
3</ 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 difference 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, 
B = ah + (o- 1) h . In finding the difference V of the potentials 
of the coating, it will be amply sufficient to regard the attracting 
matter as spread over a plane disk, with a radius s equal to 1000 
miles. On this supposition we get 



Now G = ^ TTpa, and therefore 

3F 9Ss 9 o7n- (o- -!)/ s 



- - . - . 

4 pa a 

Making the same suppositions as before with regard to the 
numerical values of a, p^h, h , and a, we get 3j" = 000147 G. 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 c at any station is expressed by a double 
integral, which is known if 8 be known, and which may be cal 
culated numerically with sufficient accuracy by dividing the 
surface into small portions and performing a summation. Theo 
retically speaking, V c 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 c could be obtained by substituting in the series the 

* The effect of tlie irregularity of the earth s surface is greater than what is 
represented by kg", for a reason which will be explained further on (Art. 25). 



156 ON THE VARIATION OF GllAVITY 

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 effect of the actual 
distribution of land and sea in its broad features. It seems 
better to form directly the expression for V c at any station. This 
expression may be calculated numerically for each station by 
using the value of 8 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 Sg" or Jcg" from one place to another. 

Let the surface be referred to polar co-ordinates originating at 
the centre, and let the angles ty, ^ be with reference to the station 
considered what 0, < were with reference to the north pole. The 
mass of a superficial element is equal to 8a 2 sin tyd-frd^, and its 
distance from the station is 2a sin ^. Hence we have 

V =afJScosWd1rdx (31) 

Let 8 m be the mean value of S throughout a circle with an 
angular radius ty, then the part of V c which is due to an annul us 
having a given infinitely small angular breadth dty is proportional 
to S m cos J-vJr, or to S m nearly when ^ is not large. If we regard 
the depth of the sea as uniform, we may suppose 8 = 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|- miles deep, and of the same radius, surround 
ing another, we get a difference of about i x 1*64 x 6 35, or 3J 
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 3J miles thick and an equal stratum of 
water. 



AT THE SURFACE OF THE EARTH. 157 

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 scrutinizing 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 ; Marian 
Islands + 6 8; Sandwich Islands + 5 2; Pulo Gaunsah Lout (a 
small island near New Guinea and almost on the equator), + 5 0. 
The largest negative errors are, California 6 ; Maranham 
5*6 ; Trinidad 5 2. 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 e, the observations will be best 
satisfied by a value of e which is too great. This is in fact pre- 

* Encyclopedia Metropolitana. Art. Figure of the Earth. 



158 ON THE VARIATION OF GRAVITY 

cisely the result of the discussion, the value of e which Mr Airy 
has obtained from the pendulum experiments ( 003535) 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 2 X cos 2 X, 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 2 X," 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 effect the result. Thus for 



AT THE SURFACE OF THE EARTH. 159 

example, Spitsbergen + 4 3, Hammerfest 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 O O, Fomentera + 2. Again, Padua + 07, Milan 
- 2 8. Again, Jamaica - O S, Trinidad - 5 2. 

24. Conceive the correction kg" calculated, and suppose it 
applied, as well as the correction g t to observed gravity reduced 
to the level of the sea, or to g, and let the result be g lt Let e /y be 
the ellipticity which would be determined by means of g n , e /t + Ae /y 
the true ellipticity. Since g tt = g g -f kg", and therefore 
9 = 9, l + 9 -ty"> we g^ by (27) 

Ki-sm 2 (/-/)} ............ (32). 

Now g = 27ro-7z = 27rS = 27r28< ; and we get from (30) and (28) 



dr 2a 

All the terms 8 4 will disappear from the second side of (32) except 
S 2 , and we therefore get 



Hence the correction Ae /y is less than that considered in Art. 18, in 
the ratio of 5 k 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 off 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 n 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- 



160 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 off, 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 g" 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 or, 
having a height from the bottom upwards which is to the depth 
of the sea as 1 to a. 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/cr. 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 x 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 as 
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 x the potential of the stratum removed. The removal of this 
stratum will depress the surface of equilibrium by the space 
G~ 1 V X : and the condition to be satisfied is, that this new 



AT THE SURFACE OF THE EARTH. 10 1 

surface of equilibrium, or else a surface of equilibrium belong 
ing to the same system, and therefore derived from the former 
by further diminishing the radius vector by the small quantity 
c t shall coincide with the actual surface. We must therefore 
have 

G- l V t +c=x-h ........................ (33). 

Let h and x be expanded in series of Laplace s functions 
/? -f 7^4- ... and ac Q + ac l + ... Then the value of V x at the sur 
face will be obtained from either of equations (28) by replacing 8 
by ax and putting r = a. We have therefore 



After substituting in (33) the preceding expressions for V x , h, 
and x, we must equate to zero Laplace s functions of the same 
order. The condition that X Q = may be satisfied by means of the 
constant c, and we shall have 



which gives, on replacing G 1 . ^TTO-CL by its equivalent 



We see that for terms of a high order # 4 is very nearly equal 
to h^ but for terms of a low order, whereby the distribution of land 
and sea would be expressed as to its broad features, a\ is sensibly 
greater than h im 

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 %g", 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 

& 

2i + 1 

and the required correction will evidently be obtained by replacing 
8 by ex. Putting for x i its value got from (35) we have 

, , v (2i -2}p , ^ f , S/a-So- 

g 3^r = 27TC72 r hi = 27T<rS (I r-^ ; 

S. II. 



162 ON THE VARIATION OF GRAVITY 

which gives, since 27r<r2^ = 27ro-A = g and G = 
_ 3<r 3/3 - So- Ji i 



If we put <r = 2J, p = 5J, a = 4000, and suppose A expressed 
in miles, we get 



.21 iA 4 +...) ...... (37). 

Had we treated the approximate correction Sg" in the same 
manner we should have had 



.429/i 3 + .333/i 4 



whereas, since k 3 (1 cr/p), we get 



kg" = S== (? x .00017 x 
(2z + 1) p 



...) ...... (38). 

The general expressions for 3/", 3g", and %" shew that the 
approximate correction kg" agrees with the true correction Sg " 
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, %g" is better represented 
by 3g" than by kg". This agrees with what has been already 
remarked in Art. 25. 

If we put g g -f ^g" g llfl and suppose G and e determined 
by means of g llfl small corrections similar to those already investi 
gated will have to be applied in consequence of the omission of the 
quantity g %g" in the value of g. The correction to 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 Q . 

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 
3g "=kg" + G x . 00017 x 
(- 6.1367/ + .455^ + .1237*,+ .056A. 3 + .032/* 4 + ...) (30), 



AT THE SURFACE OF THE EARTH. 103 

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 ln would apply to the earth so altered. 

28. Let us return to the quantity V c of Art. 19, and consider 
how the attraction of the earth s irregular coating affects the 
direction of the vertical. Let I be the latitude of the station, 
which for the sake of clear ideas may be supposed to be situated 
in the northern hemisphere, -GJ- its longitude west of a given p^ace, 
f the displacement of the zenith towards the south produced by 
the attraction of the coating, 77 its displacement towards the east. 
Then 

\_ dV, = secZ dV c 
* ~ Ga di V ~ Ga dv 

because a" 1 dVJdl and sec I . a- 1 dV e /d-& are the horizontal compo 
nents of the attraction towards the north and towards the west 
respectively, and G 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 north azimuth of the elementary ma=s 
So? sin -^rd^rd^. 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 % . dl, and therefore its linear distance, 
which was a chord ^r, or 2a sin J-^r, becomes 

2a sin |\/r a cos J>|r cos ^ . dl. 
Hence the reciprocal of the linear distance is increased by 

l/4a . cos IT/T cosec 2 Ji|r cos x . dl, 
and therefore the part of V c due to this element is increased by 

JSa cos 2 ^ cosec ^ty cos 
Hence we have 

dr. 



Bd d 

dl "2 Jj sin ^ 

112 



164 OX THE VARIATION OF GRAVITY 

Although the quantity under the integral sign in this expres 
sion becomes infinite when ^r vanishes, the integral itself has a 
finite value, at least if we suppose 8 to vary continuously in the 
immediate neighbourhood .of the station. For if 8 becomes 8 
when % becomes % + TT, we may replace 8 under the integral sign 
by 8 8 , and integrate from ^ = to % = TT, instead of inte 
grating from % = to % = 2-zr, and the limiting value of 
(8 8 ) / sin J-^r when -ty vanishes is AdS/dty, which is finite. 

To get the easterly displacement of the zenith, we have only to 
measure ^ from the west instead of from the north, or, which 
comes to the same, to write % + JTT for ^, and continue to measure 
^ from the north. We get 

sec I , c = ~ o//cos a \ty cosec J-vJr sin^. d^d% ...(41). 

20. The expressions (40) and (41) are not to be applied to 
points very near the station if 8 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 8 
vary gradually in the neighbourhood of the station, the expression 
(40) or (41) may be 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 r /o 2 r cos2 i^ cosec i^ cos X@~~ ^i) d^dfe 

where 8 t denotes the value of 8 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 cliff, 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 </ ", 
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 
g 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-v/r = ^ = s t 2a t if s 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 lf s s , s.,... and from sea 
to land at the intermediate distances s. 2 , s 4 ... On going in the 
opposite direction suppose that we pass from land to sea at the 
distances s_^ s_ 3 , s_., ... and from sea to land at the distances 
s_ 2 , s_ 4 Then we get from (40), 
dV 

_ = aS /(log S l -log 5_ t - (log * 2 - log S_ 2 ) + log S 3 - log S_ 3 

-...}cosx-<*X 

If the station be near the coast, one of the terms log^, log.9_ t 
will be large, and the zenith will be sensibly displaced towards the 



1G6 ON THE VARIATION OF GRAVITY 

sea by the irregular attraction. On account of the 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 8. 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 between 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 series 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 terms 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 the radius vector be 
calculated by Clairaut s Theorem. Let cj c be calculated gravity, 



AT THE SURFACE OF THE EARTH. 167 



r c the calculated radius vector, and put g = g c + A#, r = r c + a AM. 
Suppose A# and AM 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 A# may be got rid of by 
using a different value of G. No constant term need be taken in 
the expansion of AM, 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 AM will be determined from those in the expansion of A# by 
means of equations (8) and (12). 

Let A? = (v a + v 8 + t; 4 +...) ............... (42), 

and we shall have 

Att = v a + Jv 8 + Jv 4 + ..................... (43). 

Suppose A^ = GF (0, </>). Let -^ be the angle between the 
directions determined by the angular co-ordinates 0, (f> and & , < , 
Let (1 -2fcos^+ f 2 )* be denoted by R, and let Q t be the coef 
ficient of f* in the expansion of .ZT 1 in a series according to ascend 
ing powers of f. Then 

) Q |S in ffdffdtf, 



and therefore if f be supposed to be less than 1, and to become 1 
in the limit, we shall have 4nrAu = limit of 



f" j"F(ff, f)(5 

J o J o 



Now assume 



and we shall have 



whence we get, putting Z for -ZT 1 - Q a - ?Q, , y = 2/f " f d . $ 



168 ON THE VARIATION OF GRAVITY 

Integrating by parts, arid observing that 7 vanishes with f, we get 

7 = 2r : ^+ 3jfr-iza?. 

The last integral may be obtained by rationaliz:tion. If we 
assume R = w % , and observe that Q = 1, Q t = cos f , and that 
w = 1 when f vanishes, we shall find 



i 

.log ~ - 
1 



Whenf=lwehaveZ=(2 
and 



= - 2 sin if (1 - sin if) - cos f log [sin if (1 + sin 
Putting/ (f ) for the value of 7 when f = 1, we have 
/(f ) = cosec -|f -f 1 6 sin Jf 

5 COST/T 3 cos flog (sin -|f (1 + sin Jf)j ......... (45). 

In the expression for AM, we may suppose the line from which 
& is measured to be the radius vector of the station considered. 
We thus get, on replacing F(6 t <j> ) by G~*kg, and employing the 
notation of Art. 22, 

A = 4~Qj:fi*9-fW sm^d^d x ............ (46). 

32. Let A$r = g + A tjr. Then A ^r 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 # of which A^r 
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 different 
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 (/ respectively, so as to get the part of Aw 
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 # 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 */. 

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 EARTH. 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 could not be performed infinite terms, and 
it would be necessary in the first instance to tabulate, once for all, 
an integral of the form J7/( cos ^r) d% for values of t/r, which need 
not be numerous, from to TT. This table being made, the tabu 
lated function would take the place of f($) in (46), and the rest 
of the process would be of the same degree of difficulty as the 
quadratures expressed by the equations (31) and (46). 

34. Suppose A?* 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 AM has been calculated, find a 
number of points where Au has one of the constant values 2/3, 
/3, 0, P, 2/3 . . ., mark these points on a map of the world, and join 
by a curve those which belong to the same value of AM. 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 sufficiently 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 V c has a constant value, contour lines would be formed 



170 ON THE VARIATION OF GRAVITY 

representing 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 V x 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 II 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 V a potential V 
which satisfies equation (3) outside the earth, vanishes at an 
infinite distance, and is equal to H at the surface. Now if we 
suppose V to have the value // 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 H 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 V is to satisfy, we are enabled 
to find V just as V 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 Report of the British Association for 1849. Part n. 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 different power in different 
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 Koyal 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 sufficient 
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 






ON A MODE OF MEASURING, &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 complets 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-difference. 

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 each 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. Kefer 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 E be the retardation, measured by equivalent space in 
air, and p be the index of refraction 



The effect 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 



as above enunciated. If p be the power of the astigmatic lens, 



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 = %pp* cos 2 (0 - a) + i/p 2 cos 2 (6 - a ), 

which is the same as would be given by a single astigmatic lens of 
power PI at an azimuth a 1 , provided 

pp* cos 2 (6 - a) +p p* cos 2(0- a ) =p lP * cos 2(0- aj, 
which will be satisfied for all values of 6 provided 
p cos 2a + p cos 2/ =p l cos 2^, 
p sin 2z 4- p sin 2a = j^ sin 2^. 

These two equations geometrically interpreted give the propo 
sition enunciated above for the combination of astigmatic lenses.] 



[From the Report of the British Association for 1849. Part n. p. 11.] 



ON THE DETERMINATION OF THE WAVE LENGTH CORRESPONDING 

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//, of the refractive index in 
passing from one point of the spectrum to another is nearly propor 
tional to the increment AX" 2 of the squared reciprocal of the wave 
length. Even in the case of flint glass, the substance visually 
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 /z, as a function, not of X, but of X~ 2 , 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 B\ 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" 2 instead of \ 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 
diffraction.] 



s. ii. 



12 



[From the Transactions of the Cambridge Philosophical Society, Yol. viu. 

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 differen 
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 effect 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 iri 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 abscissae and ordinates, the differential 
equation is reduced to one containing a single constant /?, 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 (3 
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 integrals. See Art. 7, and last 
paragraph but one in the paper.] 

122 



180 DISCUSSION OF A DIFFERENTIAL EQUATION 

body, so that a small value of ft 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 {3. When /3 is not greater than 
J, 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 infinite. 
"When fi is greater than J, 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 ft is a 
good deal greater than J, 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 
T v 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 effect of that inertia is to 
preserve the bridge, so that we cannot be far wrong in regarding 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 181 

jP t as a superior limit to the actual tendency to break. When /3 is 
very large, T 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 



with the particular conditions 

y=0, ^, = 0, when* = ..................... (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 f , 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, Fits velocity on entering the bridge, 2c the length of the 
bridge, g the force of gravity, S the deflection produced by the 



182 DISCUSSION OF A DIFFERENTIAL EQUATION 

body placed at rest on the centre of the bridge, R 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 stiff 
ness, the depth to which a weight will sink when placed in succes 
sion at different points of the bridge will vary as the weight 
multiplied by (2cx - x 2 ) 2 , 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 R will be the same as if R 
were a weight resting at a distance x from the extremity of the 
bridge ; and we shall therefore have 

y = CR (2cx - x*f, 

C being a constant, which may be determined by observing that 
we must have y = S when R = Mg and x = c; whence 



MgJ 
We get therefore for the equation of motion of the body 



dx 
which becomes on observing that -5- = V 



which is the same as equation (1), a and b being defined by the 
equations 



2. To simplify equation (1) put 

x = 2c#, y = IGc afc- y, b = 
which gives 

tfy _ Q Py / 4) 

3? * ^? 1 ............. 



RELATING TO THE BREAKING 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 4 a6" x or 16$; and & on the value of which 
depends the form of the body s path, is a constant defined by the 
equation 

gc * 



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 



of which the complete integral is 

and (7) is the approximate integral of (4) for very small values of 
x. Now the second of equations (2) requires that A = 0, B = , 
so 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 # 2 ) 2 z t we get 

Since (4) gives y=(x x*)* when /3 = oo , and (5) gives = oo 
when V = 0, it follows that z is the ratio of the depression of the 
body to the equilibrium depression. It appears 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>|, the last two terms in (7) take the form x* {Ccos (qlogx) 
+ D sin (q log x) } ; and if y l denote this quantity we cannot in strictness speak of 
the limiting value of dyjdx when x = 0. If we give x a small positive value, which 
we then suppose to decrease indefinitely, dyjdx will fluctuate between the constantly 
increasing limits .r~f v /{ (|(7 + qD)* + (^D -qC)-}, or ori N /{/3(C 2 + D 2 )}, since 
q =V(j3- ). 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 2 + D 2 = 0, and therefore C=0, D=0. When /3 = the last two 
terms in (7) take the form x* (E + Flogx], and we must evidently have E = 0, F.O. 



184 DISCUSSION OF A DIFFERENTIAL EQUATION 

To integrate (8) assume then 



(9), 
and we get 

2 (i + 2) (i + 1) Atf - 22 (i + 3) (* + 2) Atf" 

+ 2(i+ 4) (i + 3) A#F* + plAtf = , 
or 

2 {[(i + 1) (f + 2) + /3] 4, - 2 (i + 1) (i + 2) 4 w 

+ (i + l)(i + 2)4 ( Jo ; = y3 ......... (10) 

where it is to be observed that no coefficients Ai with negative 
suffixes are to be taken. 

Equating to zero the coefficients of the powers 0, 1, 2... of a; in 

(10), we get 



(6 +/3) ^-12^=0, &c. 
and generally 

{( + !) (i + 2)+/3}A i -Z (i + 1) (i + 2) A M 

+ (i + l)(i+2)^ ( _ 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 = GO , if we conventionally regard A_^ as 
equal to zero, determines the constants A I} A z , A 3 ... 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 

(12); 



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 
A^ + A A.^ will consist of only a few figures. 

5. When i becomes indefinitely great, it follows from (11) 
or (12) that the relation between the coefficients A i is given by 
the equation 

4-S4..+4..-0 .................... (is), 

of which the integral is 

A t = C + (7f .......................... (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 iih is ultimately equal to x. Hence the series is 
convergent when x lies between the limits 1 and -f 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 I} A 2 ... 
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 2 J,_ 2 = 0, or, which is the same, AM. = 0. 
When i is large, A 2 ^ 4 will be small ; in fact, on substituting in the 
small term of (12) the value of A t given by (14), we see that 
Al4j is of the order i~ l . Hence A 3 ^, AM f ... will be of the orders 
2~ 2 , i~ 3 ..., so that the successive differences of A i will rapidly de 
crease. Suppose i 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 



and taking the sum between the limits i and oo we get 

(15); 



. I 

1 - x \1 - x 

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 series (9) for the particular value \ 
of x, which will be found necessary in order to determine a certain 
arbitrary constant*. 

* A mode of calculating the value of z for x=\ will presently be given, which is 
easier than that here mentioned, unless ,3 be very large. See equation (42) at the 
end of this paper. 



186 DISCUSSION OF A DIFFERENTIAL EQUATION 

7. If the constant term in equation (4) be omitted, the equa 
tion reduces itself to 



The form of this equation suggests that there may be an inte 
gral of the form y = x m (1 x) n . Assuming this expression for trial, 
we get 



i) (m+n 1)# 2 ). 
The second member of this equation will be proportional to y, if 

m + n-l = (17), 

and will be moreover equal to /3y, if 

m 2 -m + /3 = (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) 

The complete integral of (4) may now be obtained by replacing 
the constants A, B by functions R, S of x, and employing the 
method of the variation of parameters. Putting for shortness 

x m (1 - x} n = u, x n (1 - x) m = v, 
we get to determine R and 8 the equations 

dR dS 
u -=- + v -T- = 0, 
ax dx 

du dR dv dS _ ~ 
dx dx dx dx 

Since v-j u-j- = m n, we get from the above equations 

CLOG CLOO 

dR = /3v diS = j3u 

dx m n dx m n y 

whence we obtain for a particular integral of (4) 



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 x*, when # is very small, and 
therefore, as shewn in ^.rt. 3, we must have A = 0, B = 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 ft. 

8. Since [* F(x)dx=\ F (a?) dx - f * F (1 - x) dx, we have 

J Jo J 

from (20) 



in n 



x m (l-x) n x n (\-xfdx-x n (l-x) m x m (l-x) n dx] 



+j^-{x n (i-x) m f x (i-x} m x"dx-x(i-xY p "(i-*)wfcj. 

If we put f(x) for the second member of equation (20), the 
equation just written is equivalent to 

/(*)=/(! -*) + (*) ..................... (21), 

where 



Now since m + n = 1, 



At the limits x = and x = 1, we have w = x and w = 1, s = x and 
5 = 0, whence if / denote the definite integral, 



We get by integration by parts 

s m ds s m m 



188 DISCUSSION OF A DIFFERENTIAL EQUATION 

and again by a formula of reduction 



Now /3 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 J. In either case the expressions which 
are free from the integral sign vanish at the limits 5 = and s = co , 
and we have therefore, on replacing m (1 m) by its value /3, 

T _/3 ["s^ds 
~2j 1 + .9 

The function (/> (#) will have different forms according as the 
roots of (18) are real or imaginary. First suppose the roots real, 
and let m = J + r, n = J r, so that 



* = i- (23). 

In this case m is a real quantity lying between and 1, and we 
have therefore by a known formula 



f ? = . "* = -?L-... ..(24) 

o 1 +s siu ?n7r COS^ TT 



whence we get from (22), observing that the two definite integrals 
in this equation are equal to each other, 

.1 (25). 



r COS 9 TT (\ - XJ \ X 

This result might have been obtained somewhat more readily 
by means of the properties of the first and second Eulerian inte 
grals. 

When /3 becomes equal to J, r vanishes, the expression for 
$ (x) takes the form J, and we easily find 



(26). 



When /3 > J, the roots of (18) become imaginary, and r becomes 
p V 1, where 

P = V/3~i ........................... (27). 

The formula (25) becomes 



/o (e*" 7 + e-^j V 1 - 



...... (28). 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 189 

If f (x} be calculated from x = to x = J, equation (21) will 
enable us to calculate it readily from x = ^ to x = 1, since it is easy 
to calculate $>(%) 

9. A series of a simple form, which is more rapidly convergent 
than (9) when x approaches the value -J, may readily be investi 
ated. 



Let x = ^(l + w}\ then substituting in equation (8) we get 

l^{(l-V }+^=/3 .................. (29). 

Assume 

z = B +B i w* + B. 2 w*...=2B i w* i ............... (30), 

then substituting in (29) we get 

2,{2i (2t - 1) w *-* - 2 (2i -f 2) (2i + 1) w" 

+ (2i + 4) (2i + 3) U7 8H * + 4/3^ } = 4/3, 
or, 

2 {i (2i - 1) B ( - 2 [i(2i - 1) - /3] 5 M + 1(2 - 1) ^. 2 } i^ 2 = 2/3. 

This equation leaves B Q arbitrary, and gives on dividing by 
i(2i 1), and putting in succession i= 1, i = 2, &c., 



(31), 



and generally when z > 1, 

B t = B l _ i + B l _,-^?- T} B i _ l ............... (32). 

The constants B I} B^... 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 z= C w + C 1 w 3 + . . . when C it C 2 . . . will be determined 
in terms of C lt which remains arbitrary. The series will both be 
convergent between the limits w = 1 and w = l, that is, between 
the limits x = and x=\. The sum of the two series will be the 
complete integral of (29), and will be equal to (x-x^f i f(x} if the. 



190 DISCUSSION OF A DIFFERENTIAL EQUATION 

constants Z? , C be properly determined. Denoting the sums of 
the two series by F e (w), F Q (w) respectively, and writing a (x) for 

(x x 2 ) * f(x) } so that z = cr (x), we get 



and since ZF Q (w) = a (x) cr (1 x) = (x x*} 2 < (x) by (21), we get 

a.n-)-^wi-.Jto-^i^A(i) : ^ 

To determine B n we have 



which may be calculated by the series (9). 

10. The series (9), (30) will ultimately be geometric series 
with ratios #, w? 2 , or x, (2x I) 2 , respectively. Equating these 
ratios, and taking the smaller root of the resulting quadratic, we 
get x = J. Hence if we use the series (9) for the calculation of 
a (x) from x = to x \, and (30) for the calculation of cr (x) from 
x = i to x = J, we shall have to calculate series which are ulti 
mately geometric series with ratios ranging from to J. 

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 + (3)A = (3 and the general formula (12), 
calculate the coefficients A , A^ J 2 ,... as far as may be necessary. 
From the series (9), or else from the series (9) combined with the 
formula (15), calculate cr (J) or B (} , and then calculate B { , B 2 ... 
from equations (31), (32). Next calculate cr(x) from the series 
(9) for the values 02, 04,... 26 of or, and F e (w) from (30) for the 
values 04, 08..., "44 of w, and lastly (x x*f 2 </> (x) for the values 
52, 54..., 98 of x. Then we have a- (x) calculated directly from 
x = to x= 2G; equations (33) will give cr (x} from #= 28 to 
x = 72, and lastly the equation cr (x) = a (1 x) + (x x 2 f 2 $ (x) 
will give a (x) from x = 74 to x = 1. 

11. The equation (21) will enable us to express in finite terms 
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 Zcx and 16/%, and we have 
also d . 2cx/dt = V, whence, if v be the vertical velocity, we get 

d.lQSda; 8SV , 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 191 

But (21) gives f () = \ < (), whence if v c be the value of v at the 
centre, we get from (25) or (28) 

= 4,7rSVj3* S7TSVI3* 

V ~ C COS T7T ~C (6 pjr + -?") " -( I) 

according as ft < > J. 

In the extreme cases in which F is infinitely great and infinitely 
small respectively, it is evident that v c must vanish, and therefore 
for some intermediate value of F, v e must be a maximum. Since 
Foe f}-% when the same body is made to traverse the same bridge 
with different velocities, v c will be a maximum when p or q is a 
minimum, where 

p = 2~* cos TTT, = /3~* (e pw + e ^). 

Putting for cos TTT its expression in a continued product, and 
replacing r by its expression (23) in terms of j3, we get 



whence 



The same expression would have been obtained for dlogq/dfl. 
Call the second member of equation (36) F(f$), and let -3, P be 
the negative and positive parts respectively of F (P). When {3 = 0, 

N= oo , and P = - - + x ... = 1, and therefore P(/3) is nega- 
L . *2t 25 o 

tive. When /3 becomes infinite, the ratio of P to JV^ becomes 
infinite, and therefore F (ft) is positive when {$ is sufficiently large; 
and F (j3) alters continuously with /3. Hence the equation F (/3) = 
must have at least one positive root. But it cannot have more 
than one; for the rates of proportionate decrease of the quantities 
-tV, P, or - 1/JV. dXjdfr - 1/P . dPflfr are respectively 



and the several terms of the denominator of the second of these 
expressions are equal to those of the numerator multiplied by 
1 . 2 + @, 2 . 3 + /3,... respectively, and therefore the denominator is 



192 DISCUSSION OF A DIFFERENTIAL EQUATION 

equal to the numerator multiplied by a quantity greater than 
2 -f- j3, and therefore greater than /3; so that the value of the 
expression is less than I//3. Hence for a given infinitely small 
increment of /3 the change dN in N bears to N 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 
v c admits of but one maximum or minimum value, and this must 
evidently be a maximum. 

When /3 =4, N=2, andP< -^ + ^ + ... or < 1, and there 
fore F (/3) has the same sign as when @ is indefinitely small. 
Hence it is q and not p which becomes a minimum. Equating 
dq/d/3 to zero, employing (27), and putting %7rp = log e f, we find 

The real positive root of this equation will be found by trial to 
be 36-3 nearly, which gives p = 5717, /3 = J + p z = 5768. If V l be 
the velocity which gives v c a maximum, v t the maximum value of 
v c , 27 the velocity due to the height S, we get 

/7c 2 C U S7T/3 2 S jr 

F -V ira-SJ5 and v > = F+F* c F " whence 

F=-465o|?7, v =-G288tf. 

A .VI 



12. Conceive a weight TF placed at rest on a point of the 
bridge whose distance from the first extremity is to the whole 
length as x to 1. The reaction at this extremity produced by W 
will be equal to (1 x) W, and the moment of this reaction about 
a point of the bridge whose abscissa 2c^ is less than 2cx will be 
2c (1 x) x^W. This moment measures the tendency of the bridge 
to break at the point considered, and it is evidently greatest when 
x l = x, 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 (x # 2 )~ 2 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) x x (x # 2 )" 2 y, or to (x x*) 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 (x - x 2 ) z, and 1 = C (J - J), 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 t the actual 
depression. When z has been calculated in the way explained in 
Art. 10, T will be obtained by multiplying by 4 (x # 2 ), and then 
y /S will be got by multiplying T by 4 (x x 9 ). 

13. The following Table gives the values of these three quan 
tities for each of four values of /3, namely -fa, ^, , and {, to which 
correspond r = J, ?* = 0, p = J, p = 1, respectively. In performing 
the calculations I have retained five decimal places in calculating 
the coefficients A , A lt A 2 ... and B , B lt B a ... and four in calcula 
ting the series (9) and (30). In calculating c/> (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 /3, z increases from a small positive quantity 
up to GO as x increases from to 1. As far as the table goes, z is 
decidedly greater for the second of the two values of ft than for 
the first. It is easily proved however that before x attains the 
value 1, z becomes greater for the first value of ft than for the 
second. For if we suppose x very little less than !,/(! x) will 
be extremely small compared with < (a?), or, in case $ (x} contain a 
sine, compared with the coefficient of the sine. Writing x v for 

s. n. 13 



194 DISCUSSION OF A DIFFERENTIAL EQUATION 



00. 




03. 



L~-C^O5 
Oi Ii I 



Oi-HCOt~OlOrHC^-COT ICiOOt^OJOOOOlO OOOl-OO 
OOOOi 1 i IC<l(NCO-^^tiOOt^OOOiOOOOOCCO 



3 



03. 



! I 



t ICO-^HOCOCOOO 



03. 



i 1 T 1 (M CO xJH CO 



OOi |>OO>Oi I 



8O 
<M 



O>OCOT-lO 



03. 



03. 









8O O >O O >^ O >O O >O O >O O O O O O >O O CT -^ O 00 O 
O TH^HCqiMCOCO^^OOOOt-t-COCOOi01C5C5C50 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 195 

1 x, and retaining only the most important term in f(x), we get 
from (21), (25), (26), and (28) 



TT j 7T 



or 



= ^ - a?* sin (p log -} (37) 

p (* + -*) * V^ ^/ 



according as /3 < J, /3 = J, or > J ; and * will be obtained by 
dividing /(#) by x* nearly. Hence if \ > /3 2 > /3 X > 0, 2 is ultimately 
incomparably greater when /S = /3 t than when /3 = /3 2 , and when 
= #, than when = J. Since/(0) = .4 = /3 (2 + /3) 1 - (2/T 1 + I) 1 , 
/(O) increases with ft so that /(a?) is at first larger when @ = /3 a 
than when ft =j3 lt and afterwards smaller. 

When /3 > J, 2; vanishes for a certain value of x, after which it 
becomes negative, then vanishes again and becomes positive, and 
so on an infinite number of times. The same will be true of T. 
If p be small, f(x) 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, z will be sensibly equal to 1 except when x is nearly 
equal to 1, so that in this case also /(a;) will not vanish till x is 
nearly equal to 1. The table shews that when /3 = , f (x) first 
vanishes between x = -98 and x = 1, and when /3 = J between x = "94 
and x = 96. The first value of # for which f(x) vanishes is pro 
bably never much less than 1, because as /3 increases from | the 
denominator p(e p7r + e~ p ~) in the expression for < (#) becomes 
rapidly large. 

15. Since when /3>^, 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 /5 = ^ the table appears to indicate a maxi 
mum beyond x = 98. When $ = j, 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. 

1-32 



196 



DISCUSSION OF A DIFFERENTIAL EQUATION 



16. The following curves represent the trajectory of the body 
for the four values of /3 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 trajectory of the body in the ideal limit 
ing case in which {S= or V 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 j3 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*. C 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 Cyj(x - a? 2 ) 2 , where C depends on 
the length and stiffness of the bridge. Since 
this expression becomes negative with y, the 
preceding solution will not be applicable 
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 outside the full curve. 
The two curves touch each other at the point where they are cut by the line ACB, 
as is represented in the figure. 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 197 

tive. 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 AS. The preceding solution shews that 
when > 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 ft 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 ft < \ 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 , 77 
be the co-ordinates of any point of the bridge on the same scale on 
which #, 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 



When x becomes very nearly equal to 1, y varies ultimately as 
(1 a?)*"*", and therefore 77 contains terms involving (1 oi)~*~ r , and 
(dij/dx)*, and consequently (drj/dt)* contains terms involving 
(1 #)~ 3 ~ 2r . 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 fi 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 t 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 

Mk*^ 2 (}=Mgh- 7 
df\l) * 



whereas to determine the motion of a single particle whose mass is 

M Cy 



JJ/.we should have had 



Now h must be nearly equal to ^l, and /j 2 must be a little greater 
than J 2 , say equal to -|- 2 , so that the two equations are very nearly 
the same. 

Hence, /? being the quantity defined by equation (5), where S 
denotes the central statical deflection due to a weight \Mg, 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 ft 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, ft would probably be large in 
practice. When ft is so large that the coefficient ffn/p (e p7r + e~ pir ), 
or 7r/3 i e~ r si 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 x = \. 
For this value of x we have 4 (x x*) = 1, and therefore z = T = y. 
Putting C i for the (i + l) th term of the series (9), so that C i = ApT\ 
we have for x = \ 

T=G a + C l + C, + (39) 

/3 GO 



and generally, 



2) + /3 



whence Tis easily calculated. Thus for /? = 5 we have 7rft% e~ ffft = 031 
nearly, which is not large, and we get from the series (39) T= 1 27 
nearly. For ft = 10, the approximate value of the coefficient in 
<p (x) is 0048, which is very small, and we get T 1*14. In these 
calculations the inertia 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 M, the mass of the body, JJ/ , the mass of the bridge, 2c, 
the length of the bridge, S, the central statical deflection, V, the 



200 DISCUSSION OF A DIFFERENTIAL EQUATION 

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 M, M , &c., with respect 
to each of these units are given in the following table. 

Unit of length. Unit of time. Unit of mass. 
ifandJf . 001 

c and S. 1 

V. 1-1 

g. 1-20 

Now any result whatsoever concerning the problem will consist 
of a relation between certain unknown quantities x , x" ... and the 
six constants just written, a relation which may be expressed by 

/> , x", ...M, 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 M S V*\_ 

- o?y- M> ~c ) 

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 f 
of T in terms of the six constants, we shall have but one unknown 
quantity x , where x =T^ and we may take for (a), Meg, or else 
M V 2 . 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 different scale, we must alter the quantities 
M, M , &c. in such a manner that 

JlToclf, tfoc c, F 2 occ#, 

and consequently, since g is not a quantity which we can alter at 
pleasure in our experiments, V must vary as \fc. A small system 
constructed with attention to the above variations forms an exact 
dynamical model of a larger system with respect to which it may 



J* 



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 -jL-th of an inch ; and then we must give to the 
carriage the successive velocities of 3\/10, 4/v/lO, 5\/10, or 9*49, 
12-65, 15-81 miles per hour, or 13 91, 18-55, 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 /9 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 /3. 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 x = to x = 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 x 5, in which case the series are least con 
vergent, have be^n 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 



PQ 
<1 
H 



QX 



03. 



02. 



OCL 



O CO rH CO GO 1C <>1 GO CO C^l CO t- O >C GO O ^ l>- O C3 
OOrHCMCOlCt>-OOOrHrHrHT ICSt^OCprHrHO 
rH rH rH rH rH 



CMOOa5 
OOrH 



T ICO-^fOt-COO 



COOOC5Ct^CO>CCOCOOO 
Or IrHi li 1 O CS t^ C CO (M 







CirHOOCSOCOGOCiOOQOrHCOrHCOO^Ot-CO 
O CO 1C b- GO rH OJ CO -^ UC O >C >C CO Ol rH O 
C5C5C5CiCiOOOOOOO 



CC^OCrHCO +I^OOCOC^Ot^COC^OC^-^l 
Ot~rHU3CS^ltr i (ICCOOOtOOCOOOOO 
OGOCiO5Cn)OOrHT-lrHC<l(Mi-lrHOOO?OOOOO 




O C O O O >C O >C O >C O >C O >O O C O >C O >C 
OOrHrH(M(MCOCO^^xC>C ^COt^t^GOCOOiC5 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 203 

expression, for (x x?)~*<f> (#). The values of T and y/S were 
deduced from those of z, by merely multiplying twice in succes 
sion by 4# (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 z = 1 at 
this point, we get from the part of the table headed " 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 ft 
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, B, where however the depression itself is insensible. The 
greatest depression of the body, as appears from the column which 
gives y 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 
T 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 
= 3, 1-33 for {3 = 5, 119 for = 8, I ll for (3 = 12, and 1-06 for 
= 20. 

When /3 is equal to or greater than 8, the maximum value 
of T occurs so nearly when x = *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 

(42). 



. 2 + 2734- 3 . 



When ft 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 ft 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 s will be 1 and oo , and the definite integral on which 
the result depends will be 



1+5 

The formula (42) may be obtained by expanding the denomi 
nator, integrating, and expressing m in terms of ft. 

In practice the values of ft are very large, and it will be con 
venient to expand according to inverse powers of ft. This may be 
easily effected by successive substitutions. Putting for shortness 
x s? = X, equation (4) becomes by a slight transformation 



and we have for a first approximation y = X 2 , for a second 



and so on. The result of the successive substitutions may be ex 
pressed as follows : 



where each term, taken positively, is derived from the preceding by 
differentiating twice, and then multiplying by ft~ l X z . 

For such large values of ft, we need attend to nothing but the 
value of z for x = , and this may be obtained from (43) by putting 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 205 

x = J, after differentiation, and multiplying by 16. It will how 
ever be more convenient to replace x by \ (1 -f- w), which gives 
tf /da? = 4 . d*/dw* ; X 2 = T V W, where W = (1 - w 2 ) 2 . We thus get 
from (43) 



where we must put w = after differentiation, if we wish to get 
the value of z for x \. This equation gives, on performing the 
differentiations and multiplications, and then putting w = 0, 

2 = l+/3- 1 + |/3- 2 +13/3- 3 + ............... (44). 

In practical cases this series may be reduced to 1 + /3" 1 . 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 



200 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 

O 

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 I 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 affecting 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 J/ be the mass of the weight, including the arm, k the 
radius of gyration of the whole about the fulcrum, h 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, x, y 
the co-ordinates of that point at the time t, f, 77 those of any 
element of the bridge, R the reaction of the bridge against the 
weight, M f the mass of the bridge, R , 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 defection to be very small, and neglect its square. 

By D Alembert s principle the effective 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 . d% very nearly ; the horizontal effective force 
of this element will be insensible, and the vertical effective force 
will be M 1 /2c . d^jdf . dg, 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 AQ of the bridge we get, 
taking moments round Q, 



208 DISCUSSION OF A DIFFERENTIAL EQUATION 

V being the same function of f that 77 is of f. To determine K, 
let $ be the central statical deflection produced by the weight My 
resting partly on the bridge and partly on the fulcrum, which is 
equivalent to a weight h/l . My resting on the centre of the bridge. 
In this case we should have 

d*ri _ _ Mgh 
~d%~ 21 * 

Integrating this equation twice, and observing that dq/d**Q 
when f = c, and 77 = when f = 0, and that S is the value of 77 
when f = c, we get 

K- QIS ..(46). 

" Mghc 3 " 

Returning now to the bridge in its actual state, we get to de 
termine R r , by taking moments about B, 

K. 2c-B(2c -* + * 2c-!; ) d? = ...... (47). 



Eliminating R between (45) and (47), putting for A its value 
given by (46), and eliminating t by the equation dx/dt = V, we get 

d r 



(48)> 



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 as. 

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, 



We have also the equations of condition, 
77 = when x = 0, for any value of f from to 2c ; 
rj = y when f = oc t for any value of x from to 2c ; >. ..(50). 

77 = when f = . or = 2c ; y = Q and dy/doc = when a? = O 



RELATING TO THE BREAKING OF 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 rj and drj/dt; 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 77 oc y, 
% oc x x c, and 

y ISR lSM V*y ,, 78T7S y ,, tl D72 
- <* irr-n. : ,, 7 / , MtfV* ^ oc Mghl oc Rl*. 
c* Mghc 2 Mghc* c 2 

These variations give, on eliminating the variation of R, 

c z k* M P 



,_,>< 

(ol) 



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 JJ/7 2 is the same, and 
we conceive the travelling weights to move over the two bridges 
respectively, with velocities ranging from to oo , the trajectories 
described in the one case, and the deflections of the bridge, corre 
spond exactly to the trajectories and deflections in the other case, 
so that to pass from one to the other, it will be sufficient 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 corresponds 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 
h = k = l, which gives 

yxS, F 2 Soc#c 2 , JfocJf ..................... (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 @. When S oc c we get F 2 oc gc, which agrees with 
Art. 22. 

S. n. 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 #, and let y be the deflection produced at that 
point. Then, f , 77 being the co-ordinates of any point of the bridge, 
we get from (38) 



4 (1 - x) 

To obtain f x l r)dt;, we have only got to write 1 x in place 
of x. Adding together the results, and observing that, according 
to a formula referred to in Art. 1, y 16 S . F/Mg . x 2 (1 x)*, we 
obtain 

xY} ............ (53); 



and this integral expresses the mean deflection produced by the 
force F, since 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 

- g = K !i F - ft (| - f ) Fd?} = J KF (f - p ). 

Integrating this equation twice, and observing that drj/dj; = 
when =i, and 77 = when f = 0, and that (46) gives, on putting 
I = h and c = i K = 4<SS/Mg, we obtain 

f-Sf + F) ..................... (54). 



This equation gives for the mean deflection 

/ww . 

(oo); 



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 

tt = 5j?(l-3?) + 5ff 8 (l-a?) s ..................... (56). 

Putting fju for the mean deflection, expressing F 1 in terms of yu-, 
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 

^=5Mf(i-f) + ra-m ............... (57). 

If D be the central deflection, 77 = .D when f = 1 ; so that 
D : p :: 25 : 16. 

Now suppose the bridge in motion, with the mass M travelling 
over it, and let x, y be the co-ordinates of M. As before, the 
bridge would be in equilibrium under the action of the force 
M(g d z y/d?) acting vertically downwards at the point whose 
abscissa is x, and the system of forces such as l r d.cPi)/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 u, 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 l be the whole uniformly distributed force we have 



142 



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 fju be the mean deflection at the 
time t, (57) will be the equation to the bridge at that instant. 
Moreover, since the point (#, y) is a point in the bridge, we must 
have ?; = y when = x, whence y = JJLU. We have also 

_ 155 



We get from (55), F i = oMg/ji/2S. Making these various sub 
stitutions in (58), and replacing d/dt by V.d/dx, we get for the 
differential equation of motion 



155 ,,, Tr2 /KnN 

-MV* ...... (o9). 



Since //- is comparable with S, the several terms of this equa 
tion are comparable with 

%, Mg, MV 2 S, M V Z S, 

respectively. If then V 2 S be small compared with g, and likewise 
M small compared with M t 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 2 S/g = 1/16/3, which 
will be small when ft 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 ft be large, secondly, that the mass 
of the travelling body be small compared with the mass of the 
bridge. If ft be large, but M be comparable with M f , 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. 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 213 

Neglecting the third term in equation (59), and putting for u 
its value, we get 



where 

_1008J//3 



The linear equation (60) is easily integrated. Integrating, and 
determining the arbitrary constants by the conditions that /JL = 0, 
and dp/dx 0, when x = 0, we get 



24 

( 62 ); 



and we have for the equation to the trajectory 

y = ov,(x-2x* + x*)=op(X+X*) ............ (63), 

where as before X = x (1 x). 

When V = 0, q=x, and we get from (62), (63), for the 
approximate equation to the equilibrium trajectory, 

y=lOS(X+X*)* ..................... (64); 

whereas the true equation is 

(65). 



Since the forms of these equations are very different, 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 of the y in (64) over the y in (65), 
by SB, we have 

8 = - 6.Y 2 + 20Z 3 -f 10Z 4 , 



= 4 (- 3 + 1 oX + 10Z 2 ) X m 
ax dx 

Equating dS/dx to zero, we get X= 0, x 0, S = 0, a maximum; 
X = 1787, x = 233, S = 067, nearly, a minimum; and # = ^, 
8 = 023, nearly, a maximum. Hence the 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 p=p Q +p lt y = y, + y l , where ^, y Q are the values of 
//,, y for q = oo , we have 

{19 /i io\ 94 ^ 

^(l-^)-g + ^)sm^ + ^(l-cos^)j...(66), 

x)}^ .................................. (67). 



The values of ^ and y 1 may be calculated from these formulae 
for different values of q, and they are then to be added to the 
values of yit , y Q , respectively, which have to be calculated once for 
all. If instead of the mean deflection //, we wish to employ the 
central deflection D, we have only got to multiply the second sides 
of equations (62), (66) by f f , and those of (63), (67) by f , and to 
write D for /Lt. The following table contains the values of the 
ratios of D and y to 8 for ten different values of q, as well as for 
the limiting value q= oo , 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 2^/?r to which the curves respect 
ively belong, the symbol oo being written against the equilibrium 
curves. Fig. 2 represents the trajectory of the body for different 
values of q, 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 ft. 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 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 215 



TABLE III. 





Values of when is equal to 

O 7T 


X 


1 


2 


3 


4 


5 


6 


8 


10 12 


16 


00 


00 


000 


000 


000 


000 


000 


000 


000 


000 


000 


000 


000 


05 


004 


004 


005 


006 


007 


008 


014 


019 


025 


041 


156 


10 


009 


013 


022 


027 


037 


053 


081 


117 


158 


239 


307 


15 


017 


028 


048 


075 


108 


146 


234 


327 


412 


530 


449 


20 


025 


052 


099 


159 


231 


309 


469 


607 


696 


707 


580 


25 


041 


093 


177 


285 


406 


531 


746 


871 


884 


707 


696 


30 


056 


144 


282 


451 


626 


787 


1-003 


1-031 


915 


689 


794 


35 


070 


214 


418 


650 


871 


1-045 


1-180 


1-052 


845 


814 


873 


40 


100 


300 


578 


870 


1-115 


1-265 


1-238 


967 


796 


1-017 


930 


45 


134 


399 


757 


1-097 


1-332 


1-412 


1-178 


859 


856 


1-097 


965 


50 


169 


516 


947 


1-310 


1-492 


1-460 


1-036 


812 


1-004 


991 


977 


*55 


213 


640 


1-139 


1-491 


1-574 


1-403 


870 


860 


1-127 


862 


965 


60 


256 


776 


1-321 


1-619 


1-562 


1-250 


739 


969 


1-115 


872 


930 


65 


306 


913 


1-482 


1-681 


1-454 


1-027 


682 


1-054 


948 


959 


873 


70 


359 


1-050 


1-609 


1-663 


1-257 


769 


695 


1-031 


718 


924* 


794 


75 


419 


1-181 


1-691 


1-560 


990 


517 


746 


869 


549 


707 


696 


80 


475 


1-296 


1-717 


1-371 


677 


303 


777 


604 


499 


472 


580 


85 


533 


1-399 


1-681 


1-106 


350 


149 


733 


325 


516 


384 


449 


90 


586 


1-476 


1-588 


776 


037 


064 


579 


117 


-477 


385 


307 


95 


646 


1-525 


1-402 


400 


-234 


025 


321 


021 


296 


276 


156 


1-00 


699 


1-540 


1-158 


000 


-446 


019 


000 


001 


-001 


000 


000 




Values of ^ \vhen is equal to 

O 7T 


X 


1 


2 


3 


4 


5 


6 


8 


10 


12 


16 


00 


00 


000 


000 


000 


000 


000 


000 


000 


000 


000 


000 


000 


05 


001 


001 


001 


001 


001 


001 


002 


003 


004 


006 


025 


10 


003 


004 


007 


008 


012 


017 


025 


037 


050 


075 


096 


15 


008 


013 


022 


034 


050 


067 


108 


150 


190 


244 


207 


20 


015 


031 


059 


095 


137 


184 


279 


360 


414 


420 


344 


25 


029 


056 


126 


203 


290 


378 


532 


621 


630 


504 


496 


30 


045 


117 


230 


366 


509 


640 


814 


839 


744 


560 


646 


35 


063 


191 


374 


581 


778 


934 


1-054 


940 


755 


727 


780 


40 


096 


285 


550 


828 


1-062 


1-205 


1-178 


921 


759 


969 


8S6 


45 


133 


394 


748 


1-085 


1-316 


1-395 


1-164 


849 


846 


1-084 


954 


50 


169 


516 


947 


1-310 


1-492 


1-460 


1-036 


812 


1-004 


991 


977 


55 


210 


632 


1-126 


1-473 


1-555 I 1-387 


860 


850 


1-114 


852 


954 


60 


244 


739 


1-258 


1-542 


1-487 


1-191 


704 


923 


1-062 


830 


886 


65 


274 


816 


1-325 


1-502 


1-300 


917 


609 


942 


848 


857 


780 


70 


292 


854 


1-308 


1-352 


1-022 


626 


565 


839 


584 


752 


646 


75 


298 


842 


1-205 


1-111 


705 


369 


532 


619 


391 


488 


496 


80 


282 


770 


1-020 


814 


402 


180 


462 


359 


297 


280 


344 


85 


245 


644 


774 


509 


161 


069 


337 


149 


237 


178 


207 


90 


184 


463 


498 


244 


012 


020 


182 


037 


150 


121 


096 


95 


103 


243 


224 


064 


-037 


004 


051 


003 


047 


044 


025 


1-00 


000 


000 


000 


000 


000 


000 


000 


000 


000 


000 


000 



216 



DISCUSSION OF A DIFFERENTIAL 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 whenj^> is very large. 




Fig.2 



-Forms of the trajectory when jfc, is very small. 




Fig.3~Corresponding curves of deflexion 




if the ratio of M to M altered continuously from GO to 0, they 
would probably pass continuously into the oscillations which are 
so 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 M to M 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 M alone varies, j3 varies in 
versely as $, and 8 varies directly as M, so that q remains constant. 
To get rid of the apparent dependence of q on M, let $ t be the 
central statical deflection produced by a mass equal to that of the 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 217 

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 (o) for p. Let moreover r be the time the body takes 
to travel over the bridge, so that 2c = FT ; then we get 



If we suppose T expressed in seconds, and S l in inches, we must 
put g = 32 2 x 12 = 386, nearly, and we get, 





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 M=Q, where M is not 
divided by S, and replacing M/S by M jS,, and F. d/dx by d/dt. 
We thus get 



and therefore, if P be the period of the motion, or twice the time 
of oscillation from rest to rest, 

-*- ............... (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 



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 (66) to retain only the term which is divided by the 
first power of q. With this simplification we get 

25 25 . 



so that the central deflection is liable to be alternately increased 
and decreased by the fraction 25/8g of the central statical deflec 
tion. By means of the expressions (61), (69), we get 



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 difference 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 



RELATING TO THE BREAKING OF RAILWAY BRIDGES. 219 

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 w r e 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 1//3 and 2o/8q of S. It is 
only when these fractions are both small 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 | of the weight of the bridge to cause a deflection of i inch. 
These are nearly the circumstances of the Ewell bridge, mentioned 
in the report of the commissioners. In this case, S 1 = j 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 formulas (72), 

! = -0434, = 721 = 45-9 xf. 
8< 4 

The travelling load being supposed to produce a deflection of 
2 inch, we have /3 = 127, 1/0 = 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 04345 would 
be doubled. 

In the case of one of the long tubes of the Britannia bridge ft 
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 OF A DIFFERENTIAL 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 f ths of 
the same weight placed at the centre, we have in this case 
^ = 1x10 = 16; and supposing the train to be going at the rate 
of 30 miles an hour, we have r = 460 -j- 44 = 10 5, nearly. Hence 
in this case 25/8^ = 043. or -^ 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 different 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 x t 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 J. 

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 Malmsten and A. F. Svanberg of Upsala 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. 
Bat 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 NOTES ON HYDRODYNAMICS. 

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 wave. 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 type 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 Jc 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 b 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 ~b\k. 
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 b\k/bh 
or \k/h. Hence if X be very great compared with h y 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 
gp8, 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 #, and passing through points whose abscissas are x 
and x + dx. Then if rj be the elevation of the surface at 
any point of the horizontal line in which it is cut by the first 
plane, 77 + drj/dx . 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 
8 be the depth of the line PP below the surface of the fluid 
in equilibrium, the pressures at P, P will be gp (8 + 77) and 
gp (8 + 77 + drj/dx . dx) respectively ; and therefore the difference 



224 NOTES ON HYDRODYNAMICS. 

of pressures will be gp dy/dx . dx . About the line PP describe 
an infinitely thin cylindrical surface, with its generating lines per 
pendicular to the planes, and let re 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 drj/dx. dx , and the 
mass being pK dx, the accelerating force is g dr t /dx. 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, x + % the common abscissa of the same set of particles 
at the time t, so that and 77 are functions of x and t. Then 
equating the effective to the impressed accelerating force, we get 

^ = -<7^ (1). 

df 9 dx .............. 

and we have x x + ^ ............................... (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) 

tf? dTJ 



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 -f 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 f and x + f + (1 4- d/dx) dx ; the depth of the fluid contained 
between these planes is h + rj and therefore the expression for 
the volume is b (h + 77) (1 4- dg/dx) dx. Equating the two expres- 



ON WAVES. 225 



sions for the volume, dividing by bdx, and neglecting the product 
of the two small quantities, we get 



Eliminating f between (3) and (4), we get 

tfi, , d 2 rj 

-de =ffh d? 

The complete integral of this equation is 



.............. (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 *J(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 77 and drjidt, 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, corrected 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 



and denoting by ?/ the vertical displacement of any particle, we have 

d dr, 

u= Tt> v = Tf 

Substituting in (a), and integrating with respect to t, we get 



$ (*, y) denoting an arbitrary function of .r, ?/, 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 17 = 0, and does 
not vary in passing from one particle to another, and therefore ^(.r, y)=0. Inte 
grating equation (b) with respect to y from y = Q to y = li + 77, observing that is 
independent of y, and that the limits of 77 are and 97, and neglecting 77 d^dx, 
which is a small quantity of the second order, we get the equation in the 
text. 

S. II. 15 



226 NOTES ON HYDRODYNAMICS. 

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 T? and . The former is given by (6), where the functions /, F 
are 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 ; let 
\ be the length of the wave, and suppose the origin taken at the 
posterior extremity of the wave in the position it occupies when 
t = : then we may suppress the second function in (6), and 
we shall have/ (a;) = from a?=-oo to = 0, and from x = \ to 
x = + oo , and/(#) will be positive from x = to x = X. Let 

c = J(gh) .............................. (7), 

so 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 vanishes for x = + oo . 
Then we have from (4) and (6) 



=l (%<to=4 f(x-ct)dx ............... (8). 

li J n>Jx 



Consider a particle situated in front of the wave when t = 0, 
so that #>X. Since /(#) = when a?>\, we shall have 
/(#-c) = 0, until ct = x-\. Consequently from (6) and (8) 
there will be no motion until t = x-\/c, when the motion will 
commence. Suppose now that a very small portion only of the 



ON WAVES. 227 

wave, of length s, has passed over the particle considered. Then 
act = \ s , and we have from (6) and (8) 

n =/(x -),* = \\" /(x - ) d. = J f >(x - ) <fe : 

for since f(x) vanishes when x> X, we may replace the limits 

ft 

x> and s by and s. Since I f(\ s)ds is equal to s mul- 

^o 

tiplied by the mean value of /(X - s) from to 5, and this mean 
value is comparable with / (X s), it follows that f is at first very 
small compared with ?;. Hence the particle begins to move verti 
cally; and since 77 is positive the motion takes place upwards. 
As the wave advances, f becomes sensible, and goes on increasing 
positively. Hence the particle moves forwards as well as upwards. 
When the ridge of the waves reaches the particle, 77 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, 77 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, 77 = 0, so that the particle is at the same height 
from the bottom as at first ; but f is a positive constant, equal to 

*<** or to ! 



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 Russell s experiments 
described at page 342 of his second report on waves*. 

* Keport of the 14th meeting of the British Association. Mr Russell s first 
report is contained in the Report of the 7th meeting. 

152 



228 NOTES ON HYDRODYNAMICS. 

In the preceding investigation the canal has been supposed 
rectangular. 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 + dfydx) d.c, and the area of a section of the fluid is A + brj 
nearly, so that the volume is 



nearly. Equating the two expressions for the volume, we get 

A --. + bri = 0. 

ax 

Comparing this equation with (4), we sec that it is only 
necessary to write A/b for h ; so that if c be the velocity of 
propagation, 

~J( 9 4}. .(). 



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 Memoirs, 1786, p. 192. In this memoir Lagrange has obtained the 
velocity of propagation by very simple reasoning. Laplace had a little earlier (Mem. 
de V Academic for 1776, p. 542) given the expression (see equation (29) of this note) 
for the velocity of propagation of oscillatory waves, which when h is very small 
compared with X reduces itself to Lagrange s formula, but had made an unwarrant 
able extension of the application of the formula. In the Mecaniqne Analytique 
Lagrange has obtained analytically the expression (7) for the velocity of propagation 
when the depth is small, 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 WellenleJire auf Experimente fjegriindet, Leipzig, 1825. 

f Transactions of the Royal Society of Edinburgh, Vol. xiv. pp. 524, 530. 



ON WAVES. 229 

by Green*, in the particular case 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 1 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 sufficient 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 
dynamics 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 the Cambridge Philosophical Society, Vol. vn. p. 87. 
t Encyclopedia Metropolitan/!. Art. 200 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, b 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 + Judt, b +fvdt, respectively. Since the 
squares of small quantities are omitted, it is immaterial whether 
we conceive u and v to be expressed in terms of a, b, t, 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 t. 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 case 
with u and v. We may therefore assume 

u = u l sin nt + u z cos nt, v = v l sin nt + v 2 cos nt, 

where u lt u^ v l} v 2 are functions of x and y without t. Substituting 
these values of u and v in the general equations of motion, neglect 
ing the squares of small quantities, and observing that the only 
impressed force acting on the fluid is that of gravity, we get 

I dp 

-f- = nu, cos nt + nu a sin nt, 

pdx 

......... (10), 

I dp 

--**- = q nv, cos nt + nv t sin nt 



and the equation of continuity becomes 



du. dv^\ . (du dv 

-p 1 + 7 - 1 - sin nt + 7 + -T- cos nt = ........ (11. 

dx dy I \dx dy 



Eliminating p by differentiation from the two equations (10), 
we get 

f du^ 
dy dx) ~ \dy 



(du dv\ , 
cos nt - \-r - - y s 
\dy dx) 



ON WAVES, 231 

and in order that this equation may be satisfied, we must have 
separately 

^*i = 0, ^-^ = ...... (13). 

dy ax ay ax 

The first of these equations requires that u^dx + v^dy be an 
exact differential dfa, and is satisfied merely by this supposition. 
Similarly the second requires that u^dx + v^dy be an exact differ 
ential d(f> 2 . The functions (/> x , c/> 2 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 coeffi 
cients of sin nt and cos nt. Expressing u l} v l} H 2 , v 2 in terms of 
<j> l j < a in the resulting equations, we get 



with a similar equation for </> 2 . Integrating the value of dp 
given by (10), we get 



- = fjy ?i(/> 1 cos nt + nfa sin nt + tyfy) ...... (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.) 

^ + ! > + ^ = ............ (16). 

dt dx dy 

The second term in this equation is of the second order, and 
-in the third we may put for dp dy its approximate value gp. 
Consequently at the free surface, which is defined by the 
equation 

gij + nfa cos nt nfa sin nt ^r (t) = ......... (17), 

we must have 

n fa sin nt + n fa cos nt + ^ (t) - g (~^ sin nt + ^ 2 cos nty = (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 ON HYDRODYNAMICS. 

Equations (18), (14), and the corresponding equation for c/> 2 
shew that the functions < 1} </> 2 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 c 2 , which may be easily 
restored in the end by writing t + 7r/2n for t, 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 Ae ni x+m y, 
where any one of the three constants A, m, m may be positive 
or negative, real or imaginary, and m, m are connected by the 
equation m /2 + m 2 = 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 x = oo or 
for as + co , 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 m must be positive, since otherwise the 
expression considered would become infinite for y = oo . The 
equation connecting m and m gives m= m\/( 1). Unitino^ 
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 (J. e mV <- 1 > + Be~ m ^ ( -V) e m , which 
becomes, on replacing the imaginary exponentials by circular 
functions, and changing the arbitrary constants, 

(A sin mac + B cos mx) e m ->. 
Hence we must have 

(/> a = S (A sin mx + B cos mx) e m v (19), 

the sign 5 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 $ 2 , the value of <f) l given by (19), and putting y = after 
differentiation, we have 

sin nt S (n* wig) (A sin tnx + B cos m,r) + ty (t) 0. 

* See Poisson, T raite de Mecanique, Tom. n. p. 347, or Theorie dc la Chaleur, 
Chap. v. 



ox WAVES. 233 

Since no two terms such as A sin mx or B cos mx can destroy 
each other, or unite with the term -fy (t), we must have sepa 
rately ^r (t) = 0, and 

u*-my = Q (20). 

The former of these equations gives ^ (t) = k, where k is a 
constant ; but (17) shews that the mean value of the ordinate 
y of the free surface is kfg, inasmuch as </> : and t/> 2 consist of 
circular functions so far as x is concerned, and therefore we must 
have k 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 in, 

To obtain $ 2 it will be sufficient to take the expression for 
<j with new arbitrary constants. If we put </> for 

<f) l sin nt + 2 cos nt, so that <p = f(ud.v + vdy), 

we see that <f> 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 my 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 mx -\-nt 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 mx+nt, and we shall then have got only two 
terms left, involving respectively sin (mx nt) and cos (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 

< = A sin (mx - nt) e m ^ (21) ; 

and <t> determines, by its partial differential 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 

4v *,. 

<> = A cos m r . e m , <., = A sin mx . e my . 



234 NOTES ON HYDEODYNAMICS. 

Substituting in (15) and (17), putting ty (t) = 0, and replacing 
y by in the second and third terms of (17), we get 

P 
which gives the pressure at any point, and 

y = cos (mx nt) (23)*, 

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 = 2-Tr/X, n = 2-7T/7 7 , n = cm ; and therefore 
from (20) 

-y M- 

Hence the velocity of propagation varies directly, and the period 
of the wave inversely, as the square 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 (/> 
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 (f> 
adapted to that case, which will be given presently; but it may be 
shewn to be true 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 udx + vdy -f wdz 
is an exact differential. 

It will be interesting to determine the motions of the indi 
vidual particles. Let x + f , y -f 77 be the co-ordinates of the par 
ticle whose mean position has for co-ordinates x, y. Then we have 
d d(fr drj dd> 

dt dx dt dy 

and in the values of u, v we may take x, y to denote the actual 

* Equations (22), (23) may be got at once from the equations 

p fld> dip 

= - fj y -, , q\i + ~- . 
p at at 



OX WAVES. 235 

co-ordinates of any particle or their mean values indifferently, on 
account of the smallness of the motion. Hence we get from (21) 
after differentiation and integration 

in A . , mA . . 

% = sin (mx nt) e J , 77 = cos (mx nt)e y . . . (25). 

n ii 

Hence the particles describe circles about their mean places, with a 
uniform angular motion. Since 77 is a maximum at the same time 
with y in (23), and dg/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 mA/n . e m *, 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 \ the displace 
ment is to the displacement at the surface as e"* 71 " to 1, or as 1 to 
.")o5 nearly. 

If the depth of the fluid be finite, the preceding solution may 
of course be applied without sensible error, provided e m * be insensi 
ble for a negative value of y 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 measure y from the bottom 
rather than from the mean surface. Consequently we must put 
y = h, instead of y = 0, in the values of $> v </> 2 , 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 i/ = (} (26). 

ay 

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 we are at liberty to take is of the form 

Ae mu cos mx + Be my sin mx -f Ce~ my cos mx + De~ my sin mx, 
or, which is the same thing, the coefficients only being altered, 

(t m!l + e~ my ) (A cos mx + B sin mx) 
+ (e mv e~ my ) ( C cos mx -f D sin mx). 

Now (26) must be satisfied by fa and fa separately. Substituting 
then in this equation the value of (f) 1 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 C = 0, D 0; so that 

(j) l = (e my + e~ my ) (A cos mx 4- B sin mx). 

Substituting in equation (18), in which fa is supposed to be 
suppressed, and y put equal to h after differentiation, we get 

n 2 (e mh + <T >Hh ) - mg (e mh - -"*) - ......... (27), 

and ifr (t) = 0, which gives ty (t) = k. The equation (17) shews 
that this constant k must be equal to h, which is the mean value 
of y at the surface. It is easy to prove that equation (27), in 
which m is regarded as the unknown quantity, has one and but 
one positive root. For, putting mh = p, and denoting by v the 
function of //, defined by the equation 

v (& + e -M) =At ( e M_ ->) .................. (28), 

we get by taking logarithms and differentiating 
Idvl * + e-^ e^ e ^ 



Now the right-hand member of this equation is evidently positive 
when /A is positive; and since v is also positive, as appears from 
(28), it follows that dv/dfji is positive; and therefore //, and v in 
crease together. Now (28) shews that v passes from to GO as ^ 
passes from to oc , and therefore for one and but one positive 
value of /A, v is equal to the given quantity w?h/ff, which proves the 
theorem enunciated. Hence as before the most general value of 
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 = cm = c . 2ir/\ t 






If we consider only the series which is propagated in the posi 
tive direction, we may take for the same reason as before 

<f> = A (e my + e-"") sin (mx-nf) ............... (30); 

which gives 

V = g (h -ij} + nA (e wy + e 1 "*) cos (mx - nt) ....... (31 ), 

and for the equation to the free surface 

g ( y - h) = nA (e" A + - n<h ) cos (mx -nt) ......... (32). 

Equations (21), (22), (23) may be got from (30). (31), (32) by 
writing y + h for ^, Ae~" h for A, and then making h infinite. 
When X is very small compared with h, 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 the 
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 



%= - ( m + "*) sin (mx - nt), 
77 = (e m * - e m ") cos (mx - nt) ............. (33). 

7i 

Hence the particles describe elliptic orbits, the major axes of 
which 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 e~ 2my to 1 + e~ 2 " I2/ , 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 h y 
to that at the surface is equal to the ratio of 6 v + e~" <v to e mh + e~ m \ 
The ratio of the vertical displacements is that of e" ty e" 2 to 
>*_ -"i* ! The former of these ratios is greater, and the latter 



238 NOTES ON HYDRODYNAMICS. 

less- than that of ~ m(h ~ y] 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 otherwise 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 
meaus of forces applied to the surface. It follows from the latter 
character, by virtue of the theorem proved in Note IV, that 
udx + vdy is an exact differential. 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 e~ 2 "^ 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 priori, 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 Philosophical Transactions, Vol. vm. p. 441. [Ante, Vol. i. p. 197.] 



ON WAVES. 



230 



C (l + 27rV X 2 ), c 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, RN. 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 difference between the numbers given in columns 
B and C. 

In calculating the numbers in table C, I have taken g = 32 2 
feet, and expressed the velocity in knots of 1000 fathoms or 6000 
feetf. 



A 


B 


C 


D 


55 


27-0 


24-7 


2-3 


43 


24-5 


21-8 


2-7 


50 


24-0 


23-5 5 


35 to 40 


22-1 


20-4 1-7 


33 


22-1 


19-1 3-0 


57 


26-2 


25-1 1-1 


35 


22-0 


19-7 


2-3 



The mean of the numbers in column D is T94, nearly, which 
is about the one-eleventh of the mean of those in column C. The 
quantity 1*94 appears 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 

* Report for 1848, Part n. p. 38. 

t I have taken a knot to be 1000 fathoms rather than 2040 yards, because the 
former value appears to have been used in calculating the numbers in column 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 (7, in order to 
compare theory and experiment ; so that on the whole *5 or 6 
would have to be subtracted from the numbers in column C. 
But on account of the depth to which the ship sinks in the sea, 
and the rapid decrease of the factor e~ 2 "^ 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 propagated in opposite directions. 
The value of </> which belongs to the compound motion will be 



* I have been told by a naval friend that an allowance for the " heave of 
the sea" 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 
considered. 



ON WAVES. 241 

the squares of small quantities being neglected, as throughout this 
note. Since 

cos (mx nt) + cos (inx + nt + a) = 2 cos (??kz + Ja) cos (?i + Ja), 
we get by writing \A for ^, and altering the origins of x and t, so 
as to get rid of a, 

</> = A (e m * + e~ my ) cos mx . cos ?^ (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) 

\i = - mA (e my + e~ my ) sin mx cos nt} 
v = mA (e my e~ my ) cos mx cos nt 

We have also for the equation to the free surface 

nA 

y-h = (e my + e~ my ) cos mx sin nt (36). 

Equations (35) shew that for an infinite series of planes for 
which mx = 0, = + TT, = + 2?r, &c., i. e. x = 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 t = 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 
x = to x = JX, and depressed from x = JX to x = JX, which suffi 
ciently defines the form of the whole, since the planes whose 
equations are x = 0, x = JX, are planes of symmetry. When 
nt = |TT, 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 nt becomes equal to TT, 
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 the surface which were elevated becoming depressed, and 
vice versa. When nt = 27r, everything is the same as at first, 
s. ii. 16 



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 sufficient 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 Transactions of the Cambridge Philosophical Society, 
Vol. ix. p. L] 



T. ON THE DYNAMICAL THEORY OF DIFFRACTION. 

[Read November 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 

162 



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 is 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 effect of the screen is merely to stop a certain 
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 different 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 



OX THE DYNAMICAL THEORY OF DIFFRACTION. 245 

same iii 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 
p!ay 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 elastic 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 the 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 iiicompressibility 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 the 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 are 
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 different solution of the problem, which will I hope 
be found somewhat simpler and more direct than Poi.sson 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 the 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 

* Camb. Phil Trans. Vol. vn. p. 2. 



OX 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 difficulties 
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 me 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 the incident and diffracted rays I am indebted 
to the kindness of my friend Professor O Brien. 

The description of the experiments, and the discussion of the 
results, occupies Part II. of this Paper. Since in a glass grating 
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 diffracted 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. 249 

before the light reaches the grooves ; secondly, that it takes place 
after the light has passed between them; and these results are 
very 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 hypothesis. 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 sufficient 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 experimental 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 of some 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 dV to represent an element of volume, and 
write over it the sign J/J, 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 differs 
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 *da- ) dV 
= drdS = r^drda-. When the signs /// and //, referring to differen 
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 ///.// UdVda; 

2. As the operation denoted by -^ + -=- 9 + -^ will be per- 

y 
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 (#, ?/, 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 J 7 , U the function which is the subject of the 

operation, and if y be defined as the equivalent of -j-g + -^-5 + y^ , 
it is easy to prove that 



= limit of -^ dS (1), 

the integration extending throughout the surface of T, of which 



252 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

dS is an element. In fact, if l t m, n be the direction-cosines of the 
normal, we shall have 



UdU , a {{ddU dU dU, 
\\-T- dS = III - 1 ~ + m- r - + n- T - dS 
JJ dn Jj V dx dy dz 



We have also, supposing the origin of co-ordinates to be at the 
point P, as we may without loss of generality, 

dU dU\ d*U\ d z U\ d*U 



+ 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 1 1 , - dy dz, 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, 

ff/dU, dU\ . 
we must write II (-7-^ ~~^~~j dy dz, 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, 

[[fdU,, dU\j , (d z U\ [[. ... 

II ( ^ ~ 7B# d v dz = (w) JJ fc - ^ dy **. 

Bat JJ (x /f a?,) 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 



[fdU 
-j- 
JJ dn 



j T u . 

j-dS=T\ ho ) + 1:3-7) + (~J-*)f ultimately. 
dn [\darj \dy J \dz )} 

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 tlie elementary volume 
?- 2 sin 6 dr d6 dQ, or r dr dd dz, according as we wish to employ 
polar co-ordinates, or one of three rectangular co-ordinates com 
bined with polar co-ordinates in the plane 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 f 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 



(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 //4?r ; 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, 

U=-^- 



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+ If be the most general 
solution, U must satisfy the equation v U = a ^ all points, must 
nowhere become infinite, and must vanish at an infinite distance. 
But this being the case it is easy to prove that U cannot be 
different from zero. 

The solution will still hold good in certain cases when / is 
infinite at some points, or w 7 hen 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 

* U " (6). 



It is required to determine 7 by 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 ; f(r), F (r) the initial 
values of U t dU/dt. 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 



See a memoir by Poisson Mem. de rAcade mie, Tom. ill. p. 130, 
or Gregorys Examples, p. 499. 

5. Let S be a function which has given finite values within 
a finite portion of space, and vanishes elsewhere ; and let it be 
required to determine three functions f, 17, f by the conditions 

^_*? = ^_^ = *?_^l = ... ..(8) 

dy dz dz dx dx dy 



........................ . 

dx dy dz 

The functions , 77, 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), d& + i}dy+ %dz is an exact diffe 
rential d^r, and (9) gives v^ = & Hence we have by the 
formula (5) 



and T/T being known, f , 77, % will be obtained by mere differentia- 



PRELIMINARY ANALYSIS. 255 

tion. To differentiate ^ with respect to x, it will be sufficient to 
differentiate 8 under the integral sign. For draw 00 parallel to 
the axis of #, and equal to A.r, let P, P be two points similarly 
situated with respect to 0, (7, respectively, and consider the part 
of ^r and that of -^ + A-v/r 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 , and in passing from the first to the 
second 8 is changed into 8 + A8, and therefore the increment of >|r 
is simply AS/47T?*. dV. 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 A;e and passing to the limit, we get 



By employing temporarily rectangular co-ordinates in the 
triple integration, integrating by parts with respect to x t and 
observing that the quantity free from the integral sign vanishes at 
the limits, we get 



as might have been readily proved from (l6), by referring to 
a fixed origin, and then differentiating with respect to x. The 
expressions for r) and f may be written down from symmetry. 

6. Let tzr , OT", TX " 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, 77, by the condi 
tions 

#--*,. --*-, p- d -f -a." ...(is). 

dy dz dz d,c dx dy 

+ $ + -0 . ..(14). 

dx dy dz 

It is to be observed that ta- , TV", vr" are not independent. For 
differentiating equations (13) with respect to x, y, z, and adding, 
we get 



dx dy dz 



(15). 



256 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

Hence -G/, ", -ST " must be supposed given arbitrarily only in so 
far as is consistent with the above equation. 

Eliminating from (14), and the second of equations (13), 
we get 



.df " dz 

d dr 






which becomes by the last of equations (13) 

/^ z. 

V - 2 U dy 

Consequently, by equation (5), 



u dz i r 

u 

Transforming this equation in the same manner as (11), sup 
posing x, y y z measured from 0, and writing down the two equa 
tions found by symmetry, we have finally, 



(16). 



7. Let S, w , TO- , is" be as before ; and let it be required to 
determine three functions 77, f 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 f , ?7 , 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 l3 77,, and 
? 2 ^2 ?2 sucn ^ na ^ ^dx + y^y -\-%jLz may be an exact differential 
dfa, and f 2 , 17, , f a may satisfy (14). 



PROPAGATIOX OF AX ARBITRARY DISTURBANCE. 257 

Observing that & = ? -? lf 17, = i? - i? t , ? 8 = >-?i expressing 
j, 77^ in terms of ^r p and substituting in (14), we get 



where S is what 8 becomes when f , 7; , f are written for f, 77, f. 
The above equation gives 



whence t , 77^ f t , and consequently 2 , ?7 2 , ,, 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 x y y, z be the co-ordinates of any particle of the medium in 
its natural state; f, 77, 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 



dy* <& dx \dx cTu 

where a 2 , Z/ 2 , denote the two arbitrary constants. Put for shortness 



, 

-j- + ~r T j" = ^ 
dx dy dz 

S. n. 17 



258 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

and as before represent by yf the quantity multiplied by Z> 2 . Ac 
cording to this notation, the three equations of motion are 



, 72N 
(a 2 - 6 2 ) -7- 
dy 



(18). 



It is to be observed that 8 denotes the dilatation of volume of 
the element situated at the point (as, y, z). In the limiting case 
in which the medium is regarded as absolutely incompressible B 
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 2 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- , -jr> respectively, 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, arid then to be left to itself, the 
whole of the medium outside the space T being initially at rest ; 
and let it be required to determine the subsequent motion. 

Differentiating equations (18) with respect to x, y, z, respec 
tively, and adding, we get by virtue of (17) 



Again, differentiating the third of equations (18) with respect to y, 
and the second with respect to z y and subtracting the latter of the 
two resulting equations from the former, and treating in a similar 



PROPAGATION OF AX ARBITRARY DISTURBANCE. 259 

manner the first and third, and then the second and first of equa 
tions (18), we get 

Cl CT , , (I CT 72 a CT 

- = 6 2 V w f , - = b Vw , 



where CT , CT", CT " are the quantities defined by equations (13). 
These quantities express the rotations of the element of the 
medium situated at the point (JT, y, z) about axes parallel to the 
three co-ordinate axes respectively. 

Now the formula (7) enables us to express 8, CT , CT", and CT" in 
terms of their initial values and those of their differential coeffi 
cients with respect to t, which are supposed known ; and these 
functions being known, we shall determine f, 77, 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 8, d$ dt be represented 
by f(r), F (r), respectively, with the same understanding as in 
Art. 4. By the formula (7), we have 



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 lt ? 2 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 = ?\. The dilatation will then com 
mence, will last during an interval of time equal to a" 1 (r 8 rj, 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 

172 



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 b, to which are 
confined the rotations & , vr", TB" . This wave may be called for 
the sake of distinction, the wave of distortion, 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 requires that the ratio 
of b to a be not greater than that of ^/3 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 + ydy + ^dz and d/dt . dx -f- drj/dt . dy + d/dt . dz 
are initially exact differentials, there will be no wave of distortion, 
but only a wave of dilatation. By making b = we pass to the 
case of an elastic fluid, such as air. By supposing a = oo we pass 
to the case of an incompressible elastic solid. In this case we 
must have initially 8 = and dB/dt = ) 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 finite quantity, which in the end is supposed to increase 
indefinitely, we must not suppose the initial disturbance confined 

* See a memoir by Green On the reflection and refraction of Light. Caiiib. Phil. 
Trans. Vol. vii. p. 2. See also Camb. Phil Trans. Vol. vin. p. 319. [Ante, Vol. i. 
p. 128.] 



PROPAGATION OF AX 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 cfyr, where ty satisfies the 
equation y-v/r = derived from (14), and the expression 

^f 7 ^7 7 ^f 7 

- tfg -f rfy + -/ d.3 

tw dt dt 

must be initially an exact differential d^ lt where ^ satisfies the 
equation ^^ = 0. 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 we 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 involve dilatation or condensation. As no simplicity 
worth mentioning is gained by making a at once infinite, this 
constant will be retained in its 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 t are linear functions of the initial displacements 
and initial velocities ; and it appears from (21), and the corre 
sponding equations which determine , OT", and -or ", 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 which 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 elastic 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 r be an incre 
ment of t, which in the end will be supposed to vanish. The 
disturbance at the end of the time t -f r will be represented by 
f(t + r)j 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 v 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 Q will be expressed by linear functions 
of these velocities ; and consequently, if f (t) represent the dis 
turbance due to the system of velocities V Q , f(t) will represent 
the disturbance due to the system V Q . Hence the disturbance 
at the end of the time t will be represented by f(t + r) f(t). 
Now we may evidently regard the state of the material system 
immediately after the communication of the system of velocities 
V Q as its initial state, and then seek the disturbance which would 
be produced by the initial disturbance. The velocities V Q going on 
during the time T will have produced by the end of that time a 
system of displacements represented by TV O . By hypothesis, the 
system was in a position of equilibrium at the commencement of 
the time r ; 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 r vary from zero to small quan 
tities of the order r, and therefore the velocities generated by the 
end of the time T are small quantities of the order r 2 . Hence 
the velocities V Q communicated at the time destroy the pre 
viously existing velocities, except so far as regards small quantities 
of the order r 2 , which vanish in the limit, and therefore we have 
nothing to consider but the system of displacements rv Q . Hence 
the disturbance produced by a system of initial displacements TV O 
is represented by f(t + r) f(t), ultimately; and therefore the 
disturbance produced by a system of initial displacements v is 
represented by the limit of {/ (t + r) / (t)} /T, or by/ (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 
t = if the disturbance is expressed by velocities. 



2G4 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

The reader may easily, if he pleases, verify this theorem on 
some dynamical problem relating to small oscillations. 

15. Let us proceed now to determine the general values of 
f, 77, f in terms of their initial values, and those of their differential 
coefficients with respect to t. By the formulae of Section I., f, 77, f 
are linear functions of S, CT , -or", and w ", and we may therefore 
first form the part which depends upon S, and afterwards the part 
which depends upon OT , OT", -& ", 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 -sr , -or", -BT" , as well as the 
second term in equations (21), we get from equations (10) and (21), 



To understand the nature of the integration indicated in this 
equation, let be the point of space for which the value of ty is 
sought ; from draw in an arbitrary direction OP equal to r, and 
from P 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 
facilitate the transformation of the integral (22), conceive PQ 
produced to Q , let P Q = s, let d V be an element of volume, 
and replace the double integral // F . do- by the triple integral 
h 1 fffF . s~ 2 dV t taken between the limits defined by the imparities 
at< s < at 4- h, which may be done, provided h 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 PQ,Q being a point taken generally,) 
lies beyond the limits at and at -f h- t 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 ARBITRARY 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, 

dV dS 



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~ l 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 4>7rat or 47raV/r , according as r <> at. Substituting 
in (22), and omitting the accents, which are now no longer necessary, 



we get 



where the limits of integration are defined by the imparities written 
after the integrals, as will be done in similar cases. 

16. Let , v , w ot be the initial velocities ; then 

F= du + d + <fao m 
dx dij 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 x 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 x, and in the positive direction, through 
the point (0, y, z). Treating the last term in equation (23) in the 
same way, and observing that the quantities once integrated vanish 
at an infinite distance, or, to speak more properly, at the limits of 
the space T, we get 



- ( V + W + WQ*) -- (r > at). 



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 



17. To obtain the part of the displacement f due to the 
initial velocity of dilatation, we have only to differentiate ^r with 
respect to oc, and this will be effected by differentiating w , v , W Q 
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 t for the part of f due to F, 
we get 

\} dy dz 



* i r 1 1 d x d y d z , 7 T7 , . 

+ zdll(.^;3 + ^S? + w 3~= | ) d7 ( -> fl O- 



Let q Q be the initial velocity resolved along the radius vector, 
so that q = lu + mv + nw , and (q Q ) at be the value of q Q at a dis 
tance at from ; then 



mv 



d x d 



Substituting in the expression for ^, 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 a . 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 &> , &> ", &" be the initial angular velocities. Then f 2 
is given in terms of OT" and OT //X by the first of equations (16), and 
IP", is" are given in terms of <w ", &> " by the formula (7), in which 
however b must be put for a. We thus get 

* , ,,dVd<T 



PROPAGATION OF AN ARBITRARY DISTURBANCE. 267 

The integrations in this expression are to be understood as in 
Art. 15, and &> ", o> " are supposed to have the values which belong 
to the point Q, but PQ is now equal to bt instead of at. The 
quintuple integral may be transformed into a triple integral just 
as before. We get in the first place 



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= (bidder, the expressions for the integrals 

Hxr- 3 dS, ffyr~ 3 dS, ffzr~ 3 dS 

may be at once written down by observing that these integrals 
express the components of the attraction of a spherical shell, of 
radius bt and density 1, having Q for centre, on a particle situated 
at 0. Hence if x , y , z be the co-ordinates of Q, measured from 
0, and r = OQ, the integrals vanish when r < bt, and are equal to 

477 (fo)Vr - 3 , 47r (btfy r - 3 , ^ (Ufz r -\ 

respectively, when r > bt. Hence we get from (26), omitting the 
accents, whicli are now no longer necessary, since we have done 
with the point P, 



Now 

, dw Q dv _ ,, _ da dw n f , _ dv n du 
" ) = ~dy ~ ~dz W "~ ~dz ~ ~dx " &) ~ ~dx ~ ~dy 
Substituting in (27), and adding and subtracting x . dujdx under 
the integral signs, we get 

t ffff / d . d . 



. 

dx ^ dx dx )} r 

But x . d/dx + y . djdy + z . djdz is the same thing as r . d/dr, and 
we get accordingly 

d d d\ dV . 



= (M d -p dr do- (/ > bt) = - 1 1 (u^dv. 
J JJ dr JJ 



2G8 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

The second part of f 2 is precisely the expression transformed in 
the preceding article, except that the sign is changed, and b put 
for a. Hence we have 

*-Mri ~(r>U] ...(28). 

19. Adding together the expressions for and f 2 , we get for 
the disturbance due to the initial velocities 



29). 



The part of the disturbance due to the initial displacements 
may be obtained immediately by the theorem of Art. 14. Let f , 
?7 , f be the initial displacements, p the initial displacement 
resolved along a radius vector drawn from 0. The last term in 
equation (29), it will be observed, involves t in two ways, for t 
enters as a coefficient, and likewise the limits depend upon t. To 
find the part of the differential coefficient which relates to the 
variation of the limits, we have only to replace d V by r*dr do-, and 
treat the integral in the usual way. We get for the part of the 
disturbance due to the initial displacements 



(30). 



It is to be recollected that in this and the preceding equation 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 bt sub 
scribed denotes that r is supposed to be equal to at or bt after 
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 TJ 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. 2C9 

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 OT , -cr", vr " , aj , &>", w" must be separately 
equal to zero, and the expression for f will be reduced to f t , given 
by (25), and the expression thence derived which relates to the 
initial displacements. The triple integral in the expression for f t 
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 2 , 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 

d x d d 



Now 



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 Q , we obtain 

dv n x d y\ i 7 7 
-y- -o + v -,- Adxdy dz, 
dy r 3 dx r 3 J 

which, since d/dx . y/r 3 = d/dy . xjr 3 , becomes 



Treating the terms involving w in the same manner, and substitu 
ting in (31), we get 



Now the integration is to extend from r = bt 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 It. Putting dS or 6 2 a dcr for an element of 
the surface of this sphere, we obtain for the value of the second 
member of the last equation 

~ (&*) *// (k + + nwJJdS, or - ffl (q ) bt da ; 

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 a , f 6 , f c , respectively, and replace dcr by (at)~* dS 
or (lt)~* dS, as the case may be ; then 

............................ < 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 O l that the radius vector, drawn from 0, of 
any element of T makes but a very small angle with 00 lt we may 
regard I as constant in the integration, and equal to the cosine of 
the angle between 00 t 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 lt and q Q will be ultimately 
the initial velocity resolved in the direction 00 t . Hence we have 
ultimately 



where, for a given direction of Ofl, the integral receives the same 
series of values, as at increases through the value 00 19 whatever 
be the distance of from O r 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 v or in the direction of a normal to the wave. For 
a given direction of 1 ) 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 



where I, and the direction of the resolved part, q Q , of the initial 
velocity are ultimately constant, and the surface of which dS is an 
element is ultimately plane. To find the resolved part of the dis 
placement in the direction 00 lf we must suppose x measured in 



272 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

that direction, and therefore put I = 1, q = u , which gives % b = 0. 
Hence the displacement now considered takes place in a direction 
perpendicular to 00 lt or is transversal. 

For a given direction of Of), the law of disturbance is constant, 
but the magnitude of the displacements varies inversely as bt, the 
distance to which the wave has been propagated. To find the dis 
placement in any direction, OE, perpendicular to 00 X , we have 
only to take OE for the direction of the axis of x, and therefore 
put I = 0, and suppose u to refer to this direction. 

Consider, lastly, the displacement, f c , expressed by the last 
term in equation (29). The form of the expression shews that 
f c will be a small quantity of the order t/r* or 1/r 2 , 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 rt and f 6 are of the order 1/r, and therefore f c 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, H the radius of a sphere whose volume is 
equal to that of the space T, and compare the displacements f a , f w 
f c which exist, though at different times, at the same point 0, 
where O l = r. These displacements are comparable with 

vl? vtf vR 5 t 
ar " br r* 

which are proportional to 

I I R t 
a b r r 

But, in order that the triple integral in (29) may not wholly vanish, 
t/r must lie between the limits I/a and 1/b, 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 c is of the order R/r 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 
formulae 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 O lf draw in any direction the right line 
O l 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 1? and cutting 00^ produced at a distance p from O r Let p lf 
+ p 2 be the two extreme values of p 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^. Let/ M (p),f v (p), f w (p) be three 
functions of p defined by the equations 

/. (P) =//o*> . /. (P) = JIMS /. (p) = fjw a dS, ...... (34), 

and /(>), /, (p),f{ (p) three other functions depending on the initial 
displacements as the first three do on the initial velocities, so that 

ft (?) = JTfcA /, 00 "SMS, f { (p) = H&s ........ (:). 

These functions, it will be observed, vanish when the variable lies 
outside of the limits p^ and +p. 2 . They depend upon the direc 
tion Of), so that in passing to another direction their values 
change, as well as the limits of the variable between which they 
differ from 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 l to p z , that they before 
received as the variable increased from p^ to -f p 2 , 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 



(86) 



s. u. 18 



I 

274 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

The first term in the right-hand member of the first of these 
equations is got from (32) by putting 2 = 1, introducing the func 
tion f n , and replacing a t 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 t~*, and therefore of the order r~ 2 , 
but such terms are supposed to be neglected. 

The wave of dilatation will have just passed over at the end of 
the time (t"+J> t )/B* 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 (rpj/h. 
During the passage of this wave, the displacements and velocities 
will be given by the equations 



.(37). 



After the passage of the wave of distortion, which occupies an 
interval of time equal to (p l + p t )/b f 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 



dr 



Now the initial displacements and velocities are supposed to have 
finite, but otherwise arbitrary, values within the space T, and to 



PROPAGATION OF AX ARBITRARY DISTURBANCE. 27 3 

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 diffraction. 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% dr 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 differentiation with respect to t from 
which the differential coefficients just written have arisen until 
after the double integration, and we shall thus be led to the for- 
mulse 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, O v within T. But it is not mathemati 
cally true that the disturbance proceeding from even a single 
point O v when a disturbing force is supposed to act, or rather 
that part of the disturbance which is propagated with the velocity 

182 



276 ON THE DYNAMICAL THEOEY OF DIFFRACTION. 

b, is perpendicular to 00 V 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 O l be the given point, T a space comprising O x . Let the 
time t be divided into equal intervals r ; and at the beginning of 
the n ih interval let the velocity rF (n r) 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(n r) = (DT)~ l f (n r), 
so that rf (n r) is the momentum communicated at the beginning 
of the ?i th interval. Now suppose the number of intervals 
r 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 
OOjj for the square of the disturbance has been neglected all 
along. Let 00 l r; refer the displacement at to the rect 
angular axes of x, y, z ; let I, m, n be the direction-cosines of 
I , 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 00^ produced, so that 

k = II 4- 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 0, , 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 r/a. Conceive a series of spheres, 
with radii ar, 2ar. ..??ar,... described round 0, and let the n ih of 
these be the first which cuts T. Let S lt S 9 ... be the areas of the 
surfaces of the spheres, beginning with the ?i th , which lie within T \ 
then 

nr) S l + krF [t - (n 4 1) r} S 2 4 . . . 



But F(tnr), F{t (n + 1) T} ... are ultimately equal to each 
other, and to 



and ar8 t 4- arS^ 4 ... is ultimately equal to T. Hence we get, for 
the part of f which arises from the first of the double integrals, 

Zfc 



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 rja and r/b, 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 m for t in the coefficient, and 3lq Q U Q becomes ultimately 
(3lk l ) rF(t nT\ and, as well as r, is ultimately constant in the 
triple integration. Hence the triple integral ultimately becomes 

(Mk-l )T 



and we have now to perform a summation with reference to 



278 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

different values of n, which in the limit becomes an integration. 
Putting nr = t , we have ultimately 



r = dt 9 SnT.T-F(*-fiT)f 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/(), 

l] ^f( t _ r \ I -M r f _r\ 
tfr? \ a) 4*TT-DtfrJ \ b) 

t ) dl (39). 



To get 77 and f, we have only to pass from /, I to m, m and 
then to n, ri. If we take 00 l for the axis of x, and the plane 
passing through 00 l and the direction of the force for the plane 
xz, and put a for the inclination of the direction of the force to 
00 j produced, we shall have 

I = 1, Tii = 0, n 0, I = k = cos y. y m 0, ri = sin a ; 
whence 



cos a 



,/ r\ .cos a { b ,, f ,. ,/\ -,. 
f(t -- + ^~^^- tf(tt)dt 
J \ a 27rDr 5 J; J v 



?= Sln ^/ f t ^- ^3 f t f(t ~ t 



(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 f(t) equal to zero when t is negative, and we may even 
suppose f (t) periodic, so as to have finite values from t = oo to 
t = + oo . 

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 



PROPAGATION OF AX 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 f represents a disturbance which is propa 
gated from O l with a velocity a. Since there is no corresponding 
term in 77 or f, 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 O l 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 f 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 f (t) represent a force which acts 
for a short time, and then ceases, f (t t } 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 ?, and therefore 
the terms themselves will be of the order r" 2 , whereas the leading 
terms are of the order r~ l . 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 terms 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 b is extremely large, if not infinite. Conse 
quently the first term in f, which relates to normal vibrations, will 
be insensible, 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 f. 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. 



280 



ON THE DYNAMICAL THEORY OF DIFFRACTION. 



29. If we suppose a = oo , and f(t)=c sin %7rbt/X, we shall get 
from (40) 

g cX cos a. 2-7T ,, v cX 2 cos a . TTT *>- 



?r 



,= 



csna 



2ir 
T 



cXsina 



?r ,, 
C S ~ 



c\ sm QL 



3 

87r 3 D6r 



sin T" cos T" 

X X 



and we see that the most important term in f is of the order X/?rr 
compared with the leading term in , which represents the trans 
versal vibrations properly so called. Hence f, and the second and 
third terms in 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 effected 
somewhat differently by first decomposing the given functions f , 
?7 , f , and u , V Q , 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 T, 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 IX 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 
cerned. It does not at all follow that the ether is to be regarded 
as an elastic solid when large 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 in 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 IX A SECONDARY WAVE. 283 

32. Let O l be any point in the plane P; and refer the medium 
to rectangular axes passing through O t , x being measured in the 
direction of propagation of the incident light, and z in the direc 
tion of vibration. Let/ (6^ 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 r which begins at the 
end of the time t occupies a film of the medium, of thickness br, 
and consists of a displacement / (bt ) and a velocity bf (bf). 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 O r 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; I, m, n the direction-cosines of V measured from 0,, 
and therefore I, -m, n those of 00 l measured from 0; and 
let 00^ 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 b, 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 4 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 full my views on this subject in a paper On the constitution 
of the luminiferous ether, printed in the 32nd volume of the Philosophical Magazine, 
p. 349. [Ante, p. 12.] 



284 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

coefficient of these terms we must replace t by r/b ; we must put 
u = 0, v = Q, w = bf (bt r), write I, m, n for I, m, n, and 
put g = nw = nbf (bt r). The 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 dcr 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 br. To find the aggregate disturbance at correspond 
ing to a small portion, S, of the plane P lying about O lt we must 
describe spheres with radii ... r 2br, r br, r, r + 6r, r + 2br ..., 
describing as many as cut S. 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 1 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~ 2 dS, and we get accordingly 



Since Zf+wwy-f nf0, the displacement takes place in a plane 
through perpendicular to Ofl. Again, since f : 77 :: I : m, it 
takes place in a plane through Oft and the axis of z. Hence 
it takes place along a line drawn in the plane last mentioned 
perpendicular to 00 r The direction of displacement being known, 
it remains only to determine the magnitude. Let be the dis 
placement, and <p> the angle between 1 and the axis of z, so that 
n cos 0. Then sin $ will be the displacement in the direction 
of z, and equating this to f in (42) we get 



(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 DISTURBANCE IX A SECONDARY WAVE. 285 

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 , rj , f , and p in the second 
of the double integrals. We must put as before r for bt, and write 
r~*dS for da-. Moreover we have for the incident vibrations 

|=0, 77 = 0, S=f(W-x) t p = -nf(bt -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, 77, f, p are functions of r in 
consequence of being functions of x, and after differentiation we 
must put x = 0, t = t r/b. Since d/dr = I . d/dx, we get 



whence we get, remembering that the signs of I, m, n in (30) have 
to be changed, 

j. PndS ,, , N ImndS n . 

= -. -f (bt r), 77 = -- -/ (bt r), 

J v J * 



The displacement represented by these equations takes place along 
the same line as before ; and if we put f 3 for the displacement, 
and write cos 6 for I, we get 

= cos sin </>/(&<-,) ................ (). 



33. By combining the partial results obtained in the preceding 
article, we arrive at the following theorem. 

Let = 0, 77 = 0, =f(bt x) be the displacements correspond 
ing to the incident light ; let O t 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 O l which is large in compari 
son with the length of a wave ; let V =r, and let this line make 
angles 6 with the direction of propagation of the incident light, or 
the axis of x, and (f> 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 1 0, and lying in the plane zOfl; and if f be the 
displacement at 0, reckoned positive in the direction nearest to 
that in which the incident vibrations are reckoned positive, 



1/4 / 

In particular, if 



9 

f(bt - x}=c sin ^- (bt x), 

A, 



~~(l 4- cos 60 sinc/> cos ^(fo-r) ......... (46). 



we shall have 



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 draw O x f , O y , 
parallel to 0& Ojj, and let l = r , Oft y = a). Then 

dS = rdr dto = rdrdco, 

since r 2 =^ 2 -f r 2 , and p is constant. Let = s sin <. The dis 
placement % takes place in the plane zOf), and perpendicular to 
OjO; and resolving it along and perpendicular to O/, we get for 
resolved parts s sin 2 <, s sin < cos $, of which the latter is estimated 
in the direction OM, where M is the projection of O l on O y . Let 
MOO %, % being reckoned positive when M falls on that side of 
on which y is reckoned positive ; then, resolving the displace 
ment along OM parallel to O x, O y , we get for resolved parts 
s sin (f> cos (f> cos %, s sin < cos < sin ^. Hence we get for the dis 
placements f , 77, f at 

j* = s sin <f) cos ( cos ^, T; = 5 sin ^> cos (f> sin ^, ^=5 sin 2 <. 

Now produce O Oj to 2 , and refer 0^, 0^, 0^, OjO^ OjO to a 
sphere described round 1 with radius unity. Then zOf) forms a 
spherical triangle, right-angled at 2 , and 



* The corresponding expression which I have obtained for sound differs from 
this only in having cos 6 in place of sin 0, provided we suppose 6 to be the velocity, 
of propagation of sound, and f to represent a displacement in the direction OjO. 



LAW OF DISTURBANCE IX A SECONDARY WAVE. 287 

whence we get from spherical trigonometry, 

cos $ = sin 6 sin &>, sin < cos ^ = cos 0, 

sin (f> sin ^ = cos 6 tan ^ = sin cos co. 
We have therefore 

f = 5 sin 6 cos # sin o>, 77 = 5 sin 2 sin o> cos w, 



To find the aggregate disturbance at 0, we must put for s its 
value, and perform the double integrations, the limits of to being 
and 2?r, and those of r being *Jp* and cc . The positive and nega 
tive parts of the integrals which give f and 77 will evidently destroy 
each other, and we need therefore only consider f. Putting for s 
its value, and expressing 9 in terms of r, we get 



f = //(< + J) (r 5 C 3 J + / sin cos (fc - r) ~-. . ..(47). 

A, A, / 

Let us first conceive the integration performed over a large area 
A surrounding , 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 

fF(r)cos~(bt-r)dr 

Ay 

= - -~ F (r) sin ^ (bt - r) + ( --V F (r) cos ^ (bt - r) + . . . 

~7T A, \~7T/ A, 

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 \ compared with those retained. The first term, taken 
between limits, gives 

- 2 X ~ f ( P) sm f (bt +p)- ^ F (R) sin ^ (bt - /?), 

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) 



288 OX THE DYNAMICAL THEORY OF DIFFRACTION. 

When R becomes infinite, F (R) reduces itself to cos 2 w, and the 
last term in becomes 



j-j- I cos 2 co sin -^ (It R) da. 



Suppose that no finite portion of the perimeter of A is a circular 
arc with for centre, and let this perimeter be conceived to ex 
pand indefinitely, remaining similar to itself. Then, for any finite 
interval, however small, in the integration with respect to &>, the 
function sin ZirX 1 (bt R) will change sign an infinite number of 
times, having a mean value which is ultimately zero, and the limit 
of the above expression will be rigorously zero. Hence we get in 
the limit 

fcsin (btp), or = 0, 
A- 

according as p is positive or negative. Hence the disturbance 
continually transmitted across the plane P produces the same 
disturbance in front of that plane as if the wave had not been 
broken up, and does not produce any back wave, which is what 
it was required to verify. 

It may be objected 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 right 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 principle that when an infinite plane wave is conceived to 
be broken up, the aggregate effect of the secondary waves must 



LAW OF DISTURBANCE 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 oo 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)" 1 , 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 Kelland in a 
memoir On the Theoretical Investigation of tlie Absolute Intensity 
of Interfering Light, printed in the fifteenth Volume of the 
Transactions of the Eoyal 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 \r. 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. ii. 19 



290 ON THE DYNAMICAL THEOKY OF DIFFRACTION. 

In the investigations of Mr Smith and Professor Kelland, 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 <, 
so that the intensity will vary between limits which are as 1 to 
cos 2 0. If common light of the same intensity be used, the inten 
sity of the diffracted light at the given point will be proportional 
to i(l+cos 2 0). 



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 diffracted 
ray be called the angle of diffraction, and the plane containing 
these two rays the plane of diffraction; let ot it a d 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 EXPERIMENTS. 291 

dicular to the plane of diffraction, and the angle of diffraction. 
Then we easily get by a spherical triangle 

tan a d = cos 6 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, 
10,... 35 5, 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 NicoPs prisms. Suppose the plane of 
polarization of the incident light to be inclined 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 diffraction. If tzr, 
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 = sec 6 tan OT in the former event, 
and tan a = cos 6 tan OT 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 parallel to the plane of polarization, according as 
the crowding takes place towards or from the plane of diffraction. 

192 



292 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

In all ordinary cases of diffraction, the light becomes insensible 
at uch 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 diffracted 
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 diffraction 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 diffracted 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 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 themselves. 

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 diffraction was as 
sumed to be paraUel to the table, which was nearly the case; 
but the change in the direction of the incident light produced 



294 ON THE DYNAMICAL THEORY OF 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 diffracted 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 B on 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 diffracted light will not be sensible 
except in a direction BC, determined by the condition that AB + 
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 C 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, 1 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 -sr 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 under a microscope, the grooves 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 diffracted 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 off, 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. 



further 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-23400th part of an 
inch. 

* To separate the different spectra, Fraunhofer used a small prism with an 
angle of about 20, 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 
fracted 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 diffracted 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 Fresnel s 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 DYNAMICAL 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 diffraction, that is, in the manner of re 
flection, sufficient to neutralize the great crowding in the contrary 
direction produced by diffraction, 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 diffraction, 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 diffraction directly, or to the irregular reflection due to diffrac 
tion, or, far more probably, to a combination of the two. 

The experiments indicated indeed a marked crowding towards 
the plane of diffraction, 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 diffracted 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 
diffraction at reflection, a portion is diffracted 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 diffracted internally at such an angle that after regular 
reflection and refraction it enters the eye, a portion diffracted 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 diffracted 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 diffracted light being considered 
as both diffracted 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 diffracted at emergence, and a third in which the light 
has been diffracted externally. Any other light which enters the 
eye must have been at least twice diffracted, or once diffracted and 
at least three times reflected, and therefore will be comparatively 
weak, except perhaps when the angle of incidence, or else the 
angle of diffraction, 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 
first 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 hypothesis; but it is sufficient 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 regularly reflected or refracted light which forms 
the central colourless image belonging to a system of spectra is affected as to its 
polarization in the same way as if the surface were free from grooves. 



DESCRIPTION OF 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 diffraction 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 diffracted 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, trr 
denotes the inclination of the plane of polarization of the light 



302 ON THE DYNAMICAL THEOEY OF DIFFRACTION. 

incident on the grating to a vertical plane passing through the 
ray, that is, to a plane perpendicular to the plane of diffraction. 
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 diffracted light was perfectly plane- 
polarized when OT 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 diffracted light observed was 
red or reddish, on analyzation a blue light was seen at or near 
the minimum ; when the diffracted light was blue or blueish, 
a red light was seen at or near the minimum. When the angle 
of diffraction was moderately small, such as 1 5 or 20, the defect 
of polarization was small or insensible ; when the angle of 
diffraction was large, such as 50 or 60, the defect of polarization 
was considerable. For equal angles of diffraction, 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 angle 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 diffraction 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 diffracted 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 -& = + 45, on turning the 
Nicol s 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 -nr 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 r = + 45, and in the order red, blue, when 
^ = 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 is = + 45, and in the order blue, 
red, when w = 45. Hence the planes of polarization, or approxi- 



DESCRIPTION OF THE EXPERIMENTS. 303 

mate polarization, of the blue were more crowded towards the 
plane of diffraction than those of the red. 

On placing a narrow slit so as to allow a small portion only of 
the diffracted 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 diffraction 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 Nicol s 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 diffracted 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 
was made to disappear, or nearly so, by using a thin plate of mica 
in combination with the Nicol s 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 OT is the reading of the polarizer corrected for the index 
error 25. A reading such as 340 is entered indifferently in the 
column headed "w" 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 + 90, and any reading such as - 20 is entered 
indifferently as 20, +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 diffraction was large, except 
when OT 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 
6, the angle of diffraction, 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 arid 
12. In these experiments the three principal images already 
described were observed separately. The angle of diffraction is 
measured from the direction of the regularly reflected ray, so that 
i is the angle of incidence, and i + 6 the angle of reflection, or, in 
the case of the images which suffered one internal reflection, the 
angle of emergence. 

The eleven experiments which are not found in the following 
tables consist of five on diffraction by reflection, which did not 
appear worth giving on account of the superposition of different 
images ; one on diffraction 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 different images; one on 
diffraction 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 diffraction, (about 75,) with which the obser 
vations were attempted, that the details are not worth giving ; one 
on diffraction by refraction, in which the different 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 
furnishes no criterion by which to judge of the errors produced by the small 
changes in the direction of the light incident on the grating, but only of those 
which arise from the vagueness of the object observed. The reader will be much 



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 CT 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 polarized, except near the standard 
points, that is to say when -& was equal to 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 1. 

No. 18. a determined by two observations, which, generally 
speaking, agreed well with one another. For OT = 90 and 
-53- = +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 experiments given in the next section. 

s. ii. 20 



306 



ON THE DYNAMICAL THEORY OF DIFFRACTION. 



No. 11. a determined by two observations, which agreed well 
with one another. In the table, a (1), a (2), a (3) refer respec 
tively to the first, second, and third of the three principal images 
already mentioned. In this experiment the polarizer was reversed, 
that face being turned towards the mirror which in the other 
experiments was turned towards the grating, which is the reason 
why a and r increase together, although the light observed 
suffered one reflection. The same index error as before, namely 
25, is supposed to belong to the polarizer in its reversed position. 

No. 12. a. determined by three observations. The largeness 
of the angle of diffraction rendered the determination of a very 
uncertain. 



TABLE I. 



w 


a. 


57 


a. 


. 


a. 


-G7 


a 








No. 15, continued. 


Experiment, No. 1. 
Grooved face from 


No. 2, t 

- 5 


ontinued. 
+ 2020 


Experiment, No. 14. 

Grooved face from 
Polarizer. 


- 30 
- 40 


- 11555 
- 12425 


Polarizer. 


+ 5 
+ 15 


+ 3055 
+ 4055 


= 2957 . 


- 50 
- 60 


- 13341 
- 14029 


57o . 


+ 25 


+ 5045 


- 50 


+ 2225 


- 70 


- 14818 


-115 

- 70" 


- 7641 
5256 


+ 35 
+ 45 

+ 55 


+ 6145 
+ 7055 
+ 8215 


- 40 
- 30 
20 
- 10 


+ 4140 
+ 5155 
+ 6237 


- 80 
- 90 


-15250 
- 15830 


Experiment, No. 17. 


47| 
- 25 


- 6o2 





+ 10 


+ 8147 


Grooved face towards 
Polarizer. 


- 240 


+ 3751 


Experiment, No. 13. 


+ 20 


+ 9347 


= 5045 . 


+ 20 


+ 61 b 5 

_i_ ftQO^J/ 


Grooved face towards 


+ 30 


+ 10310 


- 90 


+ 7715 


+ 65 


+ 10646 


Polarizer. 
f) Qno;c<v 


+ 40 

+ 50 


+ 12242 


- 80 
- 70 


+ 8530 
+ 9312 








+ 70 


+ 143 


- 60 
- 50 


+ 10P15 

+ 10947 


Experiment. No. 2. 


- 60 


- 6 5 


+ 80 


+ 15247 


- 40 




Grooved face from 


- 50 
40 


+ 453 
+ 1552 


+ 90 
+ 100 


+ 16157 


- 30 


+ 12957 


Polarizer. 

= 5023 . 


- 30 
- 20 
- 10 


+ 25 
+ 3325 
+ 46 5 


+ 110 

+ 120 
+ 130 


+ 18252 
+ 19147 
+ 20212 


Experiment, No. 18. 

Grooved face towards 
Polarizer. 


-105 


- 80 





+ 5635 


+ 140 


+ 21142 




- 95 


- 7025 


+ 10 


+ 6750 






\) 21 39 . 


- 85 

- 75 


- 5130 


+ 20 
+ 30 


+ 7658 
+ 8755 


Experiment, No. 15. 


- 90 
45 


- 10323 
5953 


- 65 




-1- 40 


+ 9927 


Grooved face towards 




- 1258 


- 55 


- 2915 


+ 50 


+ 10830 




+ 45 


+ 3337 


- 45 


- 20 5 


+ 60 


+ 12035 


$ = 5952 . 


+ 90 


+ 7727 


- 35 


- 955 


+ 70 


+ 129 2 





- 6810 


+ 135 


+ 120 2 


- 25 


+ U 20 


+ 80 


+ 13742 


- 10 


- 81 


+ 180 


+ 16757 


- 15 


+ 10015 


-f 90 


+ 14657 


- 20 


- 9223 


+ 225 


+ 21410 



DISCUSSION OF THE NUMERICAL RESULTS. 

TABLE I. (continued). 



307 



-or 


a 


m 


a 


m 


*(D 


a (2) a (3) 


Experiment, X o. 21. 


X o. 22, continued. 


Experiment, Xo. 11. 


Grooved face towards 
Polarizer. 

Red glass used. 


-135 
- 1-20 


- 14025 
- 12445 


i=1450 ; = 22030 . 




-105 


- 11040 


-105 


- 11335 


- 117050 




a==2826 , 


- 90 


- 9655 


- 85 


-.103 5 


-101 


- 102 2(y 


- 90 


- 29 


75 


- 8332 


- 65 


- 90 


- 83 5 


- 89 


- 75 


- 16 2 


- 60 


- 69 7 


- 45 


- 7840 


- 6355 


- 7450 


- 60 


- 212 


- 45 


- 5450 


- 25 


- SSOotf 


- 44 


- 5319 


- 45 


+ 1235 


- 30 


- 3855 


- 5 


- 25 5 


- 2110 


- 2310 


- 30 


+ 2752 


- 15 


- 22W 


+ 15 


+ 1315 


+ 125 


+ 755 


1 ^0 


i A J.OJ.7 




+ 35 


+ 3835 


+ 24 5 


_i_ QOO 


10 




T 4.-1 - i 

+ 6140 

4- 7fi9 V 


Experiment, X o. 23. 


+ 55 


+ 5350 


+ 4310 


~T~ O^ 


+ 30 
+ 45 


T^ i O 2O 

+ 9218 
+ 10725 


Grooved face towards 
Polarizer. 


Experiment, Xo. 12. 


+ 60 


+ 12230 


Red glass used. 


i=9l - 0~ 5339 


+ 75 


+ 137 


= 54053 . 




+ 90 


+ 15132 





- 6W 


- 25 
- 45 


+ 535 
+ 15 


- 32 
940 


- 13045 

+ 2 


Experiment, X o. 22. 

Grooved face from 
Polarizer. 


+ 30 
+ 45 
+ 60 , 


-f 11 5 
+ 2755 
+ 42 Q 30 

+ 5822 


- 90" 
-135 


+ 2615 
+ 34030 


+ 2615 
+ 65 


+ 2615 




0=5538 . 


+ 75 + 71 5 ; 
+ 90 -i- 83 Q 22 




-180 


-187 2 


+ 105 


+ 9612 




-165 


- 17037 


+ 120 


+ 108 Q 30 




-150 


- 15430 


+ 135 


+ 12245 





SECTION II. 

Discussion of the numerical results of the experiments^ with 
reference 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 uncrystallized medium, if -or, 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 tan tzr (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- 

202 



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 first 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! 4- e. 
The regular way to determine e and m would no doubt be to 
assume an approximate value e, of e, put e = e 1 + Ae x , where Ae, is 
the small error of e lt form a series of equations of which the 
type is 

tan (a ej sec 2 (a ej Ae A = m tan r, 

and then combine the equations so as to get the most probable 
values of Ae t and m. But such a refinement would be wholly 
unnecessary 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 being 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 r selected for observation are symmetrically 
taken with respect to one of the standard points, a small error in e 
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 KESULTS. 309 

Suppose the values of a to have been determined by experiment 
corresponding to the following values of OT, 15, 0, + 15,... + 75, 
+ 90, + 105. The value of a for = 0, and the mean of the 
values for -15 and ta- = + 15, furnish two values of e; and the 
value of a. for vr = + 90, and the mean of the values for w = + 75 
and ts = + 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 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 2 be the 
readings of the analyzer for tar = 10, iv = + 5, assuming 

tti = e _ 10 - 2x t a z = e + 5 + #, we get 3x = a 2 - a 1 - 15, 

whence e, which is equal to 2 5 x, is known. The index 
error of the analyzer having been thus determined, it remains to 
get the most probable value of ra from a series of equations of the 
form (48). For facility of numerical calculation it is better to put 
this equation under the form 

log m = log tan a - log tan -& .................. (49) , 

where it is to be understood that the signs of a and w 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 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 -f 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 ) 2 , 
we get for the true equation which ought to replace (49), 



, , , . 

log m = log tan a + -: r ^ - log tan r, 

M being the modulus of the common system of logarithms. Since 
the effect of the error Aa is increased by the division by sin 2* , a 
quantity which may become very small, in combining the equations 



310 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

such as (49) I have first multiplied the several equations by 
sin 2 a , or the sine of 2 (a e) 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 2a , 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 2x for 
multiplier instead of sin 2 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 e 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 tzr 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 OT, of the 
observed angles a, -or, 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 OT there 
employed. 
Experiment No. 1.0 = 575 ; assumed index error e = 405 . 

-si a OT 

-115 -146 

- 92J - 03r 
-70 

- 47i + 033 

- 25 - 014 
_ 2J + 016 
+ 20 + 1 

+ 42i + 019 

+ Ho + 141 



DISCUSSION OF THE NUMERICAL RESULTS. 



311 



The values of a for w = 115 and CT = -f- 65 ought to differ by 
180, whereas they differ by 327 more. This angle is so large 
compared with the angles OLW given just above, that it seems 
best to reject the experiment. The experiment is sufficient 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 ra, 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. 


Experiment, No. 2. 


= 2957 


= 5023 


e=+7223 


e^ + 2412 / 


logm=+-009 


log m = + -010 




ex t<r 




a to 


o7 


calc. 


obs. 


diff. 


TZ 


calc. 


obs. 


diff. 


- 50 


-0-6 


o-o 


+ 0-6 


-105 


+ 0-3 


-0-3 


-0-6 


- 40 


-0-6 


-1-1 


-0-5 


- 95 


+ 0-1 


-0-7 


-0-8 


- 30 


-0-5 


-0-7 


-0-2 


- 85 


-0-1 


-l-5 


-l-4 


- 20 


-0-4 


-0-5 


-0-1 


- 75 


-0-3 


-l-8 


-l-5 


- 100 


- 0-2 


+ 0-2 


+ 0-4 


- 65 


-0-5 


-l-5 


-1-0 





o-o 


-l-2 


-l-2 


- 55 


-0-6 


+ 0-4 


+ 1-0 


+ 10 


+ 0-2 


-0-6 


-0-8 


- 45 


-0-7 


-0-4 


+ 0-3 


+ -20 


+ 0-4 


+ l-4 


+ 1-0 


- 35 


-0-6 


-0-2 


+ 0-4 


+ 30 


+ 0-5 


+ 0-8 


+ 0-3 


- 2o 


-0-5 


o-o 


+ 0-5 


+ 40 


+ 0-6 


+ 0-9 


+ 0-3 


- 15 


-0^-3 


o-o 


+ 0-3 


+ 50 


+ 0-6 


+ 0-3 


-0-3 


- 5 


-o-i 


o-o 


+ o-i 


+ 60 


+ 0-5 


+ 0-3 


-0-2 


+ 5 


+o-i 


+ 0-6 


+ 0-5 


+ 70 


+ 0-4 


+ 0-6 


+ 0-2 


+ 15 


+ 0-3 


+ 0-6 


+ 0-3 


+ 80 


+ 0-2 


+ 0-4 


+ 0-2 


+ 25 


+ a-5 


+ 0-4 


-o-i 


+ 90 


0-0 


-0-4 


-0-4 


+ 35 


-fO-6 


9 


? 


+ 100 


-0-2 


-0 5 


-0-3 


+ 45 


+ 0-7 


+ 6-4 


-0-3 


+ 110 


-0-4 


+ 0-5 


+ 0-9 


+ 55 


+ 0-6 


+ l-9 


+ l-3 


+ 120 


-00-5 


-0-6 


-0-1 










+ 130 


-O y -6 


-0 0< 2 


+ 0-4 










+ 140 


-0-6 


-0-7 


-0-1 











312 



ON THE DYNAMICAL THEORY OF DIFFRACTION. 



TABLE II. (continued). 



Experiment, No. 22. 
= 5538 
e= - 727 
logw = + -035 


Experiment, No. 13. 
= 3950 
c = 5650 
loRm= + -084 


w 


calc. 


a TJJ 
obs. 


diff. 


OT 


calc. 


a m 
obs. 


diff. 


-180 
-165 
-150 
-135 
- 120 
- 105 
- 90 
- 75 
- 60 
- 45 
- 30 
- 15 


o-o 

+ l-2 
+ 2-0 
+ 2-3 
+ 2-0 
+ 1-1 

o-o 

-1-1 
- 2-0 
-2-3 
-2-0 
-l-2 


o-o 

+ l-4 
+ 2-5 
+ l-6 
+ 2-2 
+ l-3 
+ 0-1 
-10-5 
-2-l 
-2-8 
-l-9 
-0-8 


0-0 

+ 0-2 
+ 0-5 
-0-7 
+ 0-2 
+ 0-2 
+ 0-1 
-0-4 
-0-1 
-0-5 

+ o-i 

+ 0-4 


- 60 
- 50 
- 40 
- 30 
- 20 
- 10 

+ 10 
+ 20 
+ 30 
+ 40 
+ 50 
+ 60 
+ 700 
+ 80 
+ 90 


-l-9 

-2-2 
-2-2 
-2-0 
-l-5 
-0-8 

o-o 

+ 0-8 
+ l-5 
+ 2-0 
+ 2-2 
+ 2-2 
+ l-9 
+ l-4 
+ 0-7 

o-o 


-0-9 
-l-9 
-1-0 
-l-8 
-3-4 
-00-7 
-0-2 
+ 1-0 
+ 0-1 
+ 10-1 
+ 2-6 
+ l-7 
+ 3-7 
+ 2-2 
+ 0-9 

+o-i 


+ 1-0 

+ 0-3 
+ l-2 
+ 0-2 
-l-9 
+ 0-1 
-0-2 
+ 0-2 
-l-4 
-0-9 
+ 0-4 
-0-5 
+ l-8 
+ 0-8 
+ 0-2 

+o-i 


Experiment, No. 18. 
= 2139 
e=-1244 
logm= + -029 


Experiment, No. 17. 
= 50 45 X 
e=+ 167015 
logm=+-122 


w 


calc. 


a TZ 
obs. 


diff. 


- 90 
- 45 

+ 45 
+ 90 
+ 135 
+ 1800 
+ 225 


o-o 

-l-9 

o-o 

+ l-9 
0-0 
-l-9 

o-o 

+ l-9 


-0-6 
-2-l 
-0-2 
+ l-3 
+ 0-2 
-2-2 
+ 0-7 
+ l-9 


-0-6 
-0-2 
-0-2 
-0-6 
+ 0-2 
-0-3 
+ 0-7 

o-o 


or 


calc. 


a! TX 
obs. 


diff. 


- 900 
- 80 
- 70 
- 60 
- 500 
- 40 
- 30 


o-o 

-2-4 
-4-6 
-6-4 
-7-6 
-b-0 
-7-4 


o-o 

- P-7 
- 4-0 
- 6-0 

7 0> 5 


o-o 

+ 00-7 
+ 0-6 
+ 0-4 
+ 00-1 
- 2-0 
+ 0-1 


Experimer 



it, No. 21. 

2826 
6049 
+ 039 


- 10-0 

- 70-3 




logra = 


Experiment, No. 23. 


OS 


calc. 


a -nr 
obs. 


diff. 




V Ui"OO 

= -727 
log m = + -082 




77 


calc. 


a! -w 
obs. 


diff. 


- 90 
- 75 
- 60 
- 45 
- 30 
- 15 

+ 15 
+ 30 
+ 45 
+ 600 
+ 75 
+ 90 


o-o 

-l-2 
-2-2 
- 2-6 
-2-3 
- P-3 

o-o 

+ l-3 
+ 2-3 
+ 2-6 
+ 2-2 
+ l-2 

o-o 


+ 0-2 
-l-6 
-3-0 
-3-2 
-2-9 
-1-0 
+ 0-8 
+ 2-6 
+ l-5 
+ l-6 
+ l-7 
+ l-2 
+ 0-7 


+ 0-2 
-0-4 
-0-8 
-0-6 
-0-6 
+ 0-3 
+ 0-8 
+ l-3 
-0-8 
-1-0 
-00-5 

o-o 

+ 0-7 



+ 15 
+ 30 
+ 45 
+ 600 
+ 75 
+ 90 
+ 1050 
+ 120 
+ 135 


o-o 

+ 2-9 
+ 4-3 
+ 5-4 
+ 4-4 
+ 2-5 

o-o 

-2-5 
-4-4 
.50.4 


+ 0-2 
+ 2-7 
+ 4-6 
+ 40-2 
+ 5-0 
+ 3-5 

o-o 

-3-l 
-4-8 
-5-6 


+ 0-2 
-0-2 
+ 0-3 
-l-2 
+ 0-6 
+ 1-0 

o-o 

-0-6 
-0-4 
-0-2 



DISCUSSION OF THE NUMERICAL RESULTS. 



313 



TABLE II. (continued}. 



Experiment, No. 15. 
= 5952 
e=- 68015 
logm=:+-225 



UJ 


calc. 


a tzr 

obs. 


diff. 





10 
20 
30 
40 
50 
60 
70 
80 
90 



0-0 
- 6-5 



- 13-4 



- 7-8 

- 4-0 

o-o 



- 2-7 

- 4-l 

- 17-7 



15-4 i 

12-2 

10-0 

4-6 

0-2 



+ 3-8 
+ 7-3 
-3-6 
-l-6 
- 2-0 
-l-2 
-2-2 
-0-6 
-0-2 



Experiment, No. 11. 
1450 ; = 2230 ; =-lo30 . 



First Image, 
log 7/i = + -289. 



uu 



calc. obs. diff. 



Second Image, 
log m = + -061. 



calc. obs. diff. 



Third Image. 



calc. obs. diff. 



-105 ! + 7-l + 6-9 -0-2 

- 85 - 2-4 - 2-6 -0-2 

- 65 -ll-5 - 9 0> 5 +2-0 

- 45 -17-8 -18-2 -0-4 

- 25 -17-2 -18-3 -1-1 
_ 50 _ 40.5 _ 40.5 -0 

+ 15 +12-5 -f!3-7 +1 2 

+ 35 +18-7 +19-1 +0-4 

+ 55 +15-2 +14-3 -0-9 



+ 2-7 H-0-8 

-0-7 -0-5 +0-2 

-2-9 -2-6 +0-3 

-4-0 -3-4 +0-6 

-3-2 -3-5 -0-3 

-0-7 -0-7 0-0 

+ 2^*1 +1^*9 0" 2 

+ 3-9 +4-6 +0-7 

+ 3-7 +3-7 0-0 



- 8-9 
- 13-3 



- 8-5 
- 14-3 



- 3-0 
+ 8-4 
+ 13-6 



- 2-7 
+ 8-4 



+ 0-4 

-0-8 
+ 0-3 

o-o 

H-0-4 



Experiment, No. 12. 
t = 9 l ; = 5339 ; e^- 



First Image. 



calc. obs. diff. 



Second Image, 
log m = + -122. 



a -m 
calc. obs. diff. 



Third Image. 
log 77i = + -366. 



CL "" 27* 

calc. obs. diff. 



+ 25 +44-4 +44-3 -0-1 

+ 45 +35-l +33-7 - 1 4 

+ 90 0-0 0-0 0-0 

+ 135 -35-l -36-7 - 1 6 



+ 6-7 +6-7 

+ 7-9 +9-l 

0-0 0-0 

-7-9 -6-2 



0-0 



0-0 



+ 22-3 +25-0 +2-7 

+ 21-7 +21-7 0-0 

0-0 0-0 0-0 

-21-7 -20-0 



314 ON THE DYNAMICAL THEORY 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 unusually large, 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 w. 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 us 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 
1 J, + 1J, +2, for the first, second, and third images respec 
tively. The largest error in the third columns, 2*7, is for w 
= + 25, third image. The three readings by which a was deter 
mined in this case were 15, 1330 , 12? Hence the error 
4- 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 l , a 2 , 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 6 the 
angle of refraction corresponding to the angle of incidence 6, so 
that sin = //, sin , JJL 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 a^ = sec 6 tan OT, and by the known formula 
applying to refraction we have tana 2 = cos (6 6 } tan a l5 tana 
= cos (6 6 } tan or 2 , whence tan a m tan OT, where 



On the second supposition, if cq be the azimuth after diffraction 
at an angle & within the glass, we have tan (X 1 = sec & tan or, 
tan a = cos (6 6 ] tan o^, whence tan a = m tan OT, where 

m = sec cos (6 6 ). 

On the third supposition we have tan a = m tan or, where 

m = sec 6. 

If we suppose the vibrations parallel to the plane of polarization, 
we shall obtain the same formulas except that cos 6, cos & will 
come in place of sec 0, sec 6 , the factor cos (6 &} 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 
6 0, m = 1, and log m = 0, in all cases. In calculating the table, 
I have supposed //, = 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 logm from theory, p being supposed 
equal to To 206. 





Vibrations supposed 


Vibrations supposed 




perpendicular to the plane 


parallel to the plane of 




of polarization. 


polarization. 


$ 


I 


II 


III 


I 


II 


III 


5" 


+ 001 


+ 001 


+ -002 


- -002 


-001 


- -002 


10 


+ 005 


+ 002 


+ -007 


- -008 


-004 


- -007 


15 


+ 011 


+ 004 


+ -015 


- -019 


-008 


- -015 


20 


+ 020 


+ 008 


+ -027 


- -034 


-015 


- -027 


25 


+ 032 


+ 012 


+ -043 


- -053 


-023 


- -043 


30 


+ 047 


+ 017 


+ -062 


- -078 


-033 


- -062 


35 


+ 065 


+ 022 


+ -087 


- -109 


-044 


- -087 


40 


+ 086 


+ 028 


+ -116 


- -146 


-058 


- -116 


45 


+ 111 


+ 033 


+ -150 


- -190 


-073 


- -150 


50 


+ 139 


+ 037 


+ -192 


- -244 


-090 


- -192 


55 


+ 173 


+ 040 


+ -241 


- -310 


-109 


- -241 


60 


+ 214 


+ 040 


+ -301 


^ -388 


-129 


- -301 


65 


+ 262 


+ 039 


+ -374 


- -486 


-151 


- -374 


70 


+ 324 


+ 034 


+ -466 


- -608 


-175 


- -466 


75 


+ 408 


+ -022 


+ -587 


- -766 


-202 


- -587 


80 


+ 533 


+ 005 


+ -760 


- -987 


-231 


- -760 


85 


+ 773 


-022 


+ 1-060 


- 1-347 


-265 


- 1-060 


90 


+ oo 


-059 


+ oo 


- 00 


-305 


- 00 



DISCUSSION OF THE NUMERICAL RESULTS. 



317 



TABLE IV. Values of log m from observation. 



Nature of Experiment. 


No. 


e 


log m 






Diffraction at refraction. 
Incidence perpendicular. 


14 


2957 


+ 009 


Grooved face of glass 


2 


rnO>>Q 


+ 010 


plate turned from the 
incident light 


22 ; 5438 +-035 


Diffraction at refraction. 
Incidence perpendicular. 
Grooved face of glass 
plate turned towards the 


18 
21 
13 
17 


2l39 
2826 
3950 
5045 


+ -029 
+ 039 
+ 034 
+ 122 


incident light. 


23 


5453 +-082 




15 5952 + -225 



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 collusiveness 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 @ be the value of OL ix for TX 45, $ = 1 when log m = 
"015, nearly; and when log m is not large,/? is nearly propor 
tional to log m. In this case yS is nearly the maximum value 
of a -GT. 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 
0=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 @ 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 THEORY OF DIFFRACTION. 



which belong to diffraction within the glass plate, are large com 
pared with the errors of observation ; for the probable cause of 
the large errors in experiment No. 15, has been already mentioned. 
In the following figure the abscissa} of the curves represent 
the angle of diffraction, and the ordinates 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 parallel,} to the 
two hypotheses respecting the direction of vibration. The dots 
represent the results of the experiments in which the grooved 
face 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 
5th 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 w r as 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 marked I. E, and the latter a very little below the 
curve II. E. 

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 should 
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. ..<>. ?h 

Let aA, bB be sections of two consecutive i 7 ^Ls . \ 

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 OR8 in the direction of a ray proceeding regu 
larly from and entering the eye ; so that OR, RS are inclined to 
the normal at angles 6, 6 , or 6 , 6, 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 diffraction at the 
edge A or B. 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 RS, 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 AB is not effective in producing illu 
mination in the direction RS. For let G be the centre of AB, 
and through C draw a plane perpendicular to RS, and then draw 
a pair of parallel planes each at a distance |X from the former 
plane, cutting AB in M t , N v another pair at a distance X, and 
cutting AB in M v N Z) 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 RS 
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 JX, 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 X ; 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. 321 

we may suppose A M = ^X. 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 be 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 A or B 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 /JL 1 ; 
and although this amount is far from insensible in the actual 
experiments, it is still not sufficiently 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 "A" 
in the first figure are well separated from those marked " E", and 
if jj, were to approach indefinitely to 1, the curves I. A and II. A 
would approach indefinitely to III. 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. n. 21 



322 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

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 i t p be the angles of refraction corresponding to the angles 
of incidence, i,i+9. Then in the case of the first image the 
tangent of the azimuth of the plane of polarization is multiplied 
by cos (i -f 6 p) sec (i + 9 + 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 (i i] sec (i + i ) in consequence 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 have 

for Istimage, log 771 = log m + log cos (i + 9 -H p) 2 log cos (i +0 p), 
for 2nd image, log m = log m + log cos (i + i ) 2 log cos (i 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 the 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 of incidence in glass, and 9 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 = 1450 , = 22 30 , whence i" = 9 41 , /> = 2330 . We 
thus get 

for 1st image, log m = + 289 - 286 = + 003, 
for 2nd image, log m = + -061 - 037 = + 024. 

The positive values of log m which result from these experi 
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 accompanied 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 /t, i, and would produce large 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, or is the angle of 
incidence, r 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 acute angle made by the regularly reflected light 
with the plane of the grating. The following 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 
e = 0-0001223 there is polarized : (Er} ( + l} , that is, the green part of 
this first spectrum, when cr = 49; (Er) (+u \ or the green part in 
the second spectrum lying on the same side of the axis, when 
cr = 40; lastly, (Er) ( ~ l) , or the green part of the first spectrum 
lying on the opposite side of the axis, when cr=69. When 
(JT) (+I) is polarized perfectly, the remaining colours of this spec 
trum are still but imperfectly polarized. This is less the case 
with (ErY +Il \ and cr can be sensibly changed while this colour still 
remains polarized. (Er) (-I) is under no angle of incidence so com 
pletely polarized (so ganz vollstandig polarisirt) as (Er) (+I} . With 
a grating in which e is greater than in that here spoken of, the 
angle of incidence would have to be quite different in order that 
the above-mentioned spectra should be polarized*." 

* Gilbert s Annalen der Physik, B. xiv. (1823) S. 364. 

212 



324 ON THE DYNAMICAL THEORY OF DIFFRACTION. 

If we suppose v v a function of v such that cr_ 1 = 69, cr +1 = 49, 
cr +2 = 40, we get by interpolation CT O = 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 yu, = 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 cr + z^V/e*, given by Fraunhofer, and 
obtained from the theory of interference; and = 90 T cr, 
where is the angle of diffraction. We thus get for green light 
polarized by reflection and the accompanying ditfraction, 

order of spectrum cr cr + 9 

-1 69 -18 13 50 47 

58 40 58 40 

+ 1 49 +17r G6l 

+ 2 40 +33 52 73 52 . 

If we suppose the formula (49) to hold good in this case, m 
becomes infinite for the angles of incidence cr and the correspond 
ing angles of reflection cr + contained in the preceding table. 

Another observation of Fraunhofer s described in the same 
paper deserves to be 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 [. 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 diffracting edge, how is it possible to build any certain con- 

* In Fraunhofer s notation the wave length is denoted by u\ 
t Gilbert s Anrialen der Pliysik, B. xiv. p. 353. 



DISCUSSION OF THE NUMERICAL RESULTS. 



,325 




elusions on an investigation in which 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 GB, HD, IE represent incident 
rays, or normals to the incident waves, 
which are supposed plane. Let BE = e, 
and BD : DE :: nil- n. Draw BK, 
DL, E^I parallel to one another in the 
direction of the spectrum of the order v 
on the one side of the axis, so that v\ is the retardation of the 
ray EM relatively to BK, and therefore sin 6 = i/X/e, being the 
angle of diffraction, or the inclination of BK to GB produced. 
Draw BN, FO, EP at an inclination 6 on the other side of the 
axis, and let L DBF = a. Then the retardation of DL relatively 
to BK is equal to nv\ or ne sin 6, and that of BN relatively to 
FO is equal to ne sin 6 + ne tan a cos 6 ne tan a, so that if we 
denote these retardations by 

M I} _R 2 , R i = ne sin 9, R^ = ne sin ne tan a versin 6. 

Let p lt p z be the greatest integers contained in the quotients of 
R lt R 2 divided by X, and let JR 1 =^i X + r i R * = P^ + r * Tn en 
the relative intensities of the two spectra of the order -f v and v 
depend on r lt r 2 : in fact, we find for the ratio of intensities, on 
the theory of interference, sin ? irrj\ : sin 2 irrJX. 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 coefficients 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 l |, gave for the double angle 
113 20 , whence /j,= 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 affected 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 empirical), 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 affirm 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 sufficient 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 anticipated 



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 moderne, 
p. 190, and will be found in the fifteenth Volume of the Comptes 
Rendus, where two short memoirs of M. Cauchy s on the subject 
are printed, the first of which begins at p. 6Qo, 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 
diffracted. 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 diffraction, 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 difficult, 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 appearance 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 (1856) 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 diffracted light, 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 
from 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.] 

* PoggendorfFs Annalen, Vol. Ill (1860) p. 315, or Philosophical Magazine, 
Vol. 21, p. 321. 



[From the Transactions of the Cambridge Philosophical Society. Vol. ix. 

Part L] 



OX 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- 

r 30 TT 

volving the square of the definite integral I cos ^ fa 3 mw] dw y 

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 = 4 to m = + 4, at 
intervals of O2, which was calculated by the method of quadratures. 
In a Supplement to the same paper ( 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 spurious rainbows, 
whereas Professor Miller was able to observe 30 of these bands. 
To attempt the computation of 30 roots of the equation W by 

* Camb. Phil. Trans. Vol. vi. p. 379. t Vol. vm. p. 595. 



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 function when m is large. It is very easy to 
deduce from this expression a formula which gives the ^ th root of 
the equation IF=0 with hardly any numerical calculation, except 
what arises from merely passing from (m/3)f, the quantity given 
immediately, to m itself. 

The ascending series in which IF may be developed belongs to 
a class of series which are of constant occurrence in physical ques 
tions. These series, like the expansions of e~ x , 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 x = oo . 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 we have got a series of this class, and let 
the series be denoted by y QT 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 t 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 
easily find what terms are of most importance when x is large: 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 coefficients 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 in /(a?) 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 iny*(#). 

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 definite 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. 

* Traite 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 W, the second to an 
infinite series which occurs in a great many physical investigations, 
the third to the integral 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 



W I cos (w 3 mw) dw (1), 

J ^ 



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 integral 



/.QO 

U= I ^( 

> o 

where 6 is supposed to lie between ?r/6 and -f Tr/6, in order that 
the integral may be convergent. 

Putting x= (cos 9 V 1 sin#) z, 

we get dx = (cos 6 V 1 sin 6) dz, and the limits of z are and oo ; 
whence, writing for shortness 

p = (cos 26 + J~l sin 20) n .................. (.3), 

.00 

we get tt = (co80-/-l sintf) I e"^~^ dz* ............ (4). 

J 



* The legitimacy of this transformation rests on the theorem that if f(x) be a 
continuous function of x, which does not become infinite for any real or imaginary, 
but finite, value of x, we shall obtain the same result for the integral of f(x)dx 
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 now 0, which hitherto has been supposed less than Tr/6, 
become equal to TT/(J. The integral obtained from (2) by putting 
= 7T/6 under the integral sign may readily be proved to be con 
vergent. But this is not sufficient in order that we may be at 
liberty to assert the equality of the results obtained from (2), (4) by 
putting 6 TT/ O before integration. It is moreover necessary that 
the convergency of the integral (2) should not become infinitely 
slow when 6 approaches indefinitely to TT 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 9 becomes equal to Tr/6*. This may be readily 
proved in the present case, since the integral (2) is even more 
convergent than the integral 



I 



DO 

- \ - 1 sin 30 x 3 - nx j 

e dx, 



which may be readily proved to be convergent. 
Putting then = 7r/6 in (2) and (4), we get 

cos (a? nx) dx \/ I | sin (x 3 nx) dx : (5), 



f. 



u= cos^-V-lsJn^ e-V-*>d* (6), 



C 



. 7T 



where p ( cos ^ -f V 1 sin ) n .................... (7). 



o 

Let u=U-\/^\U , 

and in the expression for ZJgot from (5) put 

7T\ 



w, n - 

then we get Tf = C\ U (9). 

the limits of .r are and real infinity, and accordingly we may first integrate with 
respect to z from to a large real quantity z l , 6 (which is supposed to be written 
for 6 in the expression for x) being constant, then leave z equal to z l , make 6 vary, 
and integrate from 9 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 z 1 becomes infinite, and consequently we have 
only to integrate with respect to z from to real infinity. 

* See Section m. of a paper "On the Critical Values of the sums of Periodic 
Series." Camb. Phil. Tram. Vol. vm. p. 5G1. [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), we 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 pz in (6) we should obtain u, and there 
fore U t 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) 



1 / 7T , - 7T\ 

= - cos - - V - 1 sin - , 
S\ 6 67 



which becomes by (7) 



Equating to zero the real part of the first member of this 
equation, we get 



5. We might integrate this equation by series according to 
ascending powers of n, and we 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 convenient 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 w/3 will be very small ; 
and if we regard this coefficient as constant, and &n as variable, 
we shall get for the integral of (11) 

N Bm{^(?).Bn} ... (12), 

where N, N are regarded as constants, Sn 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 f (n) in the 
true integral ; and since V(w/3) . &n must be the small increment 
of f (n), we get f(ri) =| V(n 8 /3), 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 
f </(n */3) in the 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= 6 3 N/-T{^ln + Bnft + Cn* +...}* ............ (13), 

where A, B, C... x, j3, y... are constants which have to be deter 
mined. Differentiating, and substituting in (11), we get 

a (a - 1) An*- 2 + (-)! Brf-* + ... 



+... } = 0. 

As we want a series according to descending powers of n, we 
must put 



* The idea of multiplying the circular functions by a series according to de 
scending powers of n was suggested to me by seeing in Moigno s Repertoire d optique 
moderne, p. 189, the following formulas which M. Cauchy has given for the calcu 
lation of Fresnel s integrals for large, or moderately large, values of the superior 
limit : 

cos - z-dz = \ - N cos - Hi 2 + M sin ^ m- ; 
o * * * 

f m . 7T 7T 7T 

I sin - z-dz = I, - J/ cos - m* - N sin - m- ; 
./ o 2 2 ^ 

1 1.3 1.3.5.7 1 1.3.5 

where M= ^ + . . . ; N= , _ - + . .. 

mir ii^ii 3 77i 9 7r wV 2 ? 7 jr 4 

The demonstration of these formula? will be found in the 15th Volume of the 
Comptes Eendus, pp. 554 and 573. They may be readily obtained by putting 
irz"=2x, and integrating by parts between the limits ^wm- and oo of x. 



336 ON THE NUMERICAL CALCULATION OF A CLASS OF 



whence 



F.^-Mv-Tn-Vi* 1 ^ 1 ^ [ 



1.2.3 






By changing the sign of *J( 1) both in the index of e and in the 
series, writing B for A, and adding together the results, we shall 
obtain the complete integral of (11) with its two arbitrary con 
stants. The integral will have different forms according as n is 
positive or negative. 

First, suppose n positive. Putting the function of n of which 
A is the coefficient, at the second side of (14) under the form 
P + V( 1) Q> an d 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 



U = An-* R cos + 8 sin 

...... (15), 



where 

1.5.7 .Jl_ 1 . 5 . 7 . 11 . 13 . 17 . 19 . 23 

R = ! " 1 . "2 Tl 6 2 7"3n 3 + 1 . 2.3.4 . 1 6 4 . "3V 

1.5 _ 1 .5.7.11.13.17 

o T" " i r^imi o, i 



,...(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 



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 ri 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, 
0,... a, /3, 7... in (13) were determined, it is evident that if we 
form the expression 



according 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 ri for which we are 
calculating the series numerically, and, much more, for all superior 
values. Suppose the arbitrary constants A, B in (16) determined 
by means of the ultimate form of U for n = so , and C in (17) by 
means of the ultimate form of U for n = oo . Then U t satisfies 
exactly a differential equation which differs from (11) by having 
the zero at the second side replaced by a quantity which is in 
sensible for the value of n or ri with which we are at work, and 
which is still smaller for values comprised between that and the 
particular value, (namely x ), by means of which the arbitrary 
constants were determined so as to make C^ and U agree. Hence 
L\ 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 differential equation which U^ would satisfy exactly would 
differ sensibly from (11), and the value of 7J obtained would be 
faulty. 

s. n. -22 



338 ON THE NUMERICAL CALCULATION OF A CLASS OF 

8. It remains to determine the arbitrary constants A, B, C. 
For this purpose consider the integral 



_ / 

J 



(18), 



where q is any imaginary quantity whose amplitude does not 
lie beyond the limits 7r/6 and + Tr/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 <x> through a series of imaginary values. 
Let then x = q + y, and we get 



-Q 

where the values through which y passes in the integration are 
not restricted to be such as to render x real. Putting y (3g)~* t, 
where that value of the radical is supposed to be taken which has 
the smallest amplitude, we get 

The limits of t are 3%^ and an imaginary quantity with an 
infinite modulus and an amplitude equal to Ja, where a denotes 
the amplitude of q. But we may if we please integrate up to 
a real quantity p, and then, putting t = pe 6 ^~ 1 \ and leaving p 
constant, integrate with respect to 6 from to Ja, and lastly put 
p = oo . The first part of the integral will be evidently convergent 
at the limit oo , since the amplitude of the coefficient of f in the 
index does not lie beyond the limits | TT and + JTT ; and calling 
the two parts of the integral with respect to t in (19) T, T 4 , we 
get 

(20), 



^ 

J o 



We shall evidently obtain a superior limit to either the real or 
the imaginary part of T 4 by reducing the expression under the 
integral sign to its modulus. The modulus is e~ where 

6 = (3c) ~* p* cos (30 - fa) + /a 2 cos 20, 
c being the modulus of q. The first term in this expression is 



DEFINITE INTEGRALS AND INFINITE SERIES. oo!) 

never negative, being only reduced to zero in the particular case 
in which 6 = and a- ir/G. The second term is never less than 
p 2 cos JTT or ip 2 , and is in general greater. Hence both the real 
and the imaginary parts of the expression of which T 4 is the limit 
are numerically less than ^ape-^ *, which vanishes when p = oc , 
and therefore T t = 0. Hence we have rigorously 

Q = (3q)-*eWT ........................ (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 
T v T 2 , T 3 , where T l is the integral taken from -3^ to a real 
negative quantity a, T 2 from a to a real positive quantity + & 
and T 3 from b to x ; and suppose c first to become infinite, a and b 
remaining constant, and lastly make a and b infinite. 

Changing the sign of t in T^ and the order of the limits, we get 

* dt. 



r 
= l 



Put =pe 0N/( ~ i; . Then we may integrate first from p = a to 
p = S*cr while 6 remains equal to 0, and afterwards from 6 = 
to 6 = a. while f p remains equal to 3M. 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 t 3 in the index by the product 
of t z and the greatest value (namely 3M) 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 oc . Hence the real and imaginary parts of T are both 

/* 
numerically less than I e~^ 2 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) i n the coefficient. Hence L will be 
such a limit if 

[ \-W>d6, where f(6) = 3 cos 20 - cos (3(9 - fa). 

^0 

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 



340 ON THE NUMERICAL CALCULATION OF A CLASS OF 

in which it is positive by changing the signs of a and 6. When 
# = 7r/6, the first term in f(0) is equal to f, which, being greater 
than 1, determines the sign of the whole, and therefore /(0) is 
positive; and /(0) is evidently positive from 6 = to 0=7r/6, 
since for such values cos 26 > . Also in general f(0) = 6 sin 20 
+ 3 sin (30 fa), which is evidently positive from 6 ?r/6 to 
= 7r /4< ) and the latter is the largest value we need consider, being 
the extreme value of when a has its extreme value 77/6. When 
has its extreme value fa,/(0) = 2 cos 3a, which is positive when 
a < 7T/6, and vanishes when a = Tr/6. Hence /(0) is positive when 
6 < fa ; for it has been shewn to be positive when 6 < Tr/6, which 
meets the case in which a < ?r/9 or = Tr/9, and to be constantly 
decreasing from 6 = Tr/6 to 9 = f 2, which meets the case in which 
6 > 7T/9. Hence when a < ?r/6 the limit of L for c = oo is zero, 
inasmuch as the coefficient of c 3 in the index of e is negative and 
finite ; and when a. = Tr/6 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 f(9) = S, regard 
|a as a function of S, F(B), and integrate from ^ = to 3 = ft, 
where yS is a constant which may be as small as we please. By 
what precedes, F (8) will be finite in the integration, and may 
be made as nearly as we please equal to the constant F (0) by 
diminishing ft. Hence the integral ultimately becomes 



/: 





which vanishes when c becomes infinite. Hence the limit of r l\ 
is zero. 

We have evidently T z < I e-^dt, 

which vanishes when b becomes infinite. Hence the limit of T 
is equal to that of T z . Now making c first infinite and afterwards 
a and 6, we get 

limit of T 2 = limit of f e~< 2 dt = | V* dt = VTT, 

J -a, J - oo 

and therefore we have ultimately, for very large values of c, 

a 8 (22). 



DEFINITE INTEGRALS AND INFINITE SERIES. 341 

In order to apply this expression to the integral u given by (6), 
we must put 

o 2 jf^i i fn\* JV^i 

oq = ne* , whence q = ^ ) e , 

o/ 



whence we get ultimately 



Comparing with (15) we get 



9. We cannot make ?i 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 \/( 1) in the coefficient 
of the integral and in the value of p y but not in the expression 
for n, 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 definition, 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 

fx 

shape, namely, | cos (z 3 - nx) dx, by a method similar to that employed in Art. 21. 



If x l be the positive value of x which renders x? -nx a minimum, we have a^s=:8~~*li*. 
Let the integral U be divided into three parts, by integrating separately from x=0 
to x = x l - a, from x = x l -a to x=x^+ 6, and from x=x l + b to oj=x ; then make n 
infinite while a and b remain finite, and lastly, let a and 6 vanish. In this 
manner the second of equations (23) will be obtained, by the assistance of the 
known formula 

30 



/*> / 

I vosx-dx=l 
J - x J - 



342 ON THE NUMERICAL CALCULATION OF A CLASS OF 

when n is imaginary. The differential equation (11), and conse 
quently the descending series derived from it, will also hold good 
when n is imaginary ; but since this series contains radicals, while 
U is itself a rational function of n, we might expect beforehand 
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 = r^e 1 "^" 1 
where n v is positive. Since both values of 2 (ft/3) 2 are employed 
in the series, with different arbitrary constants, we may without 
loss of generality suppose that value of n% which has fz/ for its 
amplitude to be employed in the circular functions or exponentials, 
as well as in the expression for S. In the multiplier we may 
always take z^/4 for the amplitude of n~$ by including in the 
constant coefficients the factor by which one fourth root of n differs 
from another; but then we must expect to find the arbitrary 
constants discontinuous. In fact, if we observe the forms of R 
and S, 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 n 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 STT, whereas U 
recurs when v is increased by 2?r. If we write v + 2?r for v, and 
observe that N and N recur, the expression (25) will become 

- J~^lAn~* N + J^lBn* N ; 

and since U recurs it appears that A, J5 become A/( 1) -4, A/( 1) J5, 
respectively, when v is increased by 2?r. 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 v to v = TT. 

If we put ?r 1 e 7rV( ~ 1) for n in the second member of equation (15), 
and write ft for 2 . 3~ f n*, and E I} S 1 for what E, S become when 



DEFINITE INTEGRALS AND INFINITE SERIES. 343 

?^ 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 (^ + 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 

tf= , ........... ..(26). 

2.3* 

This mode of determining the constant C is anything but satis 
factory. I have endeavoured in vain to deduce the leading term 
in Z7for 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 C 
will however be verified further on. 

10. Expressing n, U in terms of in, W by means of (8) and 
(9), putting for shortness 

-*&-< 

where the numerical values of ra and n are supposed to be taken 
when these quantities are negative, observing that 16 */(3n 3 ) = 
and reducing, we get when ra is positive 

TF= 2* (3m)- 

where 

1.5.7.11 1 . 5 . 7 . 11 . 13 . 17 . 19 . 23 

- 1.2 (720)* + 1.2.3.4(720)* 

1.5 1.5.7.11.13.17 

- J 



1.2.3 (720) 

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 "On the discontinuity 
of arbitrary constants which appear in divergent developments." (Transactions 
of the Cambridge Philosophical Society, Vol. x. p. 105.)] 



344 ON THE NUMERICAL CALCULATION OF A CLASS OF 

11. Reducing the coefficients of <j>~ 1 , 0~ 2 ... 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) 

2760793; 1-064829; 1-464775; 

0576490; 116099; 291592. 

Thus, for m = 3, in which case < = TT, 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 m=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 m = + 3 we have from (28) 

W= - 3-* (R-8) = - 3-* ("9965 - -0213) = - 5632, 
which agrees with Mr Airy s result 56322 or 56823. 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 R^Mcosty, $ = I/sin -^ and ex 
panding M and tan ^r 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 <. Putting the several terms, 
taken positively, under the form 1, ac/T 1 , a6<~ 2 , a&c<~ 3 , abcd(f>~ 4 , &c., 
and proceeding to three terms in each series, we get 



tan -\|r = a<f>~ 1 ab(c- a) <~ 3 -f ab [cd (e - a) - ab (c a)} c/T 5 . . .(32). 

The roots of the equation W=0 are required for the physical 
problem to which the integral W relates. Now equations (28), 



DEFINITE INTEGRALS AND INFINITE SERIES. 345 

(29) shew that when??* is at all large the roots of this equation are 
given very nearly by the formula (f> = (i J) TT, where i is an inte 
ger. From the definition of ^r it follows that the root satisfies 
exactly the equation 

</, = (<-!) ,r + t ...................... (S3). 

By means of this equation we may expand <$> in a series according 
to descending powers of ^>, where <J> = (i J) TT. For this purpose 
it will be convenient first to expand ^ in a series according to 
descending powers of <f>, by means of the expansion of tan" 1 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 <J>~ 5 is 

- {ab (c - a) + Ja 3 + a 2 } Q 3 
+ {ab [cd (e a) ab(c a)] 4- a 3 6 (c - a) + ia 5 

a) + Ja 3 ] + 2rt 3 J<- 5 ............ (34). 



13. To facilitate the numerical calculation of the coefficients 

let 

a , b c 



and let the coefficients of <~ 2 , $~ 4 in (31) be put under the forms 

A A 

- j-9^)i> x 2 3 4 4iy - and similarly with respect to (32), (34). 

Then to calculate W for a given value of m, we have 



where J/= 1 - ^ *- + ^- 






and for calculating the roots of the equation TF=0, we have 



346 ON THE NUMERICAL CALCULATION OF A CLASS OF 

The coefficients in these formulae are given by the equations 

A a = a (V - a ) ; A^ = a {b c (d f - 4a) + 3a 2 (2V -a)}^ 
Q = a- C s = ab (c - 3a ); O s = a V [c d (e - 5a f ) - 10 C s ] I 

E l = a ; # 3 = C 3 + 2a 2 (3D + a ) I 

# = C/. + 20a (4D + a ) (7 3 + 24a 5 + 80a 3 D (3D + 2a ) J 

14. Putting in these formulae 

a = 1 . 5 ; b = 7 . 11 ; c = 13 . 17; Z = 19 . 23; e = 25 . 29; D = 72 ; 
we get 

^1 2 =5.72; ^1 4 = 3.5.72 2 .457; ^ = 5; (7 3 = 2.5.7.11.103; 
C 3 = 4 2 .5 3 .7 2 . 11. 23861; ^=5; ^ 3 = 72.1255; # 5 =4.5 3 .72 2 . 10883; 
whence we obtain, on substituting in (36), (37), (38), 
,, - 5 2285 



_ 39655 _ 321526975 . 
a r* -* +2902376448* 



72 31104 2239488 

Keducing to decimals, having previously divided the last equation 
by TT, and put for <E> its value (i ^) TT, we get 

M =!- 034722 </>- 2 + -055097 ^> 4 ............ (40), 

tan ^ = -069444 0- 1 - -035414 4f 3 + 110781 ^> 5 ..... .(41), 

</>_. -028145 -026510 -129402 

TT" - " 4^-"T ~(4i-l) 3+ (4i-l) 5 

15. Supposing z = 1 in (42), we get 

* = -75 + -0094 - -0010 + 0005 = 7589; 

7T 

whence m = 3 (<^>/7r) 3 = 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 = 2*6. 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 table, 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 that 
this term might have been left out altogether. 

16. To determine when W is a maximum or minimum we 
must put d Wjdrii 0. We might get d W/dm by direct differen 
tiation, but the law of the series will be more easily obtained from 
the differential equation. Kesuming equation (11), and putting 
V for dU/dn, we get by dividing by n and then differentiating 



_. 

an n an 3 

This equation may be integrated by descending series just as 
before, and the arbitrary constants will be determined at once by 
comparing the result with the derivative of the second member 
of (15), in which J., B are given by (24). As the process cannot 
fail to be understood from what precedes, it will be sufficient to 
give the result, which is 



F== 3-t^y IJT cos U> + ^J + sin ( * + ^J[ (43), 

where 

-1.7.5.13 -1.7.5.13.11.19.17.25 1 

xi = 1 



1.2(72c/>) 2 1.2.3.4(720)* , 

-1.7 _ -1.7.5.13.11.19 

~ 1.2.3(72c) 3 J 



17. The expression within brackets in (43) may be reduced 
to the form NCOS ((f> + JTT ifr) just as before, and the formulas 
of Art. 13 will apply to this case if we put 

a =-1.7; 6 = 5.13; c = 11.19; &c., D = 72. 

The roots of the equation dW/dm = Q are evidently the same as 
those of V= 0. They are given approximately by the formula 
<f> = (i J) TT, and satisfy exactly the equation < = (i J)TT + ty. 
The root corresponding to any integer i may be expanded in a 
series according to the inverse odd powers of 4i 3 by the formulae 



348 ON THE NUMERICAL CALCULATION OF A CLASS OF 

of Art. 13. Putting (i f )TT for <3>, and taking the series to three 
terms only, we get 



whence </> == 3> - fa <&~ l 

or, reducing as before, 

6 . ,_. -039403 -024693 



This series will give only a rough approximation to the first 
root, but will answer very well for the others. 

For i=l the series gives Tr" 1 = 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 T0845. 
For the second and third roots we get from (45) 

for i= 2, TT" 1 = 1 25 - -00788 + 00020 = 1-24232 ; 
for i = 3, Tr" 1 (/> = 2-25 - 00438 + -00003 = 2 24565. 

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 sbries (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 i is as small as 2. By far 
the greater part of the calculation consisted in passing from the 
values of Tr" 1 cf> 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 



DEFINITE INTEGRALS AND INFINITE SERIES. 



349 



of Professor Miller s observations*. The table of the roots of the 
derived equation, which gives the maxima of W 2 , 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. 



diff. 



diff. 



1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 



2 4955 

A OA 01 

4-3631 



Q.17QQ 
84/88 



12-7395 
13-6924 
14-6132 
15-5059 



18-8502 
19-6399 
20-4139 
21-1736 
21-9199 
22-6536 
23-3757 
24-0868 



, ., 9 



10335 



. 

^991 

iff} 



26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 



26-1602 
26-8332 
27-4979 



28-8037 
29-4456 
30-0805 
30-7089 
31-3308 

32*5567 
33-1610 
33-7599 
34-3535 
34-9420 
35-5256 
36-1044 

37-2484 
3< 



38-9323 
39-4855 
40-0349 
40-5805 






-6100 
. 604 
.043 



1-0845 
3-4669 

e- 
O 



1-1914 



8 

9 

10 



1 1 - 

1 1 

-i oo 1 ~" 

12 24<o 
13-2185 



,.,-,. 
1T170 



.nill 

^-*- * -* 



* Cambridge Philosophical Transactions, Vol. vn. p. 277. 



.350 ON THE NUMERICAL CALCULATION OF A CLASS OF 
SECOND EXAMPLE. 

19. Let us take the integral 



...... (46) 



which occurs in a great many physical investigations. If we 
perform the operation x . d/dx twice in succession on the series 
we get the original series multiplied by x 2 , whence 

cP W 1 du 



20. The form of this equation shews that when x is very 
large, and receives an increment $%, which, though not necessarily 
a very small fraction itself, is very small compared with x, u is 
expressed by A cos bx + B sin &x, where under the restrictions 
specified A and B are sensibly constant (. Assume then, according 
to the plan of Art. 5, 

u = e K ^{Aaf+Bafi+ Ca?+...J ............. (48). 

On substituting in (47) we get 
J-=l {(2a + 



Since we want a descending series, we must put 
2a+l=0; =a-l; 7 = -l...; 
1) 5 = 7^1 aM ; (2y + 1) C = J 



* This integral has been tabulated by Mr Airy from x = Q to x=W, at intervals 
of 2. The table will be found in the 18th Volume of the Philosophical Magazine, 
page 1. 

t That the 1st and 3rd 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=Blogx, 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~ x *^ 2 , which would render the first term 
more important than either of the others. 



DEFINITE INTEGRALS AND INFINITE SERIES. 351 

whence = - 5 = - I J 7 = ~ f J 



22 



Substituting in (48), reducing the result to the form 



adding another solution of the form B (P J IQ), and changing 
the arbitrary constants, we get 

^ = ^.^-i(Ecos^ + /S sina;) + Bx~* (R sinoj - 5 cos #).... (49), 

I 2 3 2 I 2 3 2 5 2 7 2 
~ * 



1.2(8^) 2 1.2.3.4(8#) 4 

J. (50). 

I 2 1 2 .3 2 .5 2 



21. It remains to determine the arbitrary constants A, B. In 
equation (46) let cos 6 = 1 //,, whence 



where I/ = (2^ - ^) -i - (2^) -*, 

a quantity which does not become infinite between the limits of /*. 
Substituting in (46) we get 



w = ^ fcos}(l -^)^}/^-^ya+- I cos {(!-/*)*}#<*/*... (51). 

7? J o Tr J o 



By considering the series whose 71 th term is the part of the 
latter integral for which the limits of /z- are mraT 1 and (?i -f 1) irx~ l 
respectivel}^ it would be very easy to prove that the integral has 
a superior limit of the form HaT 1 , where H is a finite constant, 
and therefore this integral does not furnish any part of the leading 
terms in u. Putting /^c = v in the first integral in (51), so that 



352 ON THE NUMERICAL CALCULATION OF A CLASS OF 

observing that the limits of v are and x, of which the latter 
ultimately becomes co , and that 



I cos v . v ^dfji 2 I c 

-00 ->CO 

= 2 sin \ 2 d\ = I sin i/ . v - l - dv y 

J o ^ o 

we get ultimately for very large values of x 

u = (TTX)-? (cos x + sin #). 

Comparing with (49) we get 

A = B=Tr-*, 

, / 2 \* / 7T\ / 2 \i . / 7T\* ,.* 

whence ?t = - - - ) H cos ar T I + I o sin hr -- } ... (o2). 

VTTX/ V 4/ V 77 "^/ V 4/ 



For example, when # = 10 we have, retaining 5 decimal places 
in the series, 

E = 1 _ -00070 + -00001 = -99931 ; S = 01250 - -00010 = 01240 



7T 



Angle x - = 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 = Q just as before. The formulae of 
Art. 13 may be used for this purpose on putting 

a = l 2 ; & = 3 2 ; c = 5 2 ; &c.; D = 8, 

and writing x, X for c, <, where X(i ^) TT. We obtain 
J 2 = 8; J 4 = 3.8 2 .53; 0, = !; 0, = 2.8M1; 
C 6 = 3 2 . 4 2 . 5 . 1139 ; E t = 1 ; E, = 8 . 31 ; E s = 4 4 . 3779 ; 

* This expression for w, or rather an expression differing from it in nothing but 
notation and arrangement, has been already obtained in a different manner by 
Hir William E. Hamilton, in a memoir " On Fluctuating Functions." See Tramac- 
tions of the Royal Irish Academy, Vol. xix. p. 313. 



DEFINITE INTEGRALS AND INFINITE SERIES. 353 

whence we get for calculating u for a given value of x 
M=l-4,x*+Jk*-*, 

tan + = i of - ,% x 3 + ^jyfc x", 



For calculating the roots of the equation u = we have 

x-X + lX-i-fa A- + tfjfr T-. : . , 

Reducing to decimals as before, we get 

M=l- -0625 of 2 + -103516 af* ........................ (54), 

tani/r = 125 a? 1 - 064453 af 3 + 208557 sf 5 ............... (55), 

<c -050661 053041 -262051 



As before, the series (56) is not sufficiently convergent when 
i = l to give a very accurate result. In this case we get 

7T- 1 x = 75 + -017 - -002 + -001 = 766, 

whence # = 2 41. Mr Airy s table gives u = -f -0025 for x = 2 4, 
and u = 0968 for x = 2 6, whence the value of the root is 2 P 4050 
nearly. 

The value of the last term in (56) is 0000156 for z = 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 



2 f* f2 

v = -I i cos (x cos 6) xdx dO 

7T./0 Jo 



x* a; 4 2r 

* The series 1 - - + ... or -^ has been tabulated by Mr Airy from x = 

to a: = 12 at intervals of 0-2. See Camb. Phil. Trans. Vol. v. p. 291. The same 
function has also been calculated in a different manner and tabulated by M. Schwerd 

s. IT. 23 



354 ON THE NUMERICAL CALCULATION OF A CLASS OF 

which occurs in investigating the diffraction of an object-glass 
with a circular aperture. 

By performing on the series the operation denoted by 

as . d/dx . x~ l , d/dx, we get the original series with the sign 
changed, whence 

d*v Idv /~ 



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 



-tfcosa?) (59), 

where 

_-1^3.1.5 -1.3.1.5.3.7.5.9 

1 . 2~(8i) a " 1.2.3.4 (8#) 4 
-1.3 __ -1.3.1.5.3 ._7 
T78aT 1.2.3 /Q - N3 " i " " 



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 equation 



Writing down the leading terms only in this equation, we have 
x* (- A sin x + B cos x) = 7r"M (cos x + sin x) t 



whence 



(62). 



24. Putting in the formulae of Art. 13, 
a =-l .3; & = 1.5; c =3.7; d = 5 . 9; e = 7. 11; D = 8; 

in his work on diffraction. The argument in the latter table is the angle 180/7r . x, 
and the table extends from to 1125 at intervals of 15, that is, from x = to 
x = 19-63 at intervals of 0-262 nearly. 



DEFINITE INTEGRALS AND INFINITE SERIES. 



355 



we get 

J 2 = -3.8; ^ 4 = -3 3 .8Ml; (^ = -3; C 3 = - 2 . 3 2 . 5 2 ; 
C^-S .tf.o .m; ^ = -3; # 3 = -3 2 .8; E 6 = - 3 3 . 4. 8 2 . 131 ; 
whence we get for the formulae answering to those of Art. 22, 



tan VT = - af + #3 ^ 

= A - 3 Z- + T | s X- 3 + Iffi X-*, 
X being in this case equal to (i 4- J) TT. 

Reducing to decimals as before, we get for the calculation of v 
for a given value of x, 

M= 1 + 1875 x~* + -193359 af* .................... (63), 

tan ^ = --3753T 1 -f 146484 a; 3 - 348817 af 5 ...... (64), 



and for calculating the roots of the equation v = 0, 
9 . -15:1982 -015399 245835 

-- + > ^ + 1 



. 

(60) ; 



,(66). 



25. The following table contains the first 12 roots of each of 
the equations u = 0, and of 2 v = 0. The first root of the former 



i 


- for u=Q 

7T 


diff. 


- forv=0 

7T 


diff. 


1 
2 
3 

4 
5 
6 
7 
8 
9 
10 
11 
12 


7655 
1-7571 
2-7546 
3-7534 
4-7527 
5-7522 
6-7519 
7-7516 
8-7514 
9-7513 
10-7512 
11-7511 


9916 
9975 
9988 
9993 
9995 
9997 
9997 
9998 
9999 
9999 
9999 


1-2197 
2-2330 
3-2383 
4-2411 
5-2428 
6-2439 
7-2448 
8-2454 
9-2459 
10-2463 
11-2466 
12-2469 


1-0133 
1-0053 
1-0028 
1-0017 
1-0011 
1-0009 
1-0006 
1-0005 
1-0004 
1-0003 
1-0003 



232 



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 



where % is an integer, which without loss of generality may be 
supposed positive. The form under which the integral imme 
diately comes out is 



4- 



where each series will evidently contain {+1 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 + fy for 
(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 i(i+ 1) in (67) by a general constant. By 

taking gn , where g is any constant, for the independent variable 



DEFINITE INTEGRALS AND INFINITE SERIES. 357 

in place of n in the differential equations which. U, V in the first 
example satisfy, these equations are reduced to the form 



CttX/ 00 QJL/ 

and (47), (58) are in this form already. Putting now y = af a z, we 
shall reduce the last equation to the form required. 



[The four following are from the Report of the British Association for 1850, 

Part n. p. 19.] 



ON 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 
reflexion. 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 PAPEHS 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 differently 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 phenomenon 
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 difficulty in observing the phenomenon. 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 Nicol s 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 BRITISH ASSOCIATION, 1850. 361 



Ox 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 



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 
investigating 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 different colours in 
producing the brushes seen with ordinary daylight, the author 
used a telescope and prism mounted for shewing the fixed lines of 
the spectrum. The sun s light having been introduced into a 
darkened room through 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 Nicol s 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 the green, 
about the fixed line E of Fraunhofer. They became more distinct 
on passing into the blue, and were particularly strong about the 
line F. The author was able to trace them about as far as the 
line G ; 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. 3G3 

light, the brushes, when they were formed 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 
cceteris 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 account 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 
Nicol s prism, enabled him to see the brushes 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. 



Astigmatic lenses, effect of combined, 

174 
astigmatism of a defective eye, mode of 

measuring, 172 
attractions, propositions respecting, 105, 

124 

Bessel s functions, calculation of, for 
large values of the variable, 329 

Britannia Bridge, deflection liable to be 
produced in, by a travelling load, 219 

Brevrster 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, &c., 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) 

Fresnel s formula? 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 displace 
ment of, 361 

irregularities of earth s surface, effect of, 
on local gravity, 149 

Lorenz s researches on diffraction of 
polarized light, 328 

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 disappearance of central spot 
of, on increasing 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, 358 
Numerical 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, 360 

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 of, 
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 viscous 
fluid, 10 

Stanley, Capt., comparison of observa 
tions made by, on waves in open sea, 
with theory, 239 

Vortex motion, Helmholtz s propositions 
respecting, deduced from Cauchy s 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 



CAMBRIDGE: PRINTED BY c. j. CLAY AND SON, AT THE UNIVERSITY PF.ESS. 



By the same Author. 



MATHEMATICAL AND PHYSICAL PAPERS. 
Vol. I. Price 15s. 

Vol. III. in the Press, 



UNIVERSITY PRESS, CAMBRIDGE, 
January, 1884. 



CATALOGUE OF 

WORKS J 

PUBLISHED FOR THE SYNDICS 



OF THE 



Camtm&tje 




Uon&on : c. j. CLAY, M.A. AND SON. 
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

17 PATERNOSTER ROW. 
GLASGOW: 263, ARGYLE STREET. 



: DEIGHTON, BELL, AND CO. 
is: F. A. BROCKHAUS. 



16/1/84 



PUBLICATIONS OF 

Cfte Cambridge fflnibergitp 



THE HOLY SCRIPTURES, &c. 

THE CAMBRIDGE PARAGRAPH BIBLE of the Au 
thorized English Version, with the Text Revised by a Collation of its 
Early and other Principal Editions, the Use of the Italic Type made 
uniform, the Marginal References remodelled, and a Critical Intro 
duction prefixed, by F. H. A. SCRIVENER, M.A., LL.D., Editor of 
the Greek Testament, Codex Augiensis, &c., and one of the Revisers 
of the Authorized Version. Crown 4to. gilt. 2is. 

From the Times. Syndics of the Cambridge University Press, 

"Students of the Bible should be particu- an edition of the English Bible, according to 

larly grateful to (the Cambridge University the text of 1611, revised by a comparison with 

Press) for having produced, with the able as- later issues on principles stated by him in his 

sistance of Dr Scrivener, a complete critical Introduction. Here he enters at length into 

edition of the Authorized Version of the Eng- the history of the chief editions of the version, 

lish Bible, an edition such as, to use the words and of such features as the marginal notes, the 

of the Editor, would have been executed long use of italic type, and the changes of ortho- 

ago had this version been nothing more than graphy, as well as into the most interesting 

the greatest and best known of English clas- question as to the original texts from which 

sics. Falling at a time when the formal revi- our translation is produced." 
sion of this version has been undertaken by a ^ ., ,, ,, ,. . , 

distinguished company of scholars and divines, romthe Methodlst Recorder. 



From the A thenaum. " - e 

"Apart from its religious importance, the Press ls . guarantee enough for its perfection in 

English Bible has the glory, which but few outwar f form, the name of the ed.tor is equal 

sister versions indeed can claim, of being the guarantee for the worth and accuracy of its 

chief classic of the language, of having, in contents Without question it is the best 

conjunction with Shakspeare, and in an im- Paragraph Bible ever published, and its re- 

measurable degree more than he, fixed the d " ce . d pnce of a S u , mea , bnn -?? U Wlthln reach 

language beyond any possibility of important f a lar S e number of students." 

change. Thus the recent contributions to the From the London Quarterly Review. 

literature of the subject, by such workers as "The work is worthy in every respect of the 

Mr Francis Fry and Canon Westcott, appeal editor s fame, and of the Cambridge University 

to a wide range of sympathies; and to these Press. The noble English Version, to which 

may now be added Dr Scrivener, well known our country and religion owe so much, was 

for his labours in the cause of the Greek Testa- probably never presented before in so perfect a 

ment criticism, who has brought out, for the form." 

THE CAMBRIDGE PARAGRAPH BIBLE. STUDENT S 
EDITION, on good writing paper , with one column of print and wide 
margin to each page for MS. notes. This edition will be found of 
great use to those who are engaged in the task of Biblical criticism. 
Two Vols. Crown 4to. gilt. 3U. 6d. 

THE LECTIONARY BIBLE, WITH APOCRYPHA, 

divided into Sections adapted to the Calendar and Tables of 
Lessons of 1871. Crown 8vo. 3^. 6d. 

THE BOOK OF ECCLESIASTES, with Notes and In 
troduction. By the Very Rev. E. H. PLUMPTRE, D.D., Dean of 
Wells. Large Paper Edition. Demy 8vo. Js. 6d. 

" No one can say that the Old Testament is point in English exegesis of the Old Testa- 

a dull or worn-out subject after reading this ment; indeed, even Delitzsch, whose pride it 

singularly attractive and also instructive com- is to leave no source of illustration unexplored, 

mentary. Its wealth of literary and historical is far inferior on this head to Dr Plumptre." 

illustration surpasses anything to which we can Academy, Sept. 10, 1881. 



London: Cambridge University Press Warehouse, 17 Paternoster Row. 



CAMBRIDGE UNIVERSITY PRESS BOOKS. 3 

BREVIARILJM AD USUM INSIGNIS ECCLESIAE 

SARUAI. Juxta Editionem maximam pro CLAUDIO CHEVALLON 
ET FRANCISCO REGNAULT A.D. MDXXXI. in Alma Parisiorum 
Academia impressam : labore ac studio FRANCISCI PROCTER, 
A.M., ET CHRISTOPHORI WORDSWORTH, A.M. 

FASCICULUS I. In quo contmentur KALENDARIUM, et ORDO 
TEMPORALIS sive PROPRIUM DE TEMPORE TOTIUS ANNI, una cum 
ordinali suo quod usitato vocabulo dicitur PICA SIVE DIRECTOR IUM 
SACERDOTUM. Demy 8vo. i8s. 

" The value of this reprint is considerable to cost prohibitory to all but a few. .. . Messrs 

liturgical students, who will now be able to con- Procter and Wordsworth have discharged their 

suit in their own libraries a work absolutely in- editorial task with much care and judgment, 

dispensable to a right understanding of the his- though the conditions under which they have 

tory of the Prayer- Book, but which till now been working are such as to hide that fact from 

usually necessitated a visit to some public all but experts." Literary Churchman. 
library, since the rarity of the volume made its 

FASCICULUS II. In quo contmentur PSALTERIUM, cum ordinario 
Officii totius hebdomadae juxta Horas Canonicas, et proprio Com- 
pletorii, LITANIA, COMMUNE SANCTORUM, ORDINARIUM MISSAE 
CUM CANONE ET xiu MISSIS, &c. &c. Demy 8vo. i2s. 

"Not only experts in liturgiology, but all For all persons of religious tastes the Breviary, 

persons interested in the history of the Anglican with its mixture of Psalm and Anthem and 

Book of Common Prayer, will be grateful to the Prayer and Hymn, all hanging one on the 

Syndicate of the Cambridge University Press other, and connected into a harmonious whole, 

for forwarding the publication of the volume must be deeply interesting." Church Quar- 

which bears the above title, and which has terly Review. 

recently appeared under their auspices." "The editors have done their work excel- 

Notes atut Queries. lently, and deserve all praise for their labours 

"Cambridge has worthily taken the lead in rendering what they justly call this most 

with the Breviary, which is of especial value interesting Service-book more readily access- 

for that part of the reform of the Prayer- Book ible to historical and liturgical students." 

which will fit it for the wants of our time .... Saturday Review. 

FASCICULUS III. Nearly ready. 

GREEK AND ENGLISH TESTAMENT, in parallel 

Columns on the same page. Edited by J. SCHOLEFIELD, M.A. late 
Regius Professor of Greek in the University. Small Octavo. New 
Edition, with the Marginal References as arranged and revised by 
Dr SCRIVENER. Cloth, red edges. 7^. 6d. 

GREEK AND ENGLISH TESTAMENT. THE STU 
DENT S EDITION of the above, on large writing paper. 4to. 12s. 

GREEK TESTAMENT, ex editione Stephani tertia, 1550. 
Small 8vo. $s. 6d. 

THE NEW TESTAMENT IN GREEK according to the 
text followed in the Authorised Version, with the Variations adopted 
in the Revised Version. Edited by F. H. A. SCRIVENER ALA., 
D.C.L., LL.D. Crown 8vo. 6s. Morocco boards or limp. 12s. 

THE PARALLEL NEW TESTAMENT GREEK AND 

ENGLISH, being the Authorised Version set forth in 1611 Arranged 
in Parallel Columns with the Revised Version of iSSi, and with the 
original Greek, as edited by F. H. A. SCRIVENER, M.A., D.C.L., 
LL.D. Prebendary of Exeter and Vicar of Hendon. Crown 8vo. 
12s. bd. The Revised Version is the Joint Property of the Universi 
ties of Cambridge and Oxford. 



London: Cambridge Universitv Press Warehouse, 17 Paternoster Row. 

I 2 



PUBLICATIONS OF 



THE GOSPEL ACCORDING TO ST MATTHEW in 

Anglo-Saxon and Northumbrian Versions, synoptically arranged: 
with Collations of the best Manuscripts. By J. M. KEMBLE, M.A. 
and Archdeacon HARDWICK. Demy 4to. los. 

NEW EDITION. By the Rev. Professor SKEAT. [In the Press. 

THE GOSPEL ACCORDING TO ST MARK in Anglo- 
Saxon and Northumbrian Versions, synoptically arranged : with Col 
lations exhibiting all the Readings of all the MSS. Edited by the 
Rev. Professor SKEAT, M.A. late Fellow of Christ s College, and 
author of a McESO-GOTHic Dictionary. Demy 4to. IQJ. 

THE GOSPEL ACCORDING TO ST LUKE, uniform 
with the preceding, by the same Editor. Demy 4to. IDS. 

THE GOSPEL ACCORDING TO ST JOHN, uniform 
with the preceding, by the same Editor. Demy 4to. los. 

. " The Gospel according to St John> in menced by that distinguished scholar, J. M. 
Anglo-Saxon and Northumbrian Versions: Kemble, some forty years ago. Of the par- 
Edited for the Syndics of the University ticular volume now before us, we can only say 
Press, by the Rev. Walter W. Skeat, M.A., it is worthy of its two predecessors. We repeat 
Elringtpn and Bosworth Professor of Anglo- that the service rendered to the study of Anglo- 
Saxon in the University of Cambridge, com- Saxon by this Synoptic collection cannot easily 
pletes an undertaking designed and com- be overstated." Contemporary Review. 

THE POINTED PRAYER BOOK, being the Book of 
Common Prayer with the Psalter or Psalms of David, pointed as 
they are to be sung or said in Churches. Royal 241110. Cloth. 
is. 6d. 

The same in square 32mo. cloth. 6< 

"The Pointed Prayer Book deserves men- for the terseness and clearness of the direc 
tion for the new and ingenious system on which tions given for using it." Times. 
the pointing has been marked, and still more 

THE CAMBRIDGE PSALTER, for the use of Choirs and 
Organists. Specially adapted for Congregations in which the " Cam 
bridge Pointed Prayer Book" is used. Demy 8vo. cloth extra, $s. 6d. 
Cloth limp, cut flush, is. 6d. 

THE PARAGRAPH PSALTER, arranged for the use of 
Choirs by BROOKE Foss WESTCOTT, D.D., Regius Professor of 
Divinity in the University of Cambridge. Fcap. 4to. $s. 

The same in royal 32mo. Cloth Is. Leather Is. 6d. 

"The Paragraph Psalter exhibits all the and there is not a clergyman or organist in 

care, thought, and learning that those acquaint- England who should be without this Psalter 

ed with the works of the Regius Professor of as a work of reference." Morning Post. 
Divinity at Cambridge would expect to find, 

THE MISSING FRAGMENT OF THE LATIN TRANS- 

LATION OF THE FOURTH BOOK OF EZRA, discovered, 
and edited with an Introduction and Notes, and a facsimile of the 
MS., by ROBERT L. BENSLY, M.A., Reader in Hebrew, Gonville and 
Caius College, Cambridge. Demy 4to. IDS. 

"Edited with true scholarly completeness." no exaggeration of the actual fact, if by the 

Westminster Review. Bible we understand that of the larger size 

"It has been said of this book that it has which contains the Apocrypha, and if the 

added a new chapter to the Bible, and, startling Second Book of Esdras can be fairly called a 

as the statement may at first sight appear, it is part of the Apocrypha. " Saturday Review. 



London: Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 



THEOLOGY (ANCIENT). 

THE GREEK LITURGIES. Chiefly from original Autho 
rities. By C. A. SWAINSON, D.D., Master of Christ s College, Cam 
bridge. \In the Press. 

THE PALESTINIAN MISHNA. By W. H. LOWE, M.A. 

Lecturer in Hebrew at Christ s College, Cambridge. Royal 8vo. 2U. 

SAYINGS OF THE JEWISH FATHERS, comprising 
Pirqe Aboth and Pereq R. Meir in Hebrew and English, with Cri 
tical and Illustrative Notes. By CHARLES TAYLOR, D.D. Master 
of St John s College, Cambridge, and Honorary Fellow of King ? s 
College, London. Demy 8vo. los. 

"The Masseketh Aboth stands at the Jewish literature being treated in the same 

head of Hebrew non-canonical writings. It is way as a Greek classic in an ordinary critical 

of ancient date, claiming to contain the dicta edition. . . The Sayings of the Jewish Fathers 

of teachers who nourished from B.C. 200 to the may claim to be scholarly, and, moreover, of a 

same year of our era. The precise time of its scholarship unusually thorough and finished." 

compilation in its present form is, of course, in Dublin University Slagazine. 

doubt. Mr Taylors explanatory and illustra- "A careful and thorough edition which does 

tive commentary is very full and satisfactory." credit to English scholarship, of a short treatise 

Spectator. from the Mishna, containing a series of sen- 

"Ifwe mistake not, this is the first precise tences or maxims ascribed mostly to Jewish 

translation into the English language, accom- teachers immediately preceding, or immediately 

pan ied by scholarly notes, of any portion of the following the Christian era..." Contempo- 

Talmud. In other words, it is the first instance rary Review. 
of that most valuable and neglected portion of 

THEODORE OF MOPSUESTIA S COMMENTARY 
ON THE MINOR EPISTLES OF S. PAUL. The Latin Ver 
sion with the Greek Fragments, edited from the MSS. with Notes 
and an Introduction, by H. B. SWETE, D.D., Rector of Ashdon, 
Essex, and late Fellow of Gonville and Caius College, Cambridge. 
In Two Volumes. Vol. I., containing the Introduction, with Fac 
similes of the MSS., and the Commentary upon Galatians Colos- 
sians. Demy 8vo. 12s. 

"In dem oben verzeichneten Buche liegt handschriften . . . sind yortrefHiche photo- 

uns die erste Halfte einer vollstandigen, ebenso graphische Facsimile s beigegeben, wie uber- 

sorgfaltig gearbeiteten wie schon ausgestat- haupt das ganze Werk von der University 

teten Ausgabe des Commentars mit ausfuhr- Press zu Cambridge mit bekannter Eleganz 

lichen Prolegomena und reichhaltigen kritis- ausgestattet ist." Theologische Literaturzei- 

chen und erlauternden Anmerkungen vor." tung. 
LiterariscJtes Centralblatt. "It is a hopeful sign, amid forebodings 

" It is the result of thorough, careful, and which arise about the theological learning of 

patient investigation of all the points bearing the Universities, that we have before us the 

on the subject, and the results are presented first instalment of a thoroughly scientific and 

with admirable good sense and modesty." painstaking work, commenced at Cambridge 

Guardian. and completed at a country rectory." Church 

if /- /~i j J- /-\ 11 _. _!___ rrt . /~\_. |__.7._ r> /T _OO_\ 



Auf Grund dieser Quellen ist der Text Quarterly Review (Jan. 1881). 

rete mit musterhafter Akribie herge- " Hernn Swete s Leistung ist eine 

stellt. Aber auch sonst hat der Herausgeber tuchtige dass wir das Werk in keinen besseren 



mit unermiidlichem Fleisse und eingehend- Handen wissen mochten, und mit den sich- 
ster Sachkenntniss sein Werk mit alien den- ersten Erwartungen auf das Gelingen der 



jenigen Zugaben ausgerustet, welche bei einer Fortsetzung entgegen s 
solchen Text- Ausgabe nur irgend erwartet gelehrte Anzeigen (Sept. 1881). 
werden konnen. . . . Von den drei Haupt- 

VOLUME II., containing the Commentary on I Thessalonians 
Philemon, Appendices and Indices. 12s. 

"Eine Ausgabe . . . fur welche alle zugang- "Mit deiselben Sorgfalt bearbeitet die wir 

lichen Hulfsmittel in musterhafter Weise be- bei dem ersten Theile geruhmt haben." 
nutzt wurden . . . eine reife Frucht siebenjahri- Literarisches Centralblatt (July 29, 1882). 
gen Fleisses." TlieologiscJtf Literaturzeitung 
(Sept. 23, 1882). 



London: Cambridge University Press Warehouse^ 17 Paternoster Row. 



PUBLICATIONS OP 



SANCTI IREN^EI EPISCOPI LUGDUNENSIS libros 
quinque adversus Hasreses, versione Latina cum Codicibus Claro- 
montano ac Arundeliano denuo collata, prasmissa de placitis Gnos- 
ticorum prolusione, fragmenta necnon Graece, Syriace, Armeniace, 
commentatione perpetua et indicibus variis edidit W. WIGAN 
HARVEY, S.T.B. Collegii Regalis olim Socius. 2 Vols. Demy 8vo. 



M. MINUCII FELICIS OCTAVIUS. The text newly 
revised from the original MS., with an English Commentary, 
Analysis, Introduction, and Copious Indices. Edited by H. A. 
HOLDEN, LL.D. late Head Master of Ipswich School, formerly 
Fellow of Trinity College, Cambridge. Crown 8vo. js. 6d. 

THEOPHILI EPISCOPI ANTIOCHENSIS LIBRI 
TRES AD AUTOLYCUM edidit, Prolegomenis Versione Notulis 
Indicibus instruxit GULIELMUS GILSON HUMPHRY, S.T.B. Collegii 
Sancliss. Trin. apud Cantabrigienses quondam Socius. Post 8vo. 
5J. 

THEOPHYLACTI IN EVANGELIUM S. MATTH^EI 

COMMENTARIUS, edited by W. G. HUMPHRY, B.D. Prebendary 
of St Paul s, late Fellow of Trinity College. Demy Svo. 7^. 6d. 

TERTULLIANUS DE CORONA MILITIS, DE SPEC- 

TACULIS, DE 1DOLOLATRIA, with Analysis and English Notes, 
by GEORGE CURREY, D.D. Preacher at the Charter House, late 
Fellow and Tutor of St John s College. Crown Svo. 5J 1 . 



THEOLOGY-(ENGLISH). 

WORKS OF ISAAC BARROW, compared with the Ori 
ginal MSS., enlarged with Materials hitherto unpublished. A new 
Edition, by A. NAPIER, M.A. of Trinity College, Vicar of Holkham, 
Norfolk. 9 Vols. Demy Svo. ^3. 3^. 

TREATISE OF THE POPE S SUPREMACY, and a 

Discourse concerning the Unity of the Church, by ISAAC BARROW. 
Demy Svo. js. 6d. 

PEARSON S EXPOSITION OF THE CREED, edited 
by TEMPLE CHEVALLIER, B.D. late Fellow and Tutor of St Catha 
rine s College, Cambridge. New Edition. Revised by R. Sinker, 
B.D., Librarian of Trinity College. Demy Svo. 12^. 

"A new edition of Bishop Pearson s famous places, and the citations themselves have been 

work On the Creed has just been issued by the adapted to the best and newest texts of the 

Cambridge University Press. It is the well- several authors- -texts which have undergone 

known edition of Temple Chevallier, thoroughly vast improvements within the last two centu- 

overhauled by the Rev. R. Sinker, of Trinity ries. The Indices have also been revised and 

College. The whole text and notes have been enlarged Altogether this appears to be the 

most carefully examined and corrected, and most complete and convenient edition as yet 
special pains have been taken to verify the al- published of a work which has long been re- 
most innumerable references. These have been cognised in all quarters as a standard one." 
more clearly and accurately given in very many Guardian. 



London : Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 7 

AN ANALYSIS OF THE EXPOSITION OF THE 

CREED written by the Right Rev. JOHN PEARSON, D.D. late Lord 
Bishop of Chester, by W. H. MILL, D.D. late Regius Professor of 
Hebrew in the University of Cambridge. Demy 8vo. 5^. 

VVHEATLY ON THE COMMON PRAYER, edited by 
G. E. CORRIE, D.D. Master of Jesus College, Examining Chaplain 
to the late Lord Bishop of Ely. Demy Svo. js. 6d. 

CAESAR MORGAN S INVESTIGATION OF THE 

TRINITY OF PLATO, and of Philo Judaeus, and of the effefts 
which an attachment to their writings had upon the principles and 
reasonings of the Fathers of the Christian Church. Revised by H. A. 
HOLDEN, LL.D., formerly Fellow of Trinity College, Cambridge. 
Crown Svo. 4?. 

TWO FORMS OF PRAYER OF THE TIME OF QUEEN 
ELIZABETH. Now First Reprinted. Demy Svo. 6d. 

"From Collections and Notes 18671876, ker Society s volume of Occasional Forms of 
by \V. Carew Hazlitt (p. 340), we learn that Prayer, but it had been lost sight of for 200 
A very remarkable volume, in the original years. By the kindness of the present pos- 
vellum cover, and containing 25 Forms of sessor of this valuable volume, containing in all 
Prayer of the reign of Elizabeth, each with the 25 distinct publications, I am enabled to re- 
autograph of Humphrey Dyson, has lately fallen print in the following pages the two Forms 
into the hands of my friend Mr H. Pyne. It is of Prayer supposed to have been lost." Ex~ 
mentioned specially in the Preface to the Par tract front the PREFACE. 

SELECT DISCOURSES, by JOHN SMITH, late Fellow of 

Queens College, Cambridge. Edited by H. G. WILLIAMS, B.D. late 
Professor of Arabic. Royal Svo. js. 6d. 

"The Select Discourses of John Smith, with the richest lights of meditative genius... 

collected and published from his papers after He was one of those rare thinkers in whom 

his death, are, in my opinion, much the most largeness of view, and depth, and wealth of 

considerable work left to us by this Cambridge poetic and speculative insight, only served to 

School [the Cambridge Platonists]. They have evoke more fully the religious spirit, and while 

a right to a place in English literary history." he drew the mould of his thought from Plotinus, 

Mr MATTHEW ARNOLD, in the Contenipo- he vivified the substance of it from St Paul." 

retry Review. Principal TULLOCH, Rational Theology in 

"Of all the products of the Cambridge England in tJie \-jth Century. 
School, the Select Discourses are perhaps "We may instance Mr Henry Griffin Wil- 

the highest, as they are the most accessible liams s revised edition of Mr John Smith s 

and the most widely appreciated. ..and indeed Select Discourses, which have won Mr 

no spiritually thoughtful mind can read them Matthew Arnold s admiration, as an example 

unmoved. They carry us so directly into an of worthy work for an University Press to 

atmosphere of divine philosophy, luminous undertake." Times. 

THE HOMILIES, with Various Readings, and the Quo 
tations from the Fathers given at length in the Original Languages. 
Edited by G. E. CORRIE, D.D., Master of Jesus College. Demy 
Svo. 73. 6d. 

DE OBLIGATIONS CONSCIENTLE PR^ELECTIONES 

decem Oxonii in Schola Theologica habitas a ROBERTO SANDERSON, 
SS. Theologias ibidem Professore Regio. With English Notes, in 
cluding an abridged Translation, by W. WHEWELL, D.D. late 
Master of Trinity College. Demy Svo. 7.$-. 6d. 

ARCHBISHOP USHER S ANSWER TO A JESUIT, 

with other Trafts on Popery. Edited by J. SCHOLEFIELD, M.A. late 
Regius Professor of Greek in the University. Demy Svo. 7^. 6d. 



London: Cambridge University Press Warehouse, 17 Paternoster Row. 



8 PUBLICATIONS OF 

WILSON S ILLUSTRATION OF THE METHOD OF 

explaining the New Testament, by the early opinions of Jews and 
Christians concerning Christ. Edited by T. TuRTON, D.D. late 
Lord Bishop of Ely. Demy 8vo. 5^. 

LECTURES ON DIVINITY delivered in the University 
of Cambridge, by JOHN HEY, D.D. Third Edition, revised by T. 
TURTON, D.D. late Lord Bishop of Ely. 2 vols. Demy 8vo. 15^. 



ARABIC, SANSKRIT AND SYRIAC. 

POEMS OF BEHA ED DIN ZOHEIR OF EGYPT. 

With a Metrical Translation, Notes and Introduction, by E. H. 
PALMER, M.A., Barrister-at-Law of the Middle Temple, late Lord 
Almoner s Professor of Arabic, formerly Fellow of St John s College, 
Cambridge. 3 vols. Crown 4to. 

Vol. I. The ARABIC TEXT. los. 6d. ; Cloth extra. 15^. 

Vol. II. ENGLISH TRANSLATION. ios.6d.\ Cloth extra. 15.?. 

"We have no hesitation in saying that in remarked, by not unskilful imitations of the 

both Prof. Palmer has made an addition to Ori- styles of several of our own favourite poets, 

ental literature for which scholars should be living and dead." Saturday Review. 
grateful ; and that, while his knowledge of " This sumptuous edition of the poems of 

Arabic is a sufficient guarantee for his mastery Beha-ed-din Zoheir is a very welcome addition 

of the original, his English compositions are to the small series of Eastern poets accessible 

distinguished by versatility, command of Ian- to readers who are not Orientalists." Aca- 

guage, rhythmical cadence, and, as we have demy. 

KALILAH AND DIMNAH, OR, THE FABLES OF 
PILPAI ; being an account of their literary history, together with 
an English Translation of the same, with Notes, by I. G. N. KEITH- 
FALCONER, M.A., Trinity College, formerly Tyrwhitt s Hebrew 
Scholar. Demy 8vo. \In the Press. 

THE CHRONICLE OF JOSHUA THE STYLITE, com 
posed in Syriac A.D. 507 with an English translation and notes, by 
W. WRIGHT, LL.D., Professor of Arabic. Demy 8vo. los. 6d. 

" Die lehrreiche kleine Chronik Josuas hat ein Lehrmittel fur den syrischen Unterricht ; es 

nach Assemani und Martin in Wright einen erscheint auch gerade zur rechten Zeit, da die 

dritten Bearbeiter gefunden, der sich um die zweite Ausgabe von Roedigers syrischer Chres- 

Emendation des Textes wie um die Erklarung tomathie im Buchhandel vollstandig vergriffen 

der Realien wesentlich verdient gemacht hat und diejenige von Kirsch-Bernstein nur noch 

. . . Ws. Josua- Ausgabe ist eine sehr dankens- in wenigen Exemplaren vorhanden ist." 

werte Gabe und besonders empfehlenswert als Deutsche Litteratitrzeitung . 

NALOPAKHYANAM, OR, THE TALE OF NALA ; 

containing the Sanskrit Text in Roman Characters, followed by a 
Vocabulary in which each word is placed under its root, with refer 
ences to derived words in Cognate Languages, and a sketch of 
Sanskrit Grammar. By the late Rev. THOMAS JARRETT, M.A. 
Trinity College, Regius Professor of Hebrew. Demy 8vo. los. 

NOTES ON THE TALE OF NALA, for the use of 

Classical Students, by J. PEILE, M.A. Fellow and Tutor of Christ s 
College. Demy 8vo. i2s. 

CATALOGUE OF THE BUDDHIST SANSKRIT 
MANUSCRIPTS in the University Library, Cambridge. Edited 
by C. BENDALL, M.A., Fellow of Gonville and Caius College. Demy 

8VO. I2S. 

"It is unnecessary to state how the com- those concerned in it on the result . . . Mr Ben- 

pilation of the present catalogue came to be dall has entitled himself to the thanks of all 

placed in Mr Bendall s hands; from the cha- Oriental scholars, and we hope he may have 

racter of his work it is evident the selection before him a long course of successful labour in 

was judicious, and we may fairly congratulate the field he has chosen." Athenaum, 

London: Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 9 

GREEK AND LATIN CLASSICS, &c. (See also pp. 24-27.) 

SOPHOCLES : The Plays and Fragments, with Critical 
Notes, Commentary, and Translation in English Prose, by R. C. 
JEBB, M.A., LL.D., Professor of Greek in the University of Glasgow. 
Parti. Oedipus Tyrannus. Demy 8vo. 15^. 

"In undertaking, therefore, to interpret for the first class . . . The present edition of 

Sophocles to the classical scholar and to the Sophocles is to consist of eight volumes one 

British public, Professor Jebb expounds the being allowed to each play and the eighth, 

most consummate poetical artist of what com- containing the fragments and a series of short 

mon consent allows to be the highest stage in essays on subjects of general interest relating 

Greek culture ... As already hinted, Mr Jebb to Sophocles. If the remaining volumes main- 

in his work aims at two classes of readers. tain the high level of the present one, it will, 

He keeps in view the Greek student and the when completed, be truly an edition de luxe" 

English scholar who knows little or no Greek. Glasgmv Herald. 
His critical notes and commentary are meant 

AESCHYLI PABULAR IKETIAES XOH<OPOJ IN 

LIBRO MEDICEO MENDOSE SCRIPTAE EX VV. DD. 
CONIECTURIS EMENDATIUS EDITAE cum Scholiis Graecis 
et brevi adnotatione critica, curante F. A. PALEY, M.A., LL.D. 
Demy 8vo. js. 6d. 

THE AGAMEMNON OF AESCHYLUS, With a Trans 
lation in English Rhythm, and Notes Critical and Explanatory. 
New Edition Revised. By BENJAMIN HALL KENNEDY, D.D., 
Regius Professor of Greek. Crown Svo. 6^. 

"One of the best editions of the masterpiece value of this volume alike to the poetical 
of Greek tragedy." Atltenteum. translator, the critical scholar, and the ethical 

"It is needless to multiply proofs of the student." Saturday Review. 

THE THE^ETETUS OF PLATO with a Translation and 

Notes by the same Editor. Crown Svo. js. 6d. 

THE CEDIPUS TYRANNUS OF SOPHOCLES with a 
Translation and Notes by the same Editor. Crown Svo. 6s. 

"Dr Kennedy s edition of the (Edipits no more valuable contribution to the study 
Tyrannus is a worthy companion to his of Sophocles has appeared of late years." 
Agamemnon, and we may say at once that Saturday Review. 

PLATO S PH^DO, literally translated, by the late E. M. 
COPE, Fellow of Trinity College. Cambridge. Demy Svo. 5^. 

ARISTOTLE. IIEPI AIKAIO2TNH2. THE FIFTH 
BOOK OF THE NICOMACHEAN ETHICS OF ARISTOTLE. 
Edited by HENRY JACKSON, M.A., Fellow of Trinity College, Cam 
bridge. Demy Svo. 6s. 

"It is not too much to say that some of the will hope that this is not the only portion of 
points he discusses have never had so much the Aristotelian writings which he is likely to 
light thrown upon them before. . . . Scholars edit." AtJieneeum. 

ARISTOTLE. IIEPI ^TXHS. ARISTOTLE S PSY 
CHOLOGY, in Greek and English, with Introduction and Notes, 
by EDWIN WALLACE, M.A., Fellow and Tutor of Worcester College, 
Oxford. Demy Svo. iSs. 

" In an elaborate introduction Mr Wallace and to those who read it as students of philo- 

collects and correlates all the passages from the soph} ." Scotsman. 

various works of Aristotle bearing on these "The notes are exactly what such notes 

points, and this he does with a width of learn- ought to be, helps to the student, not mere 

ing that marks him out as one of our foremost displays of learning. By far the more valuable 

Aristotlic scholars, and with a critical acumen parts of the notes are neither critical nor lite- 

that is far from common." Glasgmu Herald. rary, but philosophical and expository of the 

"As a clear exposition of the opinions of thought, and of the connection of thought, in 

Aristotle on psychology, Mr Wallace s work is the treatise itself. In this relation the notes are 

of distinct value the introduction is excellently invaluable. Of the translation, it may be said 

wrought out, the translation is good, the notes that an English reader may fairly master by 

are thoughtful, scholarly, and full. We there- means of it this great treatise of Aristotle." 

fore can welcome a volume like this, which is Spectator. 
useful both to those who study it as scholars, 

London: Cambridge University Pt ess Warehouse, 17 Paternoster Row. 



io PUBLICATIONS OF 

A SELECTION OF GREEK INSCRIPTIONS, with 
Introductions and Annotations by E. S. ROBERTS, M.A. Fellow 
and Tutor of Gonville and Caius College. \In the Press. 

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. gs. 

"Mr Fennell deserves the thanks of all clas- of the vitality and development of Cambridge 

sical students for his careful and scholarly edi- scholarship, and we are glad to see that it is to 

tion of the Olympian and Pythian odes. He be continued." Saturday Review. 
brings to his task the necessary enthusiasm for "Mr C. A. M. Fennell s Pindar displays 

his author, great industry, a sound judgment, that union of laborious research and unassum- 

and, in particular, copious and minute learning ing directness of style which characterizes the 

in comparative philology. To his qualifica- best modern scholarship . . . The notes, which 

tions in this last respect every page bears wit- are in English, and at the foot of each page, are 

ness." Athenceum. clear and to the point. There is an introduc- 

" Considered simply as a contribution to the tion to each Ode. There are Greek and Eng- 

study and criticism of Pindar, Mr Fennell s lish Indices, and an Index of Quotations." 

edition is a work of great merit. . . Altogether, Westminster Review. 
this edition is a welcome and wholesome sign 

THE ISTHMIAN AND NEMEAN ODES. By the same 

Editor. Crown 8vo. gs. 

"Encouraged by the warm praise with Fennell s Pindar. " Athen&um. 

which Mr Fennell s edition of the Olympian " Mr Fennell, whose excellent edition of 

and Pythian odes was everywhere received, the Olympian and Pythian Odes of Pindar 

the Pitt Press Syndicate very properly invited appeared some four years ago, has now pub- 

him to continue his work and edit the re- lished the Nemean and Isthmian Odes, toge- 

mainder of Pindar . . . His notes are full of therwith a selection from the extant fragments 

original ideas carefully worked out, and if he of Pindar. This work is in no way inferior to 

often adopts the opinion of other editors, he the previous volume. The commentary affords 

does not do so without making it sufficiently valuable help to the study of the most difficult 

plain that he has discussed the question for of Greek authors, and is enriched with notes 

himself and decided it upon the evidence. As on points of scholarship and etymology which 

a handy and instructive edition of a difficult could only have been written by a scholar of 

classic no work of recent years surpasses Mr very high attainments." Saturday Review. 

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. i. i u. 6d. 

"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 tact, 
wishes to have a full conception of what is con- ... In every part of his work revising, sup- 
tained in the/? hetoric of Aristotle, to Mr Cope s plementing, and completing he has done ex- 
edition he must go." Academy. ceedingly well." Examiner. 

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. 6^. 

PART II. Pro Phormione, Contra Stephanum I. II.; Nicostra- 
tum, Cononem, Calliclem. Crown 8vo. js. 6d. 

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. 7^. 6d. 

"The present edition may therefore be used each paragraph of the text there is a summary 

by students more advanced than school-boys, of its subject-matter . . . The notes are uni- 

and to their purposes it is admirably suited. formly good, whether they deal with questions 

There is an excellent introduction to and ana- of scholarship or with points of Athenian law." 

lysis of each speech, and at the beginning of Saturday Review. 

London : Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 11 

THE TYPES OF GREEK COINS. By PERCY GARDNER, 

M.A., F.S.A., Disney Professor of Archaeology. With 16 Autotype 
plates, containing photographs of Coins of all parts of the Greek World. 
Impl. 4to. Cloth extra, i. i is. 6d.; Roxburgh (Morocco back), 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- medals." Saturday Review. 
ward that it may well win converts, and it may The Types of Greek Coins is a work which 
be distinctly recommended to that omnivorous is less purely and dryly scientific. Neverthe- 
class of readers men in the schools. The his- less, it takes high rank as proceeding upon a 
tory of ancient coins is so interwoven with and truly scientific basis at the same time that it 
so vividly illustrates the history of ancient States, treats the subject of numismatics in an attrac- 
that students of Thucydides and Herodotus can- tive style and is elegant enough to justify its 
not afford to neglect Professor Gardner s intro- appearance in the drawing-room .... Six- 
duction to Hellenic numismatics . . . The later teen autotype plates reproduce with marvellous 
part of Mr Gardner s useful and interesting reality more than six hundred types of picked 
volume is devoted to the artistic and archaeo- specimens of coins in every style, from the 
logical aspect of coins, and can scarcely be cabinets of the British Museum and other col- 
studied apart from photographs (like those ections. AtJtetueum. 

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. los. 6d. 

"Of the present edition of the BaccJue by Mr able advance in freedom and lightness of style. 
Sandys we may safely say that never before has . . . Under such circumstances it is superfluous 
a Greek play, in England at least, had fuller to say that for the purposes of teachers and ad- 
justice done to its criticism, interpretation, vanced students this handsome edition far sur- 
and archaeological illustration, whether for the passes all its predecessors." Atlietueum. 
young student or the more advanced scholar. " It has not, like so many such books, been 
The Cambridge Public Orator may be said to hastily produced to meet the momentary need 
have taken the lead in issuing a complete edi- of some particular examination ; but it has em- 
tion of a Greek play, which is destined perhaps ployed for some years the labour and thought 
to gain redoubled favour now that the study of of a highly finished scholar, whose aim seems 
ancient monuments has been applied to its il- to have been that his book should go forth totus 
lustration." Saturday Review. teres atque rotundus, armed at all points with 

" The volume is interspersed with well- all that may throw light upon its subject. The 

executed woodcuts, and its general attractive- result is a work which will not only assist the 

ness of form reflects great credit on the Uni- schoolboy or undergraduate in his tasks, but 

versity Press. In the notes Mr Sandys has more will adorn the library of the scholar." The 

than sustained his well-earned reputation as a Guardian. 
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 OUINQUE. The text revised and ex 
plained ; With a Translation by JAMES S. REID, M.L., Fellow and 
Assistant Tutor of Gonville and Caius College. In three Volumes. 

[In the Press. 
VOL. III. Containing the Translation. Demy Svo. 8j-. 

M. T. CICERONIS DE OFFICIIS LIBRI TRES, 

with Marginal Analysis, an English Commentary, and copious 
Indices, by H. A. HOLDEX, LL.D., late Fellow of Trinity College, 
Cambridge. Fourth Edition. Revised and considerably enlarged. 
Crown Svo. 9^. 

"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 OVIRITES With Notes Introduc 
tion and Appendices by W E HEITLAND MA, Fellow and Lecturer 
of St John s College, Cambridge. Demy Svo. Js. 6d. 

London : Cambridge University Press Warehouse, 17 Paternoster Row. 



12 PUBLICATIONS OF 

At. 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. 6d. Vol. II. i2s. 6d. 

" Such editions as that of which Prof. Mayor way admirably suited to meet the needs of the 

has given us the first instalment will doubtless student . . . The notes of the editor are all that 

do much to remedy this undeserved neglect. It could be expected from his well-known learn - 

is one on which great pains and much learning ing and scholarship." Academy. 
have evidently been expended, and is in every 

P. VERGILI MARONIS OPERA cum Prolegomenis 
et Commentario Critico pro Syndicis Preli Academici edidit BEN 
JAMIN HALL KENNEDY, S.T.P., Graecae Linguae Professor Regius. 
Extra Fcap. 8vo. $s. 

MATHEMATICS, PHYSICAL 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 1841, to the present time. Vol. I. 
Demy 8vo. i&r. [Vol.11. In the Press. 

Wherever exact science has found a fol- borne rich and abundant fruit. Twenty years 

lower Sir William Thomson s name is known as after its date the International Conference of 

a leader and a master. For a space of 40 years Electricians at Paris, assisted by the author 

each of his successive contributions to know- himself, elaborated and promulgated a series of 

ledge in the domain of experimental and mathe- rules and units which are but the detailed out- 

matical physics has been recognized as marking come of the principles laid down in these 

a stage in the progress of the subject. But, un- papers." The Times. 

happily for the mere learner, he is no writer of "We are convinced that nothing has had a 
text-books. His eager fertility overflows into greater effect on the progress of the theories of 
the nearest available journal . . . The papers in electricity and magnetism during the last ten 
this volume deal largely with the subject of the years than the publication of Sir W. Thomson s 
dynamics of heat. They begin with two or reprint of papers on electrostatics and magnet- 
three articles which were in part written at the ism, and we believe that the present volume is 
age of 17, before the author had commenced destined in no less degree to further the ad- 
residence as an undergraduate in Cambridge vancement of physical science. We owe the 
. . . No student of mechanical engineering, modern dynamical theory of heat almost wholly 
who aims at the higher levels of his profession, to Joule and Thomson, and Clausius and Ran- 
can afford to be ignorant of the principles and kine, and we have here collected together the 
methods set forth in these great memoirs . . . whole of Thomson s investigations on this sub- 
The article on the absolute measurement of ject, together with the papers published jointly 
electric and galvanic quantities (1851) has by himself and Joule." Glasgow Herald. 

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 
8vo. 15^. VOL. II. i5j. 

" The volume of Professor Stokes s papers necessary, dissertations. There nothing is 
contains much more than his hydrodynamical slurred over, nothing extenuated. We learn 
papers. The undulatory theory of light is exactly the weaknesses of the theory, and 
treated, and the difficulties connected with its the direction in which the completer theory of 
application to certain phenomena, such as aber- the future must be sought for. The same spirit 
ration, are carefully examined and resolved. pervades the papers on pure mathematics which 
Such difficulties are commonly passed over with are included in the volume. They have a severe 
scant notice in the text-books . . . Those to accuracy of style which well befits the subtle 
whom difficulties like these are real stumbling- nature of the subjects, and inspires the corn- 
blocks will still turn for enlightenment to Pro- pletest confidence in their author." The Times. 
fessor Stokes s old, but still fresh and still 

VOLUME III. In the Press. 

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. 

London: Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 13 

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. 
Part I. Demy 8vo. i6.r. 

" In this, the second edition, we notice a form within the time at our disposal would be 
large amount of new matter, the importance of utterly inadequate." Nature. 
which is such that any opinion which we could 

Part II. Demy 8vo. i8j. 

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. 12s. 

" 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- AtJienceum. 

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. I2>y. 

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. i6s. 

" It is time that Fourier s masterpiece, The process employed by the author." Contempo- 
Analytical TJteory of Heat, translated by Mr rary Revievu, October, 1878. 
Alex. Freeman, should be introduced to those "There cannot be two opinions as to the 
English students of Mathematics who do not value and importance of the Tfieorie de la Cha- 
follow with freedom a treatise in any language leur ... It is still t)ie text-book of Heat Con- 
but their own. It is a model of mathematical duction, and there seems little present prospect 
reasoning applied to physical phenomena, and of its being superseded, though it is already 
is remarkable for the ingenuity of the analytical more thaiu half a century old." Nature. 

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. i&r. 

"Every department of editorial duty ap- faction to Prof. Maxwell to see this goodly 
pears to have been most conscientiously per- volume completed before his life s work was 
formed ; and it must have been no small satis- done." Athenaum. 

AN ELEMENTARY TREATISE ON QUATERNIONS. 
By P. G. TAIT, M.A., Professor of Natural Philosophy in the Uni 
versity of Edinburgh. Second Edition. Demy Svo. 14^. 

THE MATHEMATICAL WORKS OF ISAAC BAR 
ROW, D.D. Edited by W. WHEWELL, D.D. Demy Svo. js. 6d. 

AN ATTEMPT TO TEST THE THEORIES OF 
CAPILLARY ACTION by FRANCIS BASHFORTH, B.D., late Pro 
fessor of Applied Mathematics to the Advanced Class of Royal 
Artillery Officers, Woolwich, and J. C. ADAMS, M.A., F.R.S. 
Demy 4to. i. is. 

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 410. js. 6d. 



London : Cambridge University Press Warehouse p , 17 Paternoster Row. 



PUBLICATIONS OF 



A TREATISE ON THE GENERAL PRINCIPLES OF 

CHEMISTRY, by M. M. PATTISON Mum, M.A., Fellow and Pre 
lector in Chemistry of Gonville and Caius College. Demy 8vo. 

[In the Press. 

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 
for the Year 1879. By WALTER KEEPING, M.A., F.G.S. Demy 8vo. 
icxr. 6d. 

COUNTERPOINT. A Practical Course of Study, by Pro 
fessor Sir G. A. MACFARREN, M.A., Mus. Doc. Fourth Edition, 
revised. Demy 4to. js. 6d. 

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 
1 846 to 1 860. 

ASTRONOMICAL OBSERVATIONS from 1861 to 1865. 
Vol. XXI. Royal 410. 15.?. 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. 
Strickland Curator in the University of Cambridge. Demy 8vo. i. I s. 

"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." Academy. carried it out." Notes and Queries. 

A CATALOGUE OF AUSTRALIAN FOSSILS (in 
cluding Tasmania and the Island of Timor), Stratigraphically and 
Zoologically arranged, by R. ETHERIDGE, Jun., F.G.S. , Acting Palae 
ontologist, H.M. Geol. Survey of Scotland. Demy 8vo. los. 6d. 

"The work is arranged with great clearness, consulted by the author, and an index to the 
and contains a full list of the books and papers genera." Saturday Review. 

ILLUSTRATIONS OF COMPARATIVE ANATOMY, 

VERTEBRATE AND INVERTEBRATE, for the Use of Stu 
dents in the Museum of Zoology and Comparative Anatomy. Second 
Edition. Demy 8vo. 2s. 6d. 

A SYNOPSIS OF THE CLASSIFICATION OF THE 
BRITISH PALAEOZOIC ROCKS, by the Rev. ADAM SEDGWICK, 
M.A., F.R.S., and FREDERICK M c CoY, F.G.S. One vol., Royal 410. 
Plates, l. is. 

A CATALOGUE OF THE COLLECTION OF CAM 
BRIAN AND SILURIAN FOSSILS contained in the Geological 
Museum of the University of Cambridge, by J. W. S ALTER, F.G.S. 
With a Portrait of PROFESSOR SEDGWICK. Royal 410. 7s. 6d. 

CATALOGUE OF OSTEOLOGICAL SPECIMENS con 
tained in the Anatomical Museum of the University of Cambridge. 
Demy 8vo. 2s. 6d. 

London : Cambridge University Press Warehouse, 1 7 Paternoster Row. 



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. js. 6d. 

"Prof. Clark s little book is the substance Students of jurisprudence will find much to 

of lectures delivered by him upon those ppr- interest and instruct them in the work of Prof, 

tions of Austin s work on jurisprudence which Clark," Athewzum. 
deal with the "operation of sanctions" . . . 

PRACTICAL JURISPRUDENCE, a Comment on AUSTIN. 
By E. C. CLARK, LL.D. Regius Professor of Civil Law. Crown 
Svo. 9-y. 

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 Svo. i&r. 

"Mr Willis-Bund has edited A Selection of as it may be gathered from trials before the 

Cases from the State Trials which is likely to ordinary courts. The author has very wisely 

form a very valuable addition to the standard distinguished these cases from those of im- 

literature . . . There can be no doubt, therefore, peachment for treason before Parliament, which 

of the interest that can be found in the State he proposes to treat in a future volume under 

trials. But they are large and unwieldy, and it the general head Proceedings in Parliament. " 

is impossible for the general reader to come T/ie Academy. 

across them. Mr Willis-Bund has therefore " This is a work of such obvious utility that 
done good service in making a selection that the only wonder is that no one should have un- 
is in the first volume reduced to a commodious dertaken it before ... In many respects there- 
form." TJie Examiner. fore, although the trials are more or less 
"This work is a very useful contribution to abridged, this is for the ordinary student s pur- 
that important branch of the constitutional his- pose not only a more handy, but a more useful 
tory of England which is concerned with the work than Howell s." Saturday Review. 
growth and development of the law of treason, 

VOL. II. In two parts. Price 145. each. 

" But, although the book is most interesting judicious selection of the principal statutes and 

to the historian of constitutional law, it is also the leading cases bearing on the crime of trea- 

not without considerable value to those who son ... For all classes of readers these volume? 

seek information with regard to procedure and possess an indirect interest, arising from the 

the growth of the law of evidence. We should nature of the cases themselves, from the men 

add that Mr Willis-Bund has given short pre- who were actors in them, and from the numerous 

faces and appendices to the trials, so as to form points of social life which are incidentally illus- 

a connected narrative of the events in history trated in the course of the trials. On these 

to which they relate. We can thoroughly re- features we have not dwelt, but have preferred 

commend the book. " Law Times. to show that the book is a valuable contribution 

" To a large class of readers Mr Willis- to the study of the subject with which it pro- 
Bund s compilation will thus be of great as- fesses to deal, namely, the history of the law of 
sistance, for he presents in a convenient form a treason." Athenaum. 

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 Svo. 6s. 

" In the present book we have the fruits of such a student will be interested as well as per- 
the same kind of thorough and well-ordered haps surprised to find how abundantly the ex- 
study which was brought to bear upon the notes tant fragments illustrate and clear up points 
to the Commentaries and the Institutes . . . which have attracted his attention in the Corn- 
Hitherto the Edict has been almost inac- mentanes, or the Institutes, or the Digest. 
cessible to the ordinary English student, and Law Times. 



London : Cambridge University Press Warehouse, 17 Paternoster Row. 



16 PUBLICATIONS OF 

AN INTRODUCTION TO THE STUDY OF JUS 
TINIAN S DIGEST. Containing an account of its composition 
and of the Jurists used or referred to therein, together with a full 
Commentary on one Title (de usufructu), by HENRY JOHN ROBY, 
M.A., formerly Classical Lecturer in St John s College, Cambridge, 
and Professor of Jurisprudence in University College, London. 

[/ 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. i6s. 

"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 Gaius to the studying Roman law in the original, and not a 

reader with few notes and tho"se merely by fanciful representation of it." Athenceum 

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. i6s. 

"We welcome here a valuable contribution the ordinary student, whose attention is dis- 

to the study of jurisprudence. The text of the tracted from the subject-matter by the dif- 

Institutes is occasionally perplexing, even to ficulty of struggling through the language in 

practised scholars, whose knowledge of clas- which it is contained, it will be almost indis- 

sical models does not always avail them in pensable." Spectator. 

dealing with the technicalities of legal phrase- "The notes are learned and carefully com 

ology. Nor can the ordinary dictionaries be piled, and this edition will be found useful to 

expected to furnish all the help that is wanted. students." Law Times. 
This translation will then be of great use. To 

SELECTED TITLES FROM THE DIGEST, annotated 
by B. WALKER, M.A., LL.D. Part I. Mandati vel Contra. Digest 
xvn. i. Crown 8vo. $s. 

"This small volume is published as an ex- Mr Walker deserves credit for the way in which 

periment. The author proposes to publish an he has performed the task undertaken. The 

annotated edition and translatipn of several translation, as might be expected, is scholarly." 

books of the Digest if this one is received with Law Times. 
favour. We are pleased to be able to say that 

Part II. De Adquirendo rerum dominio and De Adquirenda vel 

amittenda possessione. Digest XLI. I and n. Crown 8vo. 6s. 

Part III. De Condictionibus. Digest xn. i and 4 7 and Digest 

XIII. I 3. Crown 8vo. 6^. 

GROTIUS DE JURE BELLI ET PACIS, with the Notes 
of Barbeyrac and othefs ; 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, 6s. 



London: Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 



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. 

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, 



"He is, however, undoubtedly sound in the 
main, and his work deserves recognition as the 
result of immense industry and research in a 
field in which the labourers have hitherto been 
comparatively few." Scotsman. 

" Mr Cunningham is not likely to disap 
point any readers except such as begin by mis 
taking the character of his book. He does not 
promise, and does not give, an account of the 



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 Modern History in the University of 
Cambridge, with Portraits and Maps. 3 Vols. Demy 8vo. 48^. 



" 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 historians 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 
Stein 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." DeutscJie 
RundscJiau. 

" In a notice of this kind scant justice can 
be done to a work like the one before us ; no 
short resume can give even the most meagre 
notion of the contents of these volumes, which 
contain no page that is superfluous, and none 
that is uninteresting .... To understand the 



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." Atfiejueum. 

"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 
bridge 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 
produced a history of Germany in the Napo 
leonic age far superior to any that exists in 
German. " Examiner. 



THE UNIVERSITY OF CAMBRIDGE FROM THE 
EARLIEST TIMES TO THE ROYAL INJUNCTIONS OF 
1535, by JAMES BASS MULLINGER, M.A. Demy 8vo. (734 pp.), 12*. 



"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 



tinue his labours, and give us a history of the 
University during the troublous times of the 
Reformation and the Civil War. " A tlietueum. 
" Mr Mullinger s work is one of great learn 
ing and research, which can hardly fail to 
become a standard book of reference on the 
subject . . . We can most strongly recommend 
this book to our readers." Spectator, 



VOL. II. In the Press. 



London: Cambridge University Press Warehouse. 17 Paternoster Row. 



1 8 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. los. 

"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 useful." 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, M.A., Fellow of St John s. Two Vols. 
Demy 8vo. 24^. 

"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- still 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 Mayor has discharged his and academical, who have hitherto had to be 

editorial functions are creditable to his learning content with Dyer. " Academy. 
and industry." Athenceutn. 

HISTORY OF NEPAL, translated by MUNSHI SHEW 
SHUNKER SINGH and PANDIT SHR! GUNANAND; edited with an 
Introductory Sketch of the Country and People by Dr D. WRIGHT, 
late Residency Surgeon at Kathmandu, and with facsimiles of native 
drawings, and portraits of Sir JUNG BAHADUR, the KING OF NEPAL, 
&c. Super-royal 8vo. 2is. 

"The Cambridge University Press have interesting." Nature. 

done well in 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." Exattiiner. 
volume. The coloured lithographic plates are 

SCHOLAE ACADEMICAE: 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. 15^. 

"The general object of Mr Wordsworth s "Only those who have engaged in like la- 
book is sufficiently apparent from its title. He bours will be able fully to appreciate the 
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 genuine 
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 
Saturday Review. materials here collected." Academy. 

THE ARCHITECTURAL HISTORY OF THE UNI 
VERSITY AND COLLEGES OF CAMBRIDGE, by the late 
Professor WILLIS, M.A. With numerous Maps, Plans, and Illustra 
tions. Continued to the present time, and edited by JOHN WILLIS 
CLARK, M.A., formerly Fellow of Trinity College, Cambridge. 

\In the Press. 

London: Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 



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. Roxburgh 
(Morocco back), 2. 2s. 

"The object of the present work of Mich- 
aelis is to describe and make known the vast 
treasures of ancient 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 treasures in England are for the second 



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 
liversity Press for the liberal facilities 



time indebted for a full description of their rich 
possessions. Waagen gave to the private col 
lections of pictures the advantage of his in 
spection and 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 



of the University rress lor the liberal facilities 
afforded by them towards the production of 
this important volume by Professor Michaelis." 
Saturday Review. 

" Ancient Marbles here mean relics of 
Greek and Roman origin which have been 
imported into Great Britain from classical 
soil. How rich this island is in respect to 
these remains of ancient art, every one knows, 
but it is equally well known that these trea 
sures had been most inadequately 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 archaeology that it seems unneces 
sary to say how good a book this is." The 
A ntiquary. 



LECTURES 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 Svo. New Edition. 5.5-. 



"The lectures will be found most interest 
ing, and deserve to be carefully studied, not 
only by persons 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 after 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 
view. 

" It comprises fifteen lectures, 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 
exhaustive treatment, they yet leave very little 
of the ground uncovered ; and they combine in 
an admirable way the exposition of sound prin 
ciples with practical suggestions and illustra 
tions which are evidently derived from wide 
and varied experience, both in teaching and in 
examining." -Scotsman. 



"As principal of a training college and as a 
Government inspector of schools, Mr Fitch has 
got at his fingers ends the working of primary 
education, while as assistant commissioner to 
the late Endowed Schools Commission he has 
seen something of the machinery of our higher 
schools . . . Mr Fitch s book covers so wide a 
field and touches on so many burning questions 
that we must be content to recommend it as 
the best existing vade 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. 

"Therefore, without reviewing the book for 
the second time, we are glad to avail ourselves 
of the opportunity of calling attention to the 
re-issue of the volume in the five-shilling form, 
bringing it within the reach of the rank and 
file of the profession. We cannot let the oc 
casion pass without making special reference to 
the excellent section on punishments in the 
lecture on Discipline. " School Board Chron 
icle. 



THEORY AND PRACTICE OF TEACHING. By the 
Rev. EDWARD THRING, M.A., Head Master of Uppingham School, 
late Fellow of King s College, Cambridge. Crown 8vo. 6s. 

under the compulsion of almost passionate 



"Any attempt to summarize the contents of 
the volume would fail to give x>ur readers a 
taste of the pleasure that its perusal has given 
us." Journal of Education. 

"In his book we have something very dif 
ferent from the ordinary work on education. 
It is full of life. It comes fresh from the busy 
workshop of a teacher at once practical and 
enthusiastic, who has evidently taken up his 
pen, not for the sake of writing a book, but 



earnestness, to give expression to his views 
on questions connected with the teacher s life 
and work. For suggestiveness and clear in 
cisive statement of the fundamental problems 
which arise in dealing with the minds of chil 
dren, we know of no more useful book for any 
teacher who is willing to throw heart, and 
conscience, and honesty into his work." New 
York Evening Post. 



London : Cambridge University Press Warehouse, 1 7 Paternoster Row. 



20 PUBLICATIONS OF 

STATUTES OF THE UNIVERSITY OF CAMBRIDGE 

and for the Colleges therein, made published and approved (1878 
1882) under the Universities of Oxford and Cambridge Act, 1877. 
With an Appendix. Demy 8vo. i6s. 

THE WOODCUTTERS OF THE NETHERLANDS 

during the last quarter of the Fifteenth Century. In two parts. 
I. History of the Woodcutters. II. Catalogue of their Woodcuts. 
By WILLIAM MARTIN CONWAY. [/# the Press. 

THE DIPLOMATIC CORRESPONDENCE OF EARL 
GOWER, English Ambassador at the court of Versailles from June 
1790 to August 1792. From the originals in the Record Office with 
an introduction and Notes, by OSCAR BROWNING, M.A. [Preparing. 

A GRAMMAR OF THE IRISH LANGUAGE. By Prof. 

WINDISCH. Translated by Dr NORMAN MOORE. Crown 8vo. js. 6d. 

STATUTA ACADEMIC CANTABRIGIENSIS. Demy 
8vo. 2s. sewed. 

STATUTES OF THE UNIVERSITY OF CAMBRIDGE. 

With some Acts of Parliament relating to the University. Demy 
8vo. 3^. 6d. 

ORDINATIONES ACADEMIC CANTABRIGIENSIS. 

Demy 8vo. 3^. 6d. 

TRUSTS, STATUTES AND DIRECTIONS affecting 
(i) The Professorships of the University. (2) The Scholarships 
and Prizes. (3) Other Gifts and Endowments. Demy 8vo. 5^. 

COMPENDIUM OF UNIVERSITY REGULATIONS, 

for the use of persons in Statu Pupillari. Demy 8vo. 6d. 

CATALOGUE OF THE HEBREW MANUSCRIPTS 

preserved in the University Library, Cambridge. By Dr S. M. 
SCHiLLER-SziNESSY. Volume I. containing Section I. The Holy 
Scriptures ; Section II. Commentaries on the Bible. Demy 8vo. gs. 
Volume II. In the Press. 

A CATALOGUE OF THE MANUSCRIPTS preserved 
in the Library of the University of Cambridge. Demy 8vo. 5 Vols. 
icxr. each. 

INDEX TO THE CATALOGUE. Demy 8vo. los. 
A CATALOGUE OF ADVERSARIA and printed books 
containing MS. notes, preserved in the Library of the University of 
Cambridge. $s. 6d. 

THE ILLUMINATED MANUSCRIPTS IN THE LI 
BRARY OF THE FITZWILLIAM MUSEUM, Catalogued with 
Descriptions, and an Introduction, by WILLIAM GEORGE SEARLE, 
M.A., late Fellow of Queens College, and Vicar of Hockington, 
Cambridgeshire. Demy 8vo. js. 6d. 

A CHRONOLOGICAL LIST OF THE GRACES, 

Documents, and other Papers in the University Registry which 
concern the University Library. Demy 8vo. 2s. 6d. 

CATALOGUS BIBLIOTHEC^E BURCKHARDTIAN^. 

Demy 4to. $s. 

London: Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 21 



Cfee Cambridge $tble for ^eboote anti 



GENERAL EDITOR : THE VERY REVEREND J. J. S. PEROWNE, D.D., 
DEAN OF PETERBOROUGH. 



THE want of an Annotated Edition of trie BIBLE, in handy portions, suitable for 
School use, has long been felt. 

In order to provide Text-books for School and Examination purposes, the 
CAMBRIDGE UNIVERSITY PRESS has arranged to publish the several books of the 
BIBLE in separate portions at a moderate price, with introductions and explanatory- 
notes. 

The Very Reverend J. J. S. PEROWNE, D.D., Dean of Peterborough, has 
undertaken the general editorial supervision of the work, assisted by a staff of 
eminent coadjutors. Some of the books have been already edited or undertaken 
by the following gentlemen : 

Rev. A. CARR, M.A., Assistant Master at Wellington College. 

Rev. T. K. CHEYNE, M.A., Fellow of Balliol College, Oxford. 

Rev. S. Cox, Nottingham. 

Rev. A. B. DAVIDSON, D.D., Professor of Hebrew, Edinburgh. 

The Ven. F. W. FARRAR, D.D., Archdeacon of Westminster. 

C. D. GINSBURG, LL.D. 

Rev. A. E. HUMPHREYS, M.A., Fellow of Trinity College, Cambridge. 

Rev. A. F. KIRKPATRICK, M.A., Fellow of Trinity College, Regius Professor 
of Hebrew. 

Rev. J. J. LIAS, M.A., late Professor at St David s College, Lampeter. 

Rev. J. R. LUMBY, D.D., Norrisian Professor of Divinity. 

Rev. G. F. MACLEAR, D.D., Warden of St Augustine s College, Canterbury. 

Rev. H. C. G. MOULE, M.A., Fellow of Trinity College, Principal of Ridley 
Hall, Cambridge. 

Rev. W. F. MOULTON, D.D., Head Master of the Leys School, Cambridge. 

Rev. E. H. PEROWNE, D.D., Master of Corpus Christi College, Cambridge, 
Examining Chaplain to the Bishop of St Asaph. 

The Ven. T. T. PEROW T NE, M.A., Archdeacon of Norwich. 

Rev. A. PLUMMER, M.A., D.D., Master of University College, Durham. 

The Very Rev. E. H. PLUMPTRE, D.D., Dean of Wells. 

Rev. W. SIMCOX, M.A., Rector of Weyhill, Hants. 

ROBERTSON SMITH, M.A., Lord Almoner s Professor of Arabic. 

Rev. H. D, M. SPENCE, M.A., Hon. Canon of Gloucester Cathedral. 

Rev. A. W. STREANE, M.A., Fellofiu of Corpus Christi College, Cambridge. 



London : Cambridge University Press Warehouse, 17 Paternoster Row. 



PUBLICATIONS OF 



THE CAMBRIDGE BIBLE FOR SCHOOLS & COLLEGES. 

Continued. 

Now Ready. Cloth, Extra Fcap. 8vo. 

THE BOOK OF JOSHUA. By the Rev. G. F. MACLEAR, D.D. 

With i Maps. is. 6d. 

THE BOOK OF JUDGES. By the Rev. J. J. LIAS, M.A. 

With Map. 3-y. 6d. 

THE FIRST BOOK OF SAMUEL. By the Rev. Professor 
KIRKPATRICK, M.A. With Map. 3*. 6d. 

THE SECOND BOOK OF SAMUEL. By the Rev. Professor 
KIRKPATRICK, M.A. With 2 Maps. 3-y. 6d. 

THE BOOK OF ECCLESIASTES. By the Very Rev. E. H. 
PLUMPTRE, D.D., Dean of Wells. 5-5-. 

THE BOOK OF JEREMIAH. By the Rev. A. W. STREANE, 
M.A. With Map. 4 j. 6d. 

THE BOOKS OF OBADIAH AND JONAH. By Archdeacon 
PEROWNE. is. 6d. 

THE BOOK OF JONAH. By Archdeacon PEROWNE. is. 6d. 

THE BOOK OF MICAH. By the Rev. T. K. CHEYNE, M.A. 
is. 6d. 

THE GOSPEL ACCORDING TO ST MATTHEW. By the 
Rev. A. CARR, M.A. With i Maps. is. 6d. 

THE GOSPEL ACCORDING TO ST MARK. By the Rev. 
G. F. MACLEAR, D.D. With 2 Maps. is. 6d. 

THE GOSPEL ACCORDING TO ST LUKE. By Archdeacon 
F. W. FARRAR. With 4 Maps. 4.?. 6d. 

THE GOSPEL ACCORDING TO ST JOHN. By the Rev. 
A. PLUMMER, M.A., D.D. With 4 Maps. ^s. 6d. 

THE ACTS OF THE APOSTLES. By the Rev. Professor 
LUMBY, D.D. Part I. Chaps. I XIV. With 2 Maps. 2s. 6d. 

PART II. Chaps. XV. to end. With 2 Maps. 2s. 6d. 
PARTS I. and II., complete. With 4 Maps. 4$. 6d. 

THE EPISTLE TO THE ROMANS. By the Rev. H. C. G. 

MOULE, M.A. 3-r. 6d. 

THE FIRST EPISTLE TO THE CORINTHIANS. By the Rev. 
J. J. LIAS, M.A. With a Map and Plan. is. 

THE SECOND EPISTLE TO THE CORINTHIANS. By the 

Rev. J. J. LIAS, M.A. is. 

THE EPISTLE TO THE HEBREWS. By Archdeacon FARRAR. 
y. 6d. 

THE GENERAL EPISTLE OF ST JAMES. By the Very Rev. 
E. H. PLUMPTRE, D.D., Dean of Wells, is. 6d. 

THE EPISTLES OF ST PETER AND ST JUDE. By the 

same Editor, is. 6d. 



London : Cambridge Universitv Press Warehouse, 1 7 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 23 



THE CAMBRIDGE BIBLE FOR SCHOOLS & COLLEGES. 

Continued. 

Preparing. 

THE BOOK OF GENESIS. By ROBERTSON SMITH, M.A. 
THE BOOK OF EXODUS. By the Rev. C. D. GINSBURG, LL.D. 
THE BOOK OF JOB. By the Rev. A. B. DAVIDSON, D.D. 

THE BOOKS OF HAGGAI AND ZECHARIAH. By Arch 
deacon PEROWNE. 

THE EPISTLES OF ST JOHN. By the Rev. A. PLUMMER, 
M.A., D.D. 

THE BOOK OF REVELATION. By the Rev. W. SIMCOX, M.A. 



THE CAMBRIDGE GREEK TESTAMENT, 

FOR SCHOOLS AND COLLEGES, 

with a Revised Text, based on the most recent critical authorities, and 
English Notes, prepared under the direction of the General Editor, 

THE VERY REVEREND J. J. S. PEROWNE, D.D., 

DEAN OF PETERBOROUGH. 

Now Ready. 

THE GOSPEL ACCORDING TO ST MATTHEW. By the 
Rev. A. CARR, M.A. With 4 Maps. 4*. 6d. 

" With the Notes, in the volume before us, we are much pleased ; so far as we have searched, 
they are scholarly and sound. The quotations from the Classics are apt ; and the references to 
modern Greek form a pleasing feature." The Churchman. 

" Copious illustrations, gathered from a great variety of sources, make his notes a very valu 
able aid to the student. They are indeed remarkably interesting, while all explanations on 
meanings, applications, and the like are distinguished by their lucidity and good sense." 
Pall Mall Gazette. 

THE GOSPEL ACCORDING TO ST MARK. By the Rev. 
G. F. MACLEAR, D.D. With 3 Maps. 4*. 6d. 

"The Cambridge Greek Testament, of which Dr Maclear s edition of the Gospel according to 
St Mark is a volume, certainly supplies a want. Without pretending to compete with the leading 
commentaries, or to embody very much original research, it forms a most satisfactory introduction 
to the study of the New Testament in the original . . . Dr Maclear s introduction contains all that 
is known of St Mark s life, with references to passages in the New Testament in which he is 
mentioned ; an account of the circumstances in which the Gospel was composed, with an estimate 
of the influence of St Peter s teaching upon St Mark ; an excellent sketch of the special character 
istics of this Gospel ; an analysis, and a chapter on the text of the New Testament generally . . . 
The work is completed by two good maps, one of Palestine in the time of our Lord, the other, on 
a large scale, of the Sea of Galilee and the country immediately surrounding it." Saturday 
Review. 

"The Notes, which are admirably put together, seem to contain all that is necessary for the 
guidance of the student, as well as a judicious selection of passages from various sources illustrat 
ing scenery and manners." Academy. 

THE GOSPEL ACCORDING TO ST LUKE. By Archdeacon 
FARRAR. \_Nearly ready. 

THE GOSPEL ACCORDING TO ST JOHN. By the Rev. A. 
PLUMMER, M.A., D.D. With 4 Maps. 6s. 

"A valuable addition has also been made to The Cambridge Greek Testament for Schools, 
Dr Plummer s notes on the Gospel according to St John are scholarly, concise, and instructive, 
and embody the results of much thought and wide reading." Expositor. 

London: Cambridge University Press Warehouse, 17 Paternoster Row 



24 PUBLICATIONS OF 



THE PITT PRESS SERIES. 



I. GREEK. 

THE ANABASIS OF XENOPHON, BOOKS I. III. IV. 

and V. With a Map and English Notes by ALFRED PRETOR, M.A., Fellow 
of St Catharine s College, Cambridge ; Editor of Persius and Cicero ad Atti- 
cum Book I. is. each. 

_" In Mr Pretor s edition of the Anabasis the text of Kiihner has been followed in the main, 
while the exhaustive and admirable notes of the great German editor have been largely utilised. 
These notes deal with the minutest as well as the most important difficulties in construction, and 
all questions of history, antiquity, arid geography are briefly but very effectually elucidated." The 
Examiner. 

"We welcome this addition to the other books of the Anabasis so ably edited by Mr Pretor. 
Although originally intended for the use of candidates at the university local examinations, yet 
this edition will be found adapted not only to meet the wants of the junior student, but even 
advanced scholars will find much in this work that will repay its perusal." The Schoolmaster. 

"Mr Pretor s Anabasis of Xenophon, Book IV. displays a union of accurate Cambridge 
scholarship, with experience of what is required by learners gained in examining middle-class 
schools. The text is large and clearly printed, and the notes explain all difficulties. . . . Mr 
Pretor s notes seem to be all that could be wished as regards grammar, geography, and other 
matters." The Academy. 

BOOKS II. VI. and VII. By the same Editor. 2s. 6d. each. 

"Another Greek text, designed it would seem for students preparing for the local examinations, 
is Xenophon s Anabasis, Book II., with English Notes, by Alfred Pretor, M.A. The editor has 
exercised his usual discrimination in utilising the text and notes of Kuhner, with the occasional 
assistance of the best hints of Schneider, Vollbrecht and Macmichael on critical matters, and of 
Mr R. W. Taylor on points of history and geography. . . When Mr Pretor commits himself to 
Commentator s work, he is eminently helpful. . . Had we to introduce a young Greek scholar 
to Xenophon,, we should esteem ourselves fortunate in having Pretor s text-book as our chart and 
guide." Contemporary Review. 

THE ANABASIS OF XENOPHON, by A. PRETOR, M.A., 

Text and Notes, complete in two Volumes. js. 6d. 

AGESILAUS OF XENOPHON. The Text revised 

with Critical and Explanatory Notes, Introduction, Analysis, and Indices. 
By H. HAILSTONE, M.A., late Scholar of Peterhouse, Cambridge, Editor of 
Xenophon s Hellenics, etc. is. 6d. 

ARISTOPHANES RANAE. With English Notes and 

Introduction by W. C. GREEN, M.A., Assistant Master at Rugby School. 
3J-. 6d. 

ARISTOPHANES AVES. By the same Editor. New 

Edition. 3-y. 6d. 

"The notes to both plays are excellent. Much has been done in these two volumes to render 
the study of Aristophanes a real treat to a boy instead of a drudgery, by helping him to under 
stand the fun and to express it in his mother tongue." The Examiner. 

ARISTOPHANES PLUTUS. By the same Editor. $s.6d. 

EURIPIDES. HERCULES FURENS. With Intro 
ductions, Notes and Analysis. ByJ. T. HUTCHINSON, M.A., Christ s College, 
and A. GRAY, M.A., Fellow of Jesus College, is. 

"Messrs Hutchinson and Gray have produced a careful and useful edition." Saturday 
Review. 

THE HERACLEID^E OF EURIPIDES, with Introduc 
tion and Critical Notes by E. A. BECK, M.A., Fellow of Trinity Hall. 3.?. 6d. 

London : Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 25 

LUCIANI SOMNIUM CHARON PISCATOR ET DE 

LUCTU, with English Notes by W. E.. HEITLAND, M.A., Fellow of 
St John s College, Cambridge. New Edition, with Appendix. $s. 6d. 

OUTLINES OF THE PHILOSOPHY OF ARISTOTLE. 

Edited by E. WALLACE, M.A. (See p. 30.) 



II. LATIN. 

M. T. CICERONIS DE AMICITIA. Edited by J. S. 

REID, M.L., Fellow and Assistant Tutor of Gonville and Caius College, 
Cambridge. New Edition, with Additions. $s. 6d. 

"Mr Reid has decidedly attained his aim, namely, a thorough examination of the Latinity 

of the dialogue. The revision of the text is most valuable, and comprehends sundry 

acute corrections. . . . This volume, like Mr Reid s other editions, is a solid gain to the scholar 
ship of the country." AtJien&um. 

"A more distinct gain to scholarship is Mr Reid s able and thorough edition of the De 
Amicitia of Cicero, a work of which, whether we regard the exhaustive introduction or the 
instructive and most suggestive commentary, it would be difficult to speak too highly. . . . When 
we come to the commentary, we are only amazed by its fulness in proportion to its bulk. 
Nothing is overlooked which can tend to enlarge the learner s general knowledge of Ciceronian 
Latin or to elucidate the text." Saturday Review. 

M. T. CICERONIS CATO MAJOR DE SENECTUTE. 

Edited by J. S. REID, M.L. 3*. &/. 

" The notes are excellent and scholarlike, adapted for the upper forms of public schools, and 
likely to be useful even to more advanced students." Guardian. 

M. T. CICERONIS ORATIO PRO ARCHIA POETA. 

Edited by J. S. REID, M.L. is. 6d. 

41 It is an admirable specimen of careful editing. An Introduction tells us everything we could 
wish to know about Archias, about Cicero s connexion with him, about the merits of the trial, and 
the genuineness of the speech. The text is well and carefully printed. The notes are clear and 
scholar-like. . . . No boy can master this little volume without feeling that he has advanced a long 
step in scholarship." TJte Academy. 

M. T. CICERONIS PRO L. CORNELIO BALBO ORA- 

TIO. Edited by J. S. REID, M.L. is. 6d. 

" We are bound to recognize the pains devoted in the annotation of these two orations to the 
minute and thorough study of their Latinity, both in the ordinary notes and in the textual 
appendices." Saturday Review. 

M. T. CICERONIS PRO P. CORNELIO SULLA 

ORATIO. Edited by J. S. REID, M.L. 3 j. 6d. 

"Mr Reid is so well known to scholars as a commentator on Cicero that a new work from him 
scarcely needs any commendation of ours. His edition of the speech Pro Sulla is fully equal in 
merit to the volumes which he has already published ... It would be difficult to speak too highly 
of the notes. There could be no better way of gaining an insight into the characteristics of 
Cicero s style and the Latinity of his period than by making a careful study of this speech with 
the aid of Mr Reid s commentary . . . Mr Reid s intimate knowledge of the minutest details of 
scholarship enables him to detect and explain the slightest points of distinction between the 
usages of different authors and different periods . . . The notes are followed by a valuable 
appendix on the text, and another on points of orthography ; an excellent index brings the work 
to a close." Saturday Review. 

M. T. CICERONIS PRO CN. PLANCIO ORATIO. 

Edited by H. A. HOLDEN, LL.D., late Head Master of Ipswich School. 
4-r. 6d. 

"As a book for students this edition can have few rivals. It is enriched by an excellent intro 
duction and a chronological table of the principal events of the life of Cicero ; while in its ap 
pendix, and in the notes on the text which are added, there is much of the greatest value. The 
volume is neatly got up, and is in every way commendable." The Scotsman. 

"Dr Holden s own edition is all that could be expected from his elegant and practised 
scholarship. ... Dr Holden has evidently made up his mind as to the character of the 
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 unusually full 



and complete; and after going through it carefully, we find little or nothing to criticize. T] 
is an excellent introduction, lucidly explaining the circumstances under which the speech was 



There 
ch was 
delivered, a table of events in the life of Cicero and a useful index." Spectator, Oct. 29, 1881. 

London : Cambridge University Press Warehouse, 17 Paternoster Row. 



26 PUBLICATIONS OF 

M. T. CICERONIS IN Q. CAECILIUM DIVINATIO 

ET IN C. VERREM ACTIO PRIMA. With Introduction and Notes 
by W. E. HEITLAND, M.A., and HERBERT COWIE, M.A., Fellows of 
St John s College, Cambridge. y. 

M. T. CICERONIS ORATIO PRO L. MURENA, with 

English Introduction and Notes. By W. E. HEITLAND, M.A., Fellow 
and Classical Lecturer of St John s College, Cambridge. Second Edition, 
carefully revised. 3-5-. 

"Those students are to be deemed fortunate who have to read Cicero s lively and brilliant 
oration for L. Murena with Mr Heitland s handy edition, which may be pronounced four-square 
in point of equipment, and which has, not without good reason, attained the honours of a 
second edition." Saturday Review. 

M, T. CICERONIS IN GAIUM VERREM ACTIO 

PRIMA. With Introduction and Notes. By H. COWIE, M.A., Fellow 
of St John s College, Cambridge, is. 6d. 

M. T. CICERONIS ORATIO PRO T. A. MILONE, 

with a Translation of Asconius Introduction, Marginal Analysis and 
English Notes. Edited by the Rev. JOHN SMYTH PURTON, B.D., late 
President and Tutor of St Catharine s College. vs. 6d. 
"The editorial work is excellently done." The Academy. 

M. T. CICERONIS SOMNIUM SCIPIONIS. With In 
troduction and Notes. By W. D. PEARMAN, M.A., Head Master of Potsdam 
School, Jamaica, vs. 

P. OVIDII NASONIS FASTORUM LIBER VI. With 

a Plan of Rome and Notes by A. SIDGWICK, M.A. Tutor of Corpus Christi 
College, Oxford, is. 6d. 

" Mr Sidgwick s editing of the Sixth Book of Ovid s Fasti furnishes a careful and serviceable 
volume for average students. It eschews construes which supersede the use of the dictionary, 
but gives full explanation of grammatical usages and historical and mythical allusions, besides 
illustrating peculiarities of style, true and false derivations, and the more remarkable variations ol 
the text." Saturday Review. 

" It is eminently good and useful. . . . The Introduction is singularly clear on the astronomy of 
Ovid, which is properly shown to be ignorant and confused ; there is an excellent little map of 
Rome, giving just the places mentioned in the text and no more ; the notes are evidently written 
by a practical schoolmaster." The Academy. 

GAI IULI CAESARIS DE BELLO GALLICO COM 
MENT. I. II. With English Notes and Map by A. G. PESKETT, M.A., 
Fellow of Magdalene College, Cambridge, Editor of Caesar De Bello Gallico, 
VII. is. 6d. 

BOOKS III. AND VI. By the same Editor, is. 6d. each. 

" In an unusually succinct introduction he gives all the preliminary and collateral information 
that is likely to be useful to a young student ; and, wherever we have examined his notes, we 
have found them eminently practical and satisfying. . . The book may well be recommended for 
careful study in school or college." Saturday Review. 

"The notes are scholarly, short, and a real help to the most elementary beginners in Latin 
prose." The Examiner. 

BOOKS IV. AND V. AND BOOK VII. by the same Editor. 
2s. each. 

BOOK VIII. by the same Editor. [In the Press. 



London : Cambridge University Press Warehouse, 1 7 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 27 

P. VERGIL1 MARONIS AENEIDOS LIBRI L, II., IV., 

V., VI., VII., VIII., IX., X., XL, XII. Edited with Notes by A. 
SIDGWICK, M.A. Tutor of Corpus Christi College, Oxford, is. 6d. each. 

"Much more attention is given to the literary aspect of the poem than is usually paid to it in 
editions intended for the use of beginners. The introduction points out the distinction between 
primitive and literary epics, explains the purpose of the poem, and gives an outline of the story." 
Saturday Review. 

" Mr Arthur Sidgwick s Vergil, Aeneid, Book XII. is worthy of his reputation, and is dis 
tinguished by the same acuteness and accuracy of knowledge, appreciation of a boy s difficulties 
and ingenuity and resource in meeting them, which we have on other occasions had reason to 
praise in these pages." Tfie Academy. 

"As masterly in its clearly divided preface and appendices as in the sound and independent 
character of its annotations. . . . There is a great deal more in the notes than mere compilation 
and suggestion. ... No difficulty is left unnoticed or unhandled." Saturday Review. 

"This edition is admirably adapted for the use of junior students, who will find in it the result 
of much reading in a condensed form, and clearly expressed." Cambridge Independent Press. 

BOOKS VII. VIII. in one volume. 3* 

BOOKS IX. X. in one volume. 3*. 

BOOKS X., XL, XII. in one volume. 3*. 6d. 

QUINTUS CURTIUS. A Portion of the History. 

(ALEXANDER IN INDIA.) By W. E. HEITLAND, M. A., Fellow and Lecturer 
of St John s College, Cambridge, and T. E. RAVEN, B.A., Assistant Master 
in Sherborne School. 3^. 6d. 

"Equally commendable as a genuine addition to the existing stock of school-books is 
Alexander in India, a compilation from the eighth and ninth books of Q. Curtius, edited for 
the Pitt Press by Messrs Heitland and Raven. . . . The work of Curtius has merits of its 
own, which, in former generations, made it a favourite with English scholars, and which still 

make it a popular text-book in Continental schools The reputation of Mr Heitland is a 

sufficient guarantee for the scholarship of the notes, which are ample without being excessive, 
and the book is well furnished with all that is needful in the nature of maps, indexes, and ap 
pendices." Academy. 

M. ANNAEI LUCANI PHARSALIAE LIBER 

PRIMUS, edited with English Introduction and Xotes by \V. E. HEITLAND, 
M.A. and C. E. RASKINS, M.A., Fellows and Lecturers of St John s Col 
lege, Cambridge, is. 6d. 

"A careful and scholarlike production." Times. 

" In nice parallels of Lucan from Latin poets and from Shakspeare, Mr Haskins and Mr 
Heitland deserve praise." Saturday Re-view. 

BEDA S ECCLESIASTICAL HISTORY, BOOKS 

III., IV., the Text from the very ancient MS. in the Cambridge University 
Library, collated with six other MSS. Edited, with a life from the German of 
EBERT, and with Notes, &c. by J. E. B. MAYOR, M.A., Professor of Latin, 
and J. R. LUMBY, D.D., Norrisian Professor of Divinity. Revised edition. 
is. 6d. 

"To young students of English History the illustrative notes will be of great service, while 
the study of the texts will be a good introduction to Mediaeval Latin." The Nonconformist. 

"In Bede s works Englishmen can go back to origines of their history, unequalled for 
form and matter by any modern European nation. Prof. Mayor has done good service in ren 
dering a part of Bede s greatest work accessible to those who can read Latin with ease. He 
has adorned this edition of the third and fourth books of the Ecclesiastical History with that 
amazing erudition for which he is unrivalled among Englishmen and rarely equalled by Germans. 
And however interesting and valuable the text may be, we can certainly apply to his notes 
the expression, La sauce z aut mieux que le poisson. They are literally crammed with interest 
ing information about early English life. For though ecclesiastical in name, Bede s history treats 
of all parts of the national life, since the Church had points of contact with all." Examiner. 

BOOKS I. and II. In the Press. 



London : Cambridge University Press Warehouse, 17 Paternoster Row. 



28 P UBLICA TIONS OF 



Ml. FRENCH. 

LE BOURGEOIS GENTILHOMME, Commie-Ballet en 

Cinq Actes. Par J.-B. POQUELIN DE MOLIERE (1670). With a life of 
Moliere and Grammatical and Philological Notes. By the Rev. A. C. 
CLAPIN, M.A., St John s College, Cambridge, and Bachelier-es-Lettres of 
the University of France, is. 6d. 

LA PICCIOLA. By X. B. SAINTINE. The Text, with 

Introduction, Notes and Map, by the same Editor, is. 

LA GUERRE. By MM. ERCKMANN-CHATRIAN. With 

Map, Introduction and Commentary by the same Editor. 3^. 

LAZARE HOCHE PAR EMILE DE BONNECHOSE. 

With Three Maps, Introduction and Commentary, by C. COLBECK, M.A., 
late Fellow of Trinity College, Cambridge; Assistant Master at Harrow 
School. is. 

HISTOIRE DU SIECLE DE LOUIS XIV PAR 

VOLTAIRE. Parti. Chaps. I. XIII. Edited with Notes Philological and 
Historical, Biographical and Geographical Indices, etc. by GUSTAVE MASSON, 
B.A. Univ. Gallic., Officier d Academie, Assistant Master of Harrow School, 
and G. W. PROTHERO, M.A., Fellow and Tutor of King s College, Cam 
bridge, is. 6d. 

"Messrs Masson and Prothero have, to judge from the first part of their work, performed 
with much discretion and care the task of editing Voltaire s Siecle de Louis XIV for the Pitt 
Press Series. Besides the usual kind of notes, the editors have in this case, influenced by Vol 
taire s summary way of treating much of the history, given a good deal of historical informa 
tion, in which they have, we think, done well. At the beginning of the book will be found 
excellent and succinct accounts of the constitution of the French army and Parliament at the 
period treated of." Saturday Review. 

Part II. Chaps. XIV. XXIV. With Three Maps of the 

Period. By the same Editors, is. 6d. 

Part III. Chap. XXV. to the end. By the same Editors. 

is. 6d. 

LE VERRE D EAU. A Comedy, by SCRIBE. With a 

Biographical Memoir, and Grammatical, Literary and Historical Notes. By 
C. COLBECK, M.A., late Fellow of Trinity College, Cambridge; Assistant 
Master at Harrow School, is. 

"It may be national prejudice, but we consider this edition far superior to any of the series 
which hitherto have been edited exclusively by foreigners. Mr Colbeck seems better to under 
stand the wants and difficulties of an English boy. The etymological notes especially are admi 
rable. . . . The historical notes and introduction are a piece of thorough honest work." Journal 
of Education, 

M. DARU, par M. C. A. SAINTE-BEUVE, (Causeries du 

Lundi, Vol. IX.). With Biographical Sketch of the Author, and Notes 
Philological and Historical. By GUSTAVE MASSON. is. 

LA SUITE DU MENTEUR. A Comedy in Five Acts, 

by P. CORNEILLE. Edited with Fontenelle s Memoir of the Author, Voltaire s 
Critical Remarks, and Notes Philological and Historical. By GUSTAVE 
MASSON. is. 

LA JEUNE SIBERIENNE. LE LEPREUX DE LA 

CITfi D AOSTE. Tales by COUNT XAVIER DE MAISTRE. With Bio 
graphical Notice, Critical Appreciations, and Notes. By GUSTAVE MASSON. 



London: Cambridge University Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 29 

LE DIRECTOIRE. (Considerations sur la Revolution 

Franaise. Troisieme et quatrieme parties.) Par MADAME LA BARONNE DE 
STAEL-HOLSTEIN. With a Critical Notice of the Author, a Chronological 
Table, and Notes Historical and Philological, by G. MASSON, B.A., and 
G. W. PROTHERO, M.A. Revised and enlarged Edition, is. 

" Prussia under Frederick the Great, and France under the Directory, bring us face to face 
respectively with periods of history which it is right should be known thoroughly, and which 
are well treated in the Pitt Press volumes. The latter in particular, an extract from the 
world-known work of Madame de Stae l on the French Revolution, is beyond all praise for 
the excellence both of its style and of its matter." Times. 

DIX ANNEES D EXIL. LIVRE II. CHAPITRES i 8. 

Par MADAME LA BARONNE DE STAEL-HOLSTEIN. With a Biographical 
Sketch of the Author, a Selection of Poetical Fragments by Madame de 
Stael s Contemporaries, and Notes Historical and Philological. By GUSTAVE 
MASSON and G. W. PROTHERO, M.A. Revised and enlarged edition, is. 

FREDEGONDE ET BRUNEHAUT. A Tragedy in Five 

Acts, by N. LEMERCIER. Edited with Notes, Genealogical and Chrono 
logical Tables, a Critical Introduction and a Biographical Notice. By 
GUSTAVE MASSON. is. 

LE VIEUX CELIBATAIRE. A Comedy, by COLLIN 

D HARLEVILLE. With a Biographical Memoir, and Grammatical, Literary 
and Historical Notes. By the same Editor, is. 

" M. Masson is doing good work in introducing learners to some of the less-known French 
play-writers. The arguments are admirably clear, and the notes are not too abundant." 
A cadenty. 

LA METROMANIE, A Comedy, by PlRON, with a Bio 
graphical Memoir, and Grammatical, Literary and Historical Notes. By the 
same Editor, is. 

LASCARIS, ou LES GRECS DU XV E . SIECLE, 

Nouvelle Historique, par A. F. VILLEMAIN, with a Biographical Sketch of 
the Author, a Selection of Poems on Greece, and Notes Historical and 
Philological. By the same Editor, is. 



IV. GERMAN. 

CULTURGESCHICHTLICHE NOVELLEN, von W. H. 

RIEHL, with Grammatical, Philological, and Historical Notes, and a Com 
plete Index, by H. J. WOLSTENHOLME, B.A. (Lond.). 4^. 6d. 

ERNST, HERZOG VON SCHWABEN. UHLAND. With 

Introduction and Notes. By H. J. WOLSTENHOLME, B.A. (Lond.), 
Lecturer in German at Newnham College, Cambridge. 3-5-. 6d. 

ZOPF UND SCHWERT. Lustspiel in fiinf Aufzugen von 

KARL GUTZKOW. With a Biographical and Historical Introduction, English 
Notes, and an Index. By the same Editor. 3^. 6d. 

"We are glad to be able to notice a careful edition of K. Gutzkow s amusing comedy 
Zopf and Schwert by Mr H. J. Wolstenholme. . . . These notes are abundant and contain 
references to standard grammatical works." Academy. 



London: Cambridge University Press Warehouse, 17 Paternoster Row. 



30 PUBLICATIONS OF 

oetfye S tfnabenjaljjre. (17491759.) GOETHE S BOY- 

HOOD: being the First Three Books of his Autobiography. Arranged 
and Annotated by WILHELM WAGNER, Ph. D., late Professor at the 
Johanneum, Hamburg, is. 

HAUFF. DAS WIRTHSHAUS IM SPESSART. Edited 

by A. SCHLOTTMANN, Ph.D., Assistant Master at Uppingham School. 
3J. 6d. 

DER OBERHOF. A Tale of Westphalian Life, by KARL 

IMMERMANN. With a Life of Immermann and English Notes, by WILHELM 
WAGNER, Ph.D., late Professor at the Johanneum, Hamburg. y. 

A BOOK OF GERMAN DACTYLIC POETRY. Ar- 

ranged and Annotated by the same Editor. $s. 

l Der erfte reu$aug (THE FIRST CRUSADE), by FRIED- 
RICH VON RAUMER. Condensed from the Author s History of the Hohen- 
staufen , with a life of RAUMER, two Plans and English Notes. By 
the same Editor, is. 

"Certainly no more interesting book could be made the- subject of examinations. The story 
of the First Crusade has an undying interest. The notes are, on the whole, good." Educational 
Times. 

A BOOK OF BALLADS ON GERMAN HISTORY. 

Arranged and Annotated by the same Editor, is. 

"It carries the reader rapidly through some of the most important incidents connected with 
the German race and name, from the invasion of Italy by the Visigoths under their King Alaric, 
down to the Franco-German War and the installation of the present Emperor. The notes supply 
very well the connecting links between the successive periods, and exhibit in its various phases of 
growth and progress, or the reverse, the vast unwieldy mass which constitutes modern Germany." 
Times. 

DER STAAT FRIEDRICHS DES GROSSEN. By G. 

FREYTAG. With Notes. By the same Editor, is. 

"Prussia under Frederick the Great, and France under the Directory, bring us face to face 
respectively with periods of history which it is right should be known thoroughly, and which 
are well treated in the Pitt Press volumes." Times. 

GOETHE S HERMANN AND DOROTHEA. With 

an Introduction and Notes. By the same Editor. Revised edition by J. W. 
CARTMELL, M.A. $s. 6d. 

"The notes are among the best that we know, with the reservation that they are often too 
abundant." Academy. 

3afyr 1813 (THE YEAR 1813), by F. KOHLRAUSCH. 

With English Notes. By the same Editor, is. 



V. ENGLISH. 

OUTLINES OF THE PHILOSOPHY OF ARISTOTLE. 

Compiled by EDWIN WALLACE, M.A., LL.D. (St Andrews), Fellow and 
Tutor of Worcester College, Oxford. Third Edition Enlarged. 4^. 6d. 

THREE LECTURES ON THE PRACTICE OF EDU 
CATION. Delivered in the University of Cambridge in the Easter Term, 
1882, under the direction of the Teachers Training Syndicate, is. 

" Like one of Bacon s Essays, it handles those things in which the writer s life is most conver 
sant, and it will come home to men s business and bosoms. Like Bacon s Essays, too, it is full of 
apophthegms. " Journal of Education. 

GENERAL AIMS OF THE TEACHER, AND FORM 

MANAGEMENT. Two Lectures delivered in the University of Cambridge 
in the Lent Term, 1883, by F. W. FARRAR, D.D. Archdeacon of West 
minster, and R. B. POOLE, B.D. Head Master of Bedford Modern School. 



London: Cambridge Universitv Press Warehouse, 17 Paternoster Row. 



THE CAMBRIDGE UNIVERSITY PRESS. 31 

MILTON S TRACTATE ON EDUCATION. A fac- 

simile reprint from the Edition of 1673. Edited, with Introduction and 
Notes, by OSCAR BROWNING. M.A., Fellow and Lecturer of King s College, 
Cambridge, and formerly Assistant Master at Eton College, is. 
"A separate reprint of Milton s famous letter to Master Samuel Hartlib was a desideratum, 
and we are grateful to Mr Browning for his elegant and scholarly edition, to which is prefixed the 
careful resumi of the work given in his History of Educational Theories. " Journal of 
Education. 

LOCKE ON EDUCATION. With Introduction and Notes 

by the Rev. R. H. QUICK, M.A. y. 6d. 

"The work before us leaves nothing to be desired. It is of convenient form and reasonable 
price, accurately printed, and accompanied by notes which are admirable. There is no teacher 
too young to find this book interesting; there is no teacher too old to find it profitable." The 
School Bulletin, New York. 

THE TWO NOBLE KINSMEN, edited with Intro 
duction and Notes by the Rev. Professor SKEAT, M.A., formerly Fellow 
of Christ s College, Cambridge. $s. 6d. 

"This edition of a play that is well worth study, for more reasons than one, by so careful a 
scholar as Mr Skeat, deserves a hearty welcome." Athetueum. 

"Mr Skeat is a conscientious editor, and has left no difficulty unexplained." Times. 

BACON S HISTORY OF THE REIGN OF KING 

HENRY VII With Notes by the Rev. J. RAWS ON LUMBY, D.D., Nor- 
risian Professor of Divinity ; late Fellow of St Catharine s College. 3^. 

SIR THOMAS MORE S UTOPIA. With Notes by the 

Rev. J. RA\VSON LUMBY, D.D., Norrisian Professor of Divinity ; late Fellow 
of St Catharine s College, Cambridge. %s. 6d. 

"To Dr Lumby we must give praise unqualified and unstinted. He has done his work 

admirably Every student of history, every politician, every social reformer, every one 

interested in literary curiosities, every lover of English should buy and carefully read Dr 
Lumby s edition of the Utopia. We are afraid to say more lest we should be thought ex 
travagant, and our recommendation accordingly lose part of its force." T/ie Teacher. 

" It was originally written in Latin and does not find a place on ordinary bookshelves. A very- 
great boon has therefore been conferred on the general English reader by the managers of the 
Pitt Press Series, in the issue of a convenient little volume of More s Utopia not in the original 
Latin, but in the quaint English Translation thereof made by Raphe Robynson, which adds a 
linguistic interest to the intrinsic merit of the work. . . . All this has been edited in a most com 
plete and scholarly fashion by Dr J. R. Lumby, the Norrisian Professor of Divinity, whose name 
alone is a sufficient warrant for its accuracy. It is a real addition to the modern stock of classical 
English literature." Guardian. 

MORE S HISTORY OF KING RICHARD III. Edited 

with Notes, Glossary and Index of Names. By J. RAWSON LUMBY, D.D. 
Norrisian Professor of Divinity, Cambridge ; to which is added the conclusion 
of the History of King Richard III. as given in the continuation of Hardyng s 
Chronicle, London, 1543. y. 6d. 

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. 3^. 6d. 

"In writing this scholarly and attractive sketch, Professor Mayor has had chiefly in view 
undergraduates at the University or others who are commencing the study of the philosophical 
works of Cicero or Plato or Aristotle in the original language, but also hopes that it may be 
found interesting and useful by educated readers generally, not merely as an introduction to the 
formal history of philosophy, but as supplying a key to our present ways of thinking and judging 
in regard to matters of the highest importance. " Mind. 

"Professor Mayor contributes to the Pitt Press Series A Sketch of Ancient Philosophy in 
which he has endeavoured to give a general view of the philosophical systems illustrated by the 
genius of the masters of metaphysical and ethical science from Thales to Cicero. In the course 
of his sketch he takes occasion to give concise analyses of Plato s Republic, and of the Ethics and 
Politics of Aristotle ; and these abstracts will be to some readers not the least useful portions ot 
the book. It may be objected against his design in general that ancient philosophy is too vast 
and too deep a subject to be dismissed in a sketch* that it should be left to those who will make 
it a serious study. But that objection takes no account of the large class of persons who desire 
to know, in relation to present discussions and speculations, what famous men in the whole world 
thought and wrote on these topics. They have not the scholarship which would be necessar} for 
original examination of authorities ; but they have an intelligent interest in the relations between 
ancient and modern philosophy, and need just such information as Professor Mayor s sketch will 
give them." The Guardian. 

[Of her Volumes are in preparation^ 
London : Cambridge University Press Warehouse, 17 Paternoster Row. 



of 



LOCAL EXAMINATIONS. 

Examination Papers, for various years, with the Regulations for the 
Examination. Demy 8vo. 2s. each, or by Post, 2s. id. 

Class Lists, for various years, Boys ij., Girls 6d. 
Annual Reports of the Syndicate, with Supplementary Tables showing 
the success and failure of the Candidates. 2s. each, by Post 2s. $d. 



HIGHEE LOCAL EXAMINATIONS. 

Examination Papers for 1883, to which are added the Regulations for 

1884. Demy 8vo. 2s. each, by Post 2s. 2d. 
Class Lists, for various years, is. By post, is. 2d. 
Reports of the Syndicate. Demy 8vo. is., by Post is. 2d. 



LOCAL LECTURES SYNDICATE. 

Calendar for the years 1875 9. Fcap. 8vo. cloth. 2.5-.; for 1875 80. 2s.- 
for 188081. u. 



TEACHERS TRAINING SYNDICATE. 

Examination Papers for various years, to which are added the Regu 
lations for the Examination. Demy 8vo. 6d., by Post yd. 



CAMBRIDGE UNIVERSITY REPORTER. 

Published by Authority. 

Containing all the Official Notices of the University, Reports of 
Discussions in the Schools, and Proceedings of the Cambridge Philo 
sophical, Antiquarian, and Philological Societies, ^d. weekly. 



CAMBRIDGE UNIVERSITY EXAMINATION PAPERS. 

These Papers are published in occasional numbers every Term, and in 

volumes for the Academical year. 

VOL. X. Parts 120 to 138. PAPERS for the Year 1880 81, 15.5-. cloth. 
VOL. XI. 139 to 159. 188182, 15 s. cloth. 

1 60 to 1 76. 1882 83, i$s. cloth. 



Oxford and Cambridge Schools Examinations. 

Papers set in the Examination for Certificates, July, 1882. is. 6d. 

List of Candidates who obtained Certificates at the Examinations 
held in 1882 and 1883 ; and Supplementary Tables. 6d. 

Regulations of the Board for 1884. 6d. 

Report of the Board for the year ending Oct. 31, 1882. i s. 



ILontron: c. j. CLAY, M.A. AND SON. 

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 
17 PATERNOSTER ROW. 



CAMBRIDGE: PRINTED BY c. j. CLAY, M.A. AND SON, AT THB UNIVERSITY PRESS. 



RETURN Astronomy Mathemcrtics/Stotistks/Computer Science Library 

TO +> 1 00 Evans Hall 642-3381 



LOAN PERIOD 1 
7 DAYS 


2 


3 


4 


5 


6 



ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 



STAMPED BELOW 




JUL I* 1993 



Dua end of SPRING Semester 

Mihifirt to rorall aftar 



APR 281H! 



UNIVERSITY OF CALIFORNIA, BERKELEY 
FORM NO. DD3, 1/83 BERKELEY, CA 94720 




JVi