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Full text of "Mathematical and physical papers"


w*th. 



MATHEMATICAL 



AND 



PHYSICAL PAPERS. 



CAMBRIDGE WAREHOUSE, 

17, PATERNOSTER ROW. 




Cambrtogt: DE1GHTON, BELL, AND CO. 
lUtpjts: F. A. BROCKHAUS. 



MATHEMATICAL 



AND 



PHYSICAL PAPERS 



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. 



CambriUg? : 

AT THE UNIVERSITY PRESS. 

1880 

[The rights of translation and reproduction are reserved.] 



v/, I 




WJftth. 



CTantfcrfogc : 

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



PREFACE. 



IT is now some years since I was requested by the Syndics 
of the University Press to allow my papers on mathematical and 
physical subjects, which are scattered over various Transactions and 
scientific Journals, to be reprinted in a collected form. Many of 
these were written a long time ago, and science has in the mean 
time progressed, and it seemed to me doubtful whether it was 
worth while now to reprint a series of papers the interest of which 
may in good measure be regarded as having passed away. How 
ever, several of my scientific friends, and among them those to 
whose opinions I naturally pay the greatest deference, strongly 
urged me to have the papers reprinted, and I have accordingly 
acceded to the request of the Syndics. I regret that in con 
sequence of the pressure of other engagements the preparation 
of the first volume has been so long in hand. 

The arrangement of the papers and the mode of treating them 
in other respects were left entirely to myself, but both the Syndics 
and my friends advised me to make the reprint full, leaning rather 
to the inclusion than exclusion of a paper in doubtful cases. I 
have acted on this advice, and in the first volume, now presented 
to the public, I have omitted nothing but a few papers which 
were merely controversial. 

As to the arrangement of the papers, it seemed to me that the 
chronological order was the simplest and in many respects the 

814004 



VI PREFACE. 

best. Had an arrangement by subjects been attempted, not only 
would it have been difficult in some cases to say under what head 
a particular paper should come, but also a later paper on some one 
subject would in many cases have depended on a paper on some 
different subject which would come perhaps in some later volume, 
whereas in the chronological arrangement each paper reaches up 
to the level of the author s knowledge at the time, so that forward 
reference is not required. 

Although notes are added here and there, I have not attempted 
to bring the various papers up to the level of the present time. I 
have not accordingly as a rule alluded to later researches on the 
same subject, unless for some special reason. The notes introduced 
in the reprint are enclosed in square brackets in order to distin 
guish them from notes belonging to the original papers. To the 
extent of these notes therefore, which were specially written for 
the reprint, the chronological arrangement is departed from. The 
same is the case as regards the last paper in the first volume, 
which suggested itself during the preparation for press of the 
paper to which it relates. In reprinting the papers, any errors 
of inadvertence which may have been discovered are of course 
corrected. Mere corrections of this kind are not specified, but 
any substantial change or omission is noticed in a foot-note or 
otherwise. 

After full consideration, I determined to introduce an innova 
tion in notation which was proposed a great many years ago, for 
at least partial use, by the late Professor De Morgan, in his article 
on the Calculus of Functions in the Encyclopaedia Metropolitan^ 
though the proposal seems never to have been taken up. Mathe 
maticians have been too little in the habit of considering the 
mechanical difficulty of setting up in type the expressions which 
they so freely write with the pen ; and where the setting up can 
be facilitated with only a trifling departure from existing usage as 
regards the appearance of the expression, it seems advisable to 
make the change. 

Now it seems to me preposterous that a compositor should be 
called on to go through the troublesome process of what printers 
call justification, merely because an author has occasion to name 



PREFACE. Vll 

some simple fraction or differential coefficient in the text, in which 
term I do not include the formal equations which are usually 
printed in the middle of the page. The difficulty may be avoided 
by using, in lieu of the bar between the numerator and denomi 
nator, some symbol which may be printed on a line with the type. 
The symbol " : " is frequently used in expressing ratios ; but for 
employment in the text it has the fatal objection that it is appro 
priated to mean a colon. The symbol " -r- " is certainly distinctive, 
but it is inconveniently long, and dy -r dx for a differential coef 
ficient would hardly be tolerated. Now simple fractions are fre 
quently written with a slant line instead of the horizontal bar 
separating the numerator from the denominator, merely for the 
sake of rapidity of writing. If we simply consent to allow the 
same to appear in print, the difficulty will be got over, and a 
differential coefficient which we have occasion to name in the text 
may be printed as dyjdx. The type for the slant line already 
exists, being called a solidus. 

On mentioning to some of my friends my intention to use 
the "solidus" notation, it met with a good deal of approval, and 
some of them expressed their readiness to join me in the use of it, 
amongst whom I may name Sir William Thomson and the late 
Professor Clerk Maxwell. 

In the formal equations I have mostly preserved the ordinary 
notation. There is however one exception. It frequently happens 
that we have to deal with fractions of which the numerator and 
denominator involve exponentials the indices of which are fractions 
themselves. Such expressions are extremely troublesome to set 
up in type in the ordinary notation. But by merely using the 
solidus for the fractions which form the indices, the setting up 
of the expression is made comparatively easy, while yet there 
is not much departure from the appearance of the expressions 
according to the ordinary notation. Such exponential expressions 
are commonly associated with circular functions; and though it 
would not otherwise have been necessary, it seemed desirable 
to employ the solidus notation for the fraction under the symbol 
"sin" or "cos," in order to preserve the similarity of appearance 
between the exponential and circular functions. 



Vlll PREFACE. 

In the use of the solidus it seems convenient to enact that 
it shall as far as possible take the place of the horizontal bar 
for which it stands, and accordingly that what stands immediately 
on the two sides of it shall be regarded as welded into one. Thus 
sin mrx/a means sin (mrx -f- a), and not (sin mrx} + a. This welding 
action may be arrested when necessary by a stop : thus sin nO . /r n 
means (sin nd) -f- r n and not sin (n9 -r- r n ). 

The only objection that I have heard suggested against the 
solidus notation on the ground of its being already appropriated 
to something else, relates to a condensed notation sometimes 
employed for factorials, according to which x (x + a) . . . to n 
factors is expressed by x nla or by x nja . I do not think the ob 
jection is a serious one. There is no risk of the solidus notation, 
as I have employed it, being mistaken for the expression of 
factorials; of the two factorial notations just given, that with 
the separating line vertical seems to be the more common, and 
might be adhered to when factorials are intended ; and if a 
greater distinction were desired, a factorial might be printed 
in the condensed notation as x n ^ a , where the " ( " would serve 
to recall the parentheses in the expression written at length. 



G. G. STOKES. 



CAMBKIDGE, 

August 16, 1880. 



CONTENTS. 



PAGE 

On the Steady Motion of Incompressible Fluids 1 

On some cases of Fluid Motion 17 

On the Motion of a Piston and of the Air in a Cylinder 69 

On the Theories of the Internal Friction of Fluids in Motion, and of the 

Equilibrium and Motion of Elastic Solids 75 

SECTION I. Explanation of the Theory of Fluid Motion proposed. Form 
ation of the Differential Equations. Application of these Equations 

to a few simple cases 78 

SECTION II. Objections to Lagrange s proof of the theorem that if 
udx+vdy + wdz is an exact differential at any one instant it is always 
so, the pressure being supposed equal in all directions. Principles of 
M. Cauchy s proof. A new proof of the theorem. A physical inter 
pretation of the circumstance of the above expression being an exact 

differential 10G 

SECTION III. Application of a method analogous to that of Section I. to 
the determination of the equations of equilibrium and motion of 

elastic solids 113 

SECTION IV. Principles of Poisson s theory of elastic solids, and of the 
oblique pressures existing in fluids in motion. Objections to one of 
his hypotheses. Keflections on the constitution, and equations of 

motion of the luminiferous ether in vacuum 110 

On the Proof of the Proposition that (Mx + Ny)~ l is an Integrating Factor of 

the Homogeneous Differential Equation M+Ndyf dx = . . . 130 

On the Aberration of Light 134 

On Fresnel s Theory of the Aberration of Light 141 

On a Formula for determining the Optical Constants of Doubly Refracting 

Crystals 143 

On the Constitution of the Luminiferous Ether, viewed with reference to 

the Aberration of Light 153 



X CONTENTS. 

PAftT! 

v Report on Recent Researches on Hydrodynamics 157 

* I. General theorems connected with the ordinary equations of Fluid 

Motion 158 

II. Theory of waves, including tides 161 

III. The discharge of gases through small orifices 176 

IV. Theory of sound 178 

V. Simultaneous oscillations of fluids and solids 179 

VI. Formation of the equations of motion when the pressure is not sup 
posed equal in all directions 182 

* Supplement to a Memoir on some cases of Fluid Motion 188 

^ On the Theory of Oscillatory Waves 197 

On the Resistance of a Fluid to two Oscillating Spheres 230 

On the Critical Values of the Sums of Periodic Series 237 

SECTION I. Mode of ascertaining the nature of the discontinuity of a 
function which is expanded in a series of sines or cosines, and of 
obtaining the developments of the derived functions .... 239 

SECTION II. Mode of ascertaining the nature of the discontinuity of 
the integrals which are analogous to the series considered in Section 
I., and of obtaining the developments of the derivatives of the 
expanded functions 271 

SECTION III. On the discontinuity of the sums of infinite series, and of 

the values of integrals taken between infinite limits .... 279 

SECTION IV. Examples of the application of the formulas proved in the 

preceding sections 286 

j Supplement to a paper on the Theory of Oscillatory Waves .... 314 
Index , 327 



ERRATA. 

P. 103, 1. 14, for their read there. 
P. 193, 1. 3, for p*~* read p y ^. 



MATHEMATICAL AND PHYSICAL PAPEES. 



{From the Transactions of the Cambridge Philosophical Society, 
Vol. vii. p. 439.] 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

[Bead April 25, 1842.] 

IN this paper I shall consider chiefly the steady motion of 
fluids in two dimensions. As however in the more general case 
of motion in three dimensions, as well as in this, the calculation 
is simplified when udx + vdy + wdz is an exact differential, I 
shall first consider a class of cases where this is true. I need 
not explain the notation, except where it may be new, or liable 
to be mistaken. 

To prove that udx + vdy + wdz is an exact differential, in 
the case of steady motion, when the lines of motion are open 
curves, and when the fluid in motion has come from an expanse 
of fluid of indefinite extent, and where, at an indefinite distance, 



ential. JNow from the way in which this equation is obtained, 
lf S. 1 



CONTENTS. 



v Eeport on Eecent Eesearches on Hydrodynamics ...... 157 

* I. General theorems connected with the ordinary equations of Fluid 

Motion ............ 158 

II. Theory of waves, including tides ....... 161 

III. The discharge of gases through small orifices ..... 176 

IV. Theory of sound .......... 178 

V. Simultaneous oscillations of fluids and solids ..... 179 

VI. Formation of the equations of motion when the pressure is not sup 
posed equal in all directions ........ 182 

* Supplement to a Memoir on some cases of Fluid Motion ..... 188 
y On the Theory of Oscillatory Waves ........ 197 

On the Eesistance of a Fluid to two Oscillating Spheres ..... 230 

On the Critical Values of the Sums of Periodic Series ..... 237 

SECTION I. Mode of ascertaining the nature of the discontinuity of a 
function which is expanded in a series of sines or cosines, and of 
obtaining the developments of the derived functions .... 239 

SECTION II. Mode of ascertaining the nature of the discontinuity of 
the integrals which are analogous to the series considered in Section 
I., and of obtaining the developments of the derivatives of the 
expanded functions .......... 271 

SECTION III. On the discontinuity of the sums of infinite series, and of 

the values of integrals taken between infinite limits .... 279 

SECTION IV. Examples of the application of the formulae proved in the 

preceding sections .......... 286 

j Supplement to a paper on the Theory of Oscillatory Waves .... 314 

Index , 327 



EEEATA. 

P. 103, 1. 14, for their read there. 
P. 193, 1. 3, for "-* read p^. 



EEEATUM. 



P. 318, Equations (17) and (18). For - read + before the terms multiplied 
by sin 30 and cos 30. 



MATHEMATICAL AND PHYSICAL PAPEES. 



{From the Transactions of the Cambridge Philosophical Society, 
Vol. vir. p. 439.] 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

[Bead April 25, 1842.] 

IN this paper I shall consider chiefly the steady motion of 
fluids in two dimensions. As however in the more general case 
of motion in three dimensions, as well as in this, the calculation 
is simplified when udx + vdy + wdz is an exact differential, I 
shall first consider a class of cases where this is true. I need 
not explain the notation, except where it may be new, or liable 
to be mistaken. 

To prove that udx + vdy + wdz is an exact differential, in 
the case of steady motion, when the lines of motion are open 
curves, and when the fluid in motion has come from an expanse 
of fluid of indefinite extent, and where, at an indefinite distance, 
the velocity is indefinitely small, and the pressure indefinitely 
near to what it would be if there were no motion. 

By integrating along a line of motion, it is well known that 
we get the equation 

P^V-i^ + v + w^+C (1), 

where dV= Xdx-\- Ydy + Zdz, which I suppose an exact differ 
ential. Now from the way in which this equation is obtained, 
\ \ s. 1 



ON THE STEAD I MOTION OF INCOMPRESSIBLE FLUIDS. 



it appears that G need only be constant for the same line of 
motion, and therefore in general will be a function of the para 
meter of a line of motion. I shall first shew that in the case 
considered C is absolutely constant, and then that whenever it 
is, udx + vdy + wdz is an exact differential *. 

To determine the value of C for any particular line of motion, 
it is sufficient to know the values of p, and of the whole velocity, 
at any point along that line. Now if there were no motion we 
should have 



t ............................ (2), 

P! being the pressure in that case. But considering a point in 
this line at an indefinite distance in the expanse, the value of 
p at that point will be indefinitely nearly equal to p^ and the 
velocity will be indefinitely small. Consequently C is more nearly 
equal to G t than any assignable quantity : therefore C is equal to 
Cj ; and this whatever be the line of motion considered ; therefore 
C is constant. 

In ordinary cases of steady motion, when the fluid flows in 
open curves, it does come from such an expanse of fluid. It is 
conceivable that there should be only a canal of fluid in this 
expanse in motion, the rest being at rest, in which case the 
velocity at an infinite distance might not be indefinitely small. 
But experiment shews that this is not the case, but that the 
fluid flows in from all sides. Consequently at an indefinite dis 
tance the velocity is indefinitely small, and it seems evident that 
in that case the pressure must be indefinitely near to what it 
would be if there were no motion. 

Differentiating therefore (1) with respect to x, we get 

1 dp ^ du dv dw 

- ^r = ^-u- r -v- r -w- r ; 
p dx dx dx ax 

1 dp ^ du du du 

- = -- V - W 



dv du\ dw du 
whence 



[* See note, page 3.] 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 3 

. ., , (dw dv\ fdu dv\ 

Similarly, w (- r -- r ) + u (_- =0, 
\dy dzj \dy dx) 

fdu dw\ (dv dw\ _ 
\dz dx) \dz dy) 

i dv du dw dv du dw 

whence* JT=-J- > j~ = ^-j ~r ~ ~j~ > 

dx dy dy dz dz dx 

and therefore udx + vdy + wdz is an exact differential. 

When udx -f vdy + wdz is an exact differential, equation (1) 
may be deduced in another wayf% from which it appears that 
C is constant. .Consequently, in any case, udx -{ vdy + wdz is, or 
is not, an exact differential, according as C is, or is not, constant. 



Steady Motion in Two Dimensions. 

I shall first consider the more simple case, where udx + vdy 
is an exact differential. In this case u and v are given by the 
equations 

J* + ^ = ........................... (3), 

dx dy 

----0 M- 

dy dx~~ " () > 

and p is given by the equation 



The differential equation to a line of motion is 

dy = v_ 
dx u 

* [This conclusion involves an oversight (see Transactions, p. 465) since the 
three preceding equations are not independent, as may readily be seen. I have not 
thought it necessary to re-write this portion of the paper, since in the two classes 
of steady motion to which the paper relates, namely those of motion in two dimen 
sions, and of motion symmetrical about an axis, the three analogous equations are 
reduced to one, and the proposition is true. None of the succeeding results are 
affected by this error, excepting that the second paragraph of p. 11 must be re 
stricted to the two cases above mentioned.] 

t See Poisson, Traite de Mecanique. 

12 



4 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

Now from equation (3) it follows that udy vdx is always 
the exact differential of a function of x and y. Putting then 

d U = udy vdx, 

U=G will be the equation to the system of lines of motion, 
C being the parameter. U may have any value which allows 
d U/dy and d Ujdoc to satisfy the equations which u and v satisfy. 
The first equation has been already introduced ; the second leads 
to the equation which U is to satisfy ; viz. 



The integral of this equation may be put under different forms. 
By integrating according to the general method, we get 



Now it will be easily seen that U must be wholly real for all 
values of x and y t at least within certain limits. But ^(a) may 
be put under the form F l (a) 4- \f^l F z (a), where F l (a) and F 3 (a) 
are wholly real. Making this substitution in the value of U, we 
get a result, which, without losing generality, may be put under 
the form 

U = F(^ + V^l y)+F(x - V^l y] 



=l y} -f(x- a#)}, 

changing the functions. 

If we develope these functions in series ascending according 
to integral powers of y, by Taylor s Theorem, which can always 
be done as long as the origin is arbitrary, we get a series which 
I shall write for shortness, 



2 cos (A y) F(x) - 2 sin (^ y]f (), 



the same result as if we had integrated at once by series by 
Maclaurin s Theorem. 

It has been proved that the general integral of (5) may be 
put under the form 

U= 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 5 

where a 2 + {3* = 0. Consequently a and /3 must be, one real, the 
other imaginary, or both partly real and partly imaginary. Putting 
then a = 1 + V-la 2 , /3 = ^ + V- 1 /3 2 , introducing the condition 
that a 2 + /3 2 = 0, and replacing imaginary exponentials by sines and 
cosines, we find that the most general value of U is of the form 

U = 2Ue w ( cos Y-*- sin Y- y +a \ cos n (sin 7 . x + cos 7 . y + 6), 

where A, n, 7, a and & have any real values, the value of U being 
supposed to be real. 

If we take the value of U 



and develope each term, such as ax n , in F (x) or f (x), in a series, 
and then sum the series by the formula 

cos nO + V- 1 sin nO = cos n <9 (l + j V^T tan -... V 
we find that the general value of U takes the form 



As long as the origin of x is arbitrary, only integral powers 
of x will enter into the development F (x) and f(x), and there 
fore the above series will contain only integral values of n. For 
particular positions of the origin however, fractional powers may 
enter. The equation 

d 2 U I dU 1 d*U _ 
dr* + r ~dr + r* dP " 

which (5) becomes when transferred to polar co-ordinates, is satis 
fied by the above value of U, whatever n be, even if it be 
imaginary, in which case the value of U takes the form 

U = 2Ar m e nd cos (mO - log e r n + B). 

We may employ equation (5), to determine whether a proposed 
system of lines can be a system in which fluid can move, the 
motion being of the kind for which udx + vdy is an exact 
differential. 

Let / (x, y) = U^ C be the equation to the system, C being 
the parameter. Then, if the motion be possible, some value of 



6 ON THE STEADY MOTION OF INCOMPKESSIBLE FLUIDS. 

U which satisfies (5) must be constant for all values of x and y 
for which U^ is constant. Consequently this value must be a 
function of U . Let it=^(C^). Then, substituting this value 
in (5), and performing the differentiations, we get 




\ dx \dy 

Now, if the motion be possible, the second term of this equa 
tion must be a function of U l ; a?, y and U^ being connected by 
the equation f(x, y}= U^. Consequently, if by means of this 
latter equation we eliminate x or y from the second term of (6), 
the other must disappear. If it does not, the motion is impossible ; 
if it does, the integration of equation (6), in which the variables 
are separated, will give < (U^) under the form 



A and B being the arbitrary constants. The values of u and v 
will immediately be got by differentiation, and then p will be 
known. Nothing will be left arbitrary but a constant multiplying 
the values of u and v, and another added to the value of p. 

I shall mention a few examples. Let U = ar 2 cos J^. In this 
case the lines of motion are similar parabolas a,bout the same 
focus. The velocity at any point varies inversely as the square 
root of the distance from the focus. 

Again, let U = axy. In this case the lines of motion are 
rectangular hyperbolas about the same asymptotes. Also, 

dU dU 

u = -j = ax, and v = -j = ay. 
dy dx 

In this case therefore the velocity varies as the distance from the 
centre, and the particles in a section parallel to either of the axes 
remain in a section parallel to that axis. 

I shall now consider the general case, where udx -f vdy need 
not be an exact differential. 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

In this case p, u and v, are given by the equations 

I dp du du 

- -^=X-u- j -- V-Y- ................... (7), 

p dx dx dy 

1 dp v dv dv 

--T- = Y-u-j -- v -j- .............. ..(8) 

p dy dx dy 

du dv 



We still have ^ = - , for the differential equation to a line of 

motion, where udy vdx is still an exact differential, on account 
of equation (9). Eliminating p by differentiation from (7) and 
(8), and expressing the result in terms of U, we get the equation 
which U is to satisfy, viz. 



dU d^ (d*U <PU\ __ dU d^ (d*U d*U\ _ 
dy dx (da? " h dy 2 ) dx dy ( dx 2 + ~dtf) ~ 0) 



or, for shortness, 



d__dU _ 

(dy dx dx dy) (~%? + ~df) ......... (10) * 

* [This equation may be applied to prove an elegant theorem due to Mr F. D. 
Thomson {see the Oxford, Cambridge, and Dublin Messenger of Mathematics, Vol. 
in. (I860), p. 238, and Vol. iv. p. 37}, that if a vessel bounded by a cylindrical sur 
face of any kind and by two planes perpendicular to its generating lines be filled 
with homogeneous liquid, and the whole be revolving uniformly about a fixed axis 
parallel to its generating lines, then if the vessel be suddenly arrested the motion 
of the liquid will be steady. 

If w be the angular velocity, we shall have for the motion before impact 



dx) - J w (ce 2 + ?/ 2 ) = - s wr 2 , 

omitting the constant as unnecessary. If u, v be the components of the change of 
velocity produced by impact, it follows from the equations of impulsive motion that 
udx + vdy will be a perfect differential d<p, where satisfies the partial differential 

d 2 d 2 
equation V0 = 0, V standing for ^ 2 + ^. If U be the 17-function corresponding 

to this motion and such a function exists by virtue of the equation of continuity 
whether the motion be steady or not we have 

, r fd<t d(f> , \ /Yd0 1 d<t> . \ 

U = n~ dy--^- dx\= ( ~rd6 -/ dr } , 
J \dx b dy ) J \dr r dd J 

where the quantity under the sign J is a perfect differential by virtue of the equa- 



8 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

In this case, since p = I (~ dx -\--jr dy\ , equations (7) and 
(8) give 

tion V0 = ; an ^ we see at once *^ at V^ = - Hence for the whole motion just 
after impact 



which satisfies the equation of steady motion (10); and as the condition at the 
boundary, namely that the fluid shall slide along it, is satisfied, being satisfied ini 
tially, it follows that the initial motion after impact will be continued as steady 
motion. 

To actually determine the function </> or U , and thereby the motion in any given 
case, we must satisfy not only the general equation y0=0 but also the equation 
of condition at the boundary, namely that there shall be no velocity in a direction 
normal to the surface, which gives 

(-)*-(!?)*- ................................. <> 

at any point of the boundary. If f(x,y)=Q be the equation of the boundary, we 
must substitute - df / dx -r- df I dy for dy/dx in (a), and the resulting equation will 
have to be satisfied when/=0 is satisfied. 

There are but few forms of boundary for which the solution of the problem can 
be actually effected analytically, among which may be mentioned in particular the 
case of a rectangle. But by taking particular solutions of the equation V0 = > 
substituting in (a) and integrating, which gives 



or what comes to the same thing taking particular solutions of the equation v^ = 
and substituting in (/3), which gives the general equation of the lines of motion, we 
may synthetically obtain an infinity of examples in which the conditions of the 
problem are satisfied, any one of the lines of motion being taken as the boundary 
of the fluid. 

Thus for U = fcr*cos 30 we have for the lines of motion 

-iwr 2 + fcr 3 cos30=C ....................................... (7), 

or -Icor 2 + &{4(rcos0) 3 -3r 2 .rcos0}=<7 ................ (5), 

which therefore are cubic curves, recurring when is increased by 120. (5) is 

satisfied by 

r cos 6 = a, 
giving a straight line, provided 



Hence when Tc has the above value the cubic curve (7) breaks up, for the particular 
value of the parameter G above written, into three straight lines forming the sides 
of an equilateral triangle, and the vessel may therefore be supposed to be an equi 
lateral triangular prism. The various lines of motion correspond to values of the 
parameter C from to - -f wa 2 . This case is given by Mr Thomson. 

U = kr*cos 20 leads to the case of steady motion in similar and concentric ellipses 
considered in the text a little further on, which therefore may be conceived to have 
been produced from motion about a fixed axis as pointed out by Mr Thomson. In 
fact, any case of steady motion in two dimensions in which yU= const, may be 
conceived to have been so produced.] 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 



= V - L _^ *F_\ dx 

P J\\dy dxdy dx dy* J 

fdU d*U dUd*U\ } 
\dx dxdy dy dx 2 ) *) 

AT 1,7 f/^Y , (dU\*\ fdU d 2 U , dU d z U\ , 
Now Ja-u-j-J + -j- }\= -j -- j-^ + ~j- -j, T\dx 
\\dxj \dy J } \dx dx dy dxdy) 

(dU d 2 U dU d 2 U 
^\dx dxdy + dy dtf 
whence, 



dU d 2 U , dU d*U , dU d z U . dUd*U, 
-j- , , dx + -j -j =- av -- =- -=- dx -- Y- -y-g- ctv 

dx dy dy dx 



V.AH7V) fd*U d*U\ (dU , , dU . \ 

I + TT" ) r ~ ^7""2" + ^j^" ^j~ "* + ~j~ dy ; 
/ v dy J J \ cfo dy J \ dx dy y j 

and therefore 

^ 2 ^7 rf?7\ /dtr , . dU 
dy 



It will be observed that -, 2 +-j- a = %(^)> ^ s a nrs ^ integral 

of (10). Consequently this latter term, which is the value of C in 
(1), comes out a function of the parameter of a line of motion as 
it should. 

We may employ equation (10), precisely as before, to enquire 
whether a proposed system of lines can, under any circumstances, 
be a system of lines of motion. Let f (x, y) = U^ = C, be the 
equation to the system; then, putting as before, U = (f>(U l ), 
we get 

+ 



dy dx dx dy dx dy J 

U d dU d\fU. 



or, P<f>" ( U^} + Qf ( fTJ = 0, suppose. 



10 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

Hence, as before, if we express y in terms of x and U lt from 
the equation f(x, y) = U lt and substitute that value in p, the 

result must not contain x. If it does, the proposed system of lines 
cannot be a system of lines of motion ; if not, the integration of 
the above equation will give < (Z7J, under the form 



and we can immediately get the values of u, v and p, with the 
same arbitrary constants as in the previous case. 

One case in which the motion is possible is where the lines of 
motion are a system of similar ellipses or hyperbolas about the 
same centre, or a system of equal parabolas having the same axis. 
In the case of the ellipse, the particles in a radius vector at any 
time remain in a radius vector, and the value of p has the form 



When however the ellipse becomes a circle, P and Q vanish in the 
equation P</>" (tTj + Q<f> (UJ = 0. Consequently the form of < 
may be any whatever. The value of U^ being x* + # 2 , we have 



whence, v? + v* = 4 (</> ( U,)} 9 (a? + /) = 4 U, {< ( U,)} 9 . 

Hence, the velocity may be any function of the distance from the 
centre. It is evident that we may conceive cylindrical shells of 
fluid > having a common axis, to be revolving about that axis with 
any velocities whatever, if we do not consider friction, or whether 
such a mode of motion would be stable. The result is the same if 
we enquire in what way fluid can move in a system of parallel lines. 

In any case where the motion in a certain system of lines is 
possible, if we suppose two of these lines to be the bases of bound 
ing cylindrical surfaces, and if we suppose the velocity and direc 
tion of motion, at each point of a section of the entering, and also 
of the issuing fluid, to be what that case requires, I have not 
proved that the fluid must move in that system of lines. When 
the above conditions are given there may still perhaps be different 
modes of steady motion ; and of these some may be stable, and 
others unstable. There may even be no stable steady mode of 



ON THE STEADY MOTION OF INCOMPKESSIBLE FLUIDS. 11 

motion possible, in which case the fluid would continue perpetually 
eddying. 

In the case of rectangular hyperbolas, the fluid appeared, on 
making the experiment, to move in hyperbolas when the end 
at which the fluid entered was broad and the other end narrow, 
but not when the end by which the fluid entered was narrow. 
This may, I think, in some measure be accounted for. Suppose 
fluid to flow out of a vessel where the pressure is p { into one where 
it is p z) through a small orifice. Then, the motion being steady, 
we have, along the same line of motion, p/p C J-y 2 , where v is 
the whole velocity. At a distance from the orifice, in the first 
vessel, the pressure will be approximately p : , and the velocity 
nothing. At a distance in the second vessel, the pressure will 

/2 If) q} } 

be approximately^, and therefore the velocity = A/ ", 

nearly. The result is the same if forces act on the fluid. Hence 
the velocity must be approximately constant ; and therefore, the 
fluid which came from the first vessel, instead of spreading out, 
must keep to a canal of its own of uniform breadth. This is found 
to agree with experiment. Hence we might expect that in the 
case of the hyperbolas, if the end at which the fluid entered were 
narrow, the entering fluid would have a tendency to keep to a 
canal of its own, instead of spreading out. 

In ordinary cases of steady motion, when the lines of motion 
are open curves, the fluid is supplied from an expanse of fluid, and 
consequently udx + vdy + wdz is an exact differential. Conse 
quently, cases of open curves for which it is not an exact differen 
tial do not ordinarily occur. We may, however, conceive such 
cases to occur ; for we may suppose the velocity and direction of 
motion, at each point of a section of the entering, and also of the 
issuing stream, to be such as any case requires, by supposing the 
fluid sent in and drawn out with the requisite velocity and in the 
requisite direction through an infinite number of infinitely small 
tubes. 

In the case of closed curves however, in whatever manner the 
fluid may have been put in motion, it seems probable that, if we 
neglect the friction against the sides of the vessel, the fluid will 
have a tendency to settle down into some steady mode of motion. 
Consequently, taking account of the friction against the sides of 



12 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

the vessel, it seems probable that the motion may in some cases 
become approximately steady, before the friction has caused it to 
cease altogether. 



Motion symmetrical about an axis, the lines of motion being 
in planes passing through the aods. 

Before considering this case, it may be well to prove a prin 
ciple which will a little simplify our equations. 

The general equations of motion are, 





I dp 


yr dU 


du 


du 


ni) 




p dx 
I dp _ 


dx 

y dV 


dy 
dv 


dz 

dv 


\ ij v> 

(12) 




pdy 
Idp _ 


dx 
~ dw 


dy 

dw 

11 I/) 


dz" 
dw 


n^\ 


And 


p dz 
the equation of 


dx 
continuity is 


dy 


dz 


\ J 



Putting tsTj, trr 2 , r 3 , for the last three terms in (11), (12), (13), 
respectively, we have 

^ = V - / (efjdx + *r t dy + OT 8 dz). 

Hence the pressure consists of two parts, the firs,t, p V, the same 
as if there were no motion, the second, the part due to the velocity. 
Now the velocities are given by equation (14), and by the three 
equations which result on eliminating p from (11), (12), and (13). 
These latter equations, as well as (14), will be the same as if there 
were no forces since 



_^ = ^^. 

dy ~ dx dz ~ dx dz dy 

and therefore we shall not lose generality by omitting the forces 
in (11), (12) and (13), since we shall only have to add pV to the 
value of p so determined. 

When the motion is symmetrical about an axis, and in planes 
passing through that axis, let z be measured along the axis, and 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 13 

r be the perpendicular distance from the axis, and s be the ve 
locity perpendicular to the axis. Then, transforming the co-ordi 
nates to z and r, and omitting the forces, it will be found that 
equations (11), (12) and (13) are equivalent to only two separate 
equations, which are 

1 dp ds ds 

~-r = -s-r-w-r 

p dr dr dz 

1 dp dw dw 

--T- = S-J- W-T- 
p dz dr dz 

and the equation of continuity becomes 

J+ s - + ^ = 0.. ,..(17). 

dr r dz 

In the case where udx + vdy + wdz is an exact differential, it 
will be found that the three equations 

du _ dv du _dw dv _ dw 

dy ~ dx dz dx dz dy 

t/ i7 

are equivalent to only one equation, which is 

ds dw 



In the general case we get, by eliminating p from (15) 

and (16), 

d f ds ds\ _ d f dw 

dz \ dr dz) dr\ dr w dz , 

ds ds ds dw d?t 



or ~~ ~~ 



jj ^^j 

dr dz dz dz drdz 

dw dw dw ds d*w d z io 



The differential equation, between z and r, to a line of 
motion is 

dz _w 
dr s 

Let ju be a factor which renders sdz wdr an exact differential, 



,, diis _ 

then ~ + - = 0, 

dr dz 






14 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 
/n ^_v du, da S 

or, using (17), s + w = f ,.. 

whence we easily see that //, = r is one such factor. 
Let then dU= rsdz rwdr, 

IdU IdU 

so that s = 7 , w = --- , . 

r dz r a/ 

The equation which U is to satisfy will be got by expressing s 
and w in terms of U, and substituting in (19) in the general case, 
or by substituting in (18), in the case where udx + vdy + wdz is 
an exact differential. 

In the latter case the equation which Z7is to satisfy is 

idu 



7 Q T o 7 

dz dr r dr 



In the general case, the equation is what I shall write 

d L _dUd\(l(d^7 d^U_ldJA\_ 
dz dr dr dz) \r z \dz z * dr* r dr)} ~ " ( > 

The value of p is given by the equation 

p ((( ds ds\ 7 f dw dw\ 7 ) 
*L--\\{s-^+w-j-}dr + \8- r + to- T -\dz\. 

p J [\ dr dz) \ dr dz) } 

Now 

1 , , 2N ^5 7 dw j ds j dw 7 
i-d (s 2 + w ) = s -j- dr + w -j- dz + s -j- dz + w -j- dr ; 
dr dz dz < dr 

and therefore 

ds ds\ -, / dw dw 



= J d (s 2 + w 2 ) + j- (wdr - sc?z) + -T- 



/P 1/2 2\ [fds dw\ 1 7 77 

whence = % (s + w } -f I ( -j- =- -7- 1 - au 

p J V^ ar/ r 



ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 15 

Hence the quantity under the integral sign must be a function 
of U. And in fact, we can easily shew by trial that 

d*U Id 



is a first integral of (21). The last term of (22) is the value of 
the constant in (1). 

By expanding U in a series ascending according to integral 
powers of z, which may be done as long as the origin is arbitrary, 
it will be found that the integral of (20) may be written under the 
form 

U= cos ( V *) F (r) + sin ( v *) vYM, 

where y a .F(r) denotes (-3-5 -- -*-jF(r) t and y 2n F (r) denotes 
that the operation -^ 2 ^- is repeated n times on F (r). 

We may employ equations (21) or (20) just as before, to 
determine whether the motion in a proposed system of lines is 
possible. If F(r, z) = U t = C be the equation to the system, we 
must have, as before, U (f> ( U^ ; whence we get, in the general 
case, 



^ 
dz y ^ 2 dr* ~r dr J] j 

and in the more restricted case where udx -f vdy + wdz is an exact 
differential, we get 

ff U ffU Id U\ . 



As before, the ratio of the coefficients of 0" ( U^ and $ ( U^) must 
be a function of U^ alone, when 3, r and C^ are connected by the 
equation F (r, z) = Z7 r If the motion be possible, it will in general 
be determinate, U being of the form Af (r, z] + B. If U^ = r how 
ever, the form of remains arbitrary. In this case the fluid may 
be conceived to move in cylindrical shells parallel to the axis, the 
velocity being any function of the distance from the axis. 



16 ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. 

Particular cases are, where the lines of motion are right lines 
directed to a point in the axis, and where they are equal parabolas 
having the axis of z for a common axis. In these cases 

udx + vdy + wdz 
is an exact differential. 

We may employ equations (20) and (21) to determine whether 
the hypothesis of parallel sections can be strictly true in any case. 
In this case, the sections being perpendicular to the axis of z, we 
must have 

IdU 

w --- T -F(is) ; 
r dr 

dU , N 

y rl-W; 

U1 + *(*)+/(*). 

Substituting this value in (21), we find, by equating to zero 
coefficients of different powers of r, that the most general case cor 
responds to 



If udoc -\- vdy 4- wdz be an exact differential, the most general 
case corresponds to 

U= (a + bz) 



[From the Transactions of the Cambridge Philosophical Society, 
Vol. vin. p. 105.] 

ON SOME CASES OF FLUID MOTION. 

[Read May 29, 1843.] 

THE equations of Hydrostatics are founded on the principles 
that the mutual action of two adjacent elements of a fluid is normal 
to the surface which separates them, and that the pressure is equal 
in all directions. The latter of these is a necessary consequence 
of the former, as has been shewn by Mr Airy*. An exactly simi 
lar proof may be employed in Hydrodynamics, by which it may 
be shewn that, if the mutual action of two adjacent elements of a 
fluid in motion is normal to their common surface, the pressure 
must be equal in all directions, in order that the accelerating force 
which acts on the centre of gravity of an element may not become 
infinite, when we suppose the dimensions of the element indefi 
nitely diminished. In Hydrostatics, the accurate agreement of the 
results of our calculations with experiments, (those phenomena 
which depend on capillary attraction being excepted), fully justifies 
our fundamental assumption. The same assumption is made in 
Hydrodynamics, and from it are deduced the fundamental equa 
tions of fluid motion. But the verification of our fundamental law 
in the case of a fluid at rest, does not at all prove it to be true 
in the case of a fluid in motion, except in the very limited case of 
a fluid moving as if it were solid. Thus, oil is sufficiently fluid to 
obey the laws of fluid equilibrium, (at least to a great extent), 
yet no one would suppose that oil in motion ought to be considered 
a perfect fluid. It would appear from the following consideration, 
that the fluidity of water and other such fluids is not quite perfect. 

* See also Professor Miller s Hydrostatics, page 2. 
S. 2 



18 ON SOME CASES OF FLUID MOTION. 

When a mass of water contained in a vessel of the form of a solid 
of revolution is stirred round, and then left to itself, it presently 
comes to rest. This, no doubt, is owing to the friction against the 
sides of the vessel. But if the fluidity of water were perfect, it 
does not appear how the retardation due to this friction could be 
transmitted through the mass. It would appear that in that case 
a thin film of fluid close to the sides of the vessel would remain at 
rest, the remaining part of the fluid being unaffected by it. And 
in this respect, that part of Poisson s solution of the problem of an 
oscillating sphere, which relates to friction, appears to me in some 
degree unsatisfactory. A term enters into the equation of motion 
of the sphere depending on the friction of the fluid on the sphere, 
while no such term enters into the equations of motion of the 
fluid, to express the equal and opposite friction of the sphere on 
the fluid. In fact, as long as we regard the fluidity of the fluid as 
perfect, no such term can enter. The only way by which to esti 
mate the extent to which the imperfect fluidity of fluids may 
modify the laws of their motion, without making any hypothesis 
as to the molecular constitution of fluids, appears to be, to calculate 
according to the hypothesis of perfect fluidity some cases of fluid 
motion, which are of such a nature as to be capable of being accu 
rately compared with experiment. The cases of that nature which 
have hitherto been calculated, are by no means numerous. My 
object in the present paper which I have the honour to lay before 
the Society, has been partly to calculate some such cases which 
may be useful in determining how far we are justified in regarding 
fluids as perfectly fluid, and partly to give examples of the methods 
by which the solution of problems depending on partial differential 
equations may be effected. 

In the first seven articles, I have mentioned and explained 
some general principles, which are afterwards applied. Some of 
these are not new, but it was convenient to state them for the 
sake of reference. Others are I believe new, at least in their 
development. In the remaining articles, I have given different 
problems, of which I have succeeded in obtaining the solutions. 
As the problem to be solved is usually stated at the head of each 
article, I shall here only mention some of the results. As a parti 
cular case of the problem given in Art. 8, I find that, when a 
cylinder oscillates in an infinitely extended fluid, the effect of the 
inertia of the fluid is to increase the mass of the cylinder by that of 



ON SOME CASES OF FLUID MOTION. 10 

the fluid displaced. In part of Art. 9, I find that when a ball pen 
dulum oscillates in a concentric spherical envelope, the effect of the 

b s 4- 2a 3 

inertia of the fluid is to increase the mass of the ball by -^jJ- 3T 

2i(Jb a ) 

times that of the fluid displaced, a being the radius of the ball, b 
that of the envelope. Poisson, in his solution of the problem of the 
sphere, arrives at the strange result that the envelope does not at 
all retard the oscillating sphere. I have pointed out the errone 
ous step by which he was led to this conclusion, which I am clearly 
called upon to do, in venturing to differ from so high an authority. 
Of the different cases of fluid motion which I have given, that 
which appears to be capable of the most accurate and varied com 
parison with experiment, is the motion of fluid in a rectangular 
box which is closed on all sides, given in Art. 13. The experiment 
consists in comparing the calculated and observed times of oscil 
lation. I find that when the motion is small, the effect of the 
fluid on the motion of the box is the same as that of a solid 
having the same mass, centre of gravity, and principal axes, but 
having different moments of inertia, these moments being given 
by infinite series, which converge with great rapidity. I have also 
in Art. 11, given some cases of progressive motion, deduced on the 
supposition that the same particles of fluid remain in contact with 
the solid, which do not at all agree with experiment. 

In almost all the cases given in this paper, the problem of 
finding the permanent state of temperature in the several solids 
considered, supposing the surfaces of those solids kept up to con 
stant temperatures varying from point to point, may be solved by 
a similar analysis. I find that some of these cases have been 
already solved by M. Duhamel in a paper inserted in the 22nd 
Cahier of the Journal de lEcole Poly technique. The cases alluded 
to are those of the temperature in a solid sphere, and in a rect 
angular parallelepiped. Since, however, the application of the 
formulae in the two cases of fluid motion and of the permanent 
state of temperature is different, as well as the formulae themselves 
to a certain extent, I thought it might be worth while to give 
them. 



1. The investigations in this paper apply directly to incom 
pressible fluids, as the fluids spoken of will be supposed to be, 

22 



20 ON SOME CASES OF FLUID MOTION. 

unless the contrary is stated. The motions of elastic fluids may 
in most cases be divided into two classes, one consisting of those 
condensations on which sound depends, the other, of those motions 
which the fluid takes in consequence of the motion of solid bodies 
in it. Those motions of the fluid, which take place in consequence of 
very rapid motions of solids, (such as those of bullets), form a con 
necting link between these two classes. The motions of the second 
class are, it is true, accompanied by condensations, and propagated 
with the velocity of sound, but if the motions of the solids are not 
great we may, without sensible error, suppose the motions of the 
fluid propagated instantaneously to distances where they cease to 
be sensible, and may neglect the condensation. The investigations 
in this paper will apply without sensible error to this kind of 
motion of elastic fluids. 

In all cases also the motion will be supposed to begin from 
rest, which allows us to suppose that udx + vdy + wdz is an exact 
differential d(f>, where u, v and w are the components, parallel to 
the axes of x, y, and z, of the whole velocity of any particle. In 
applying our investigations however to fluids such as they exist in 
nature, this principle must not be strained too far. When a body 
is made to revolve continually in a fluid, the parts of the fluid 
near the body will soon acquire a rotatory motion, in consequence, 
in all probability, of the mutual friction of the parts of the fluid ; 
so that after a time udx + vdy + wdz could no longer be taken an 
exact differential. It is true that in motion in two dimensions 
there is one sort of rotatory motion for which that quantity is an 
exact differential ; but if a close vessel, filled with fluid at first at 
rest, be made to revolve uniformly round a fixed axis, the fluid 
will soon do so too, and therefore that quantity will cease to be an 
exact differential. For the same reason, in the progressive motion 
of a solid in a fluid, the effect of friction continually accumulating, 
the motion might at last be sensibly different from what it would 
be if there were no friction, and that, even if the friction were 
very small. In the case of small oscillatory motions however it 
would appear that the effect of friction in the forward oscillation, 
supposing that friction small, would be counteracted by its effect 
in the backward oscillation, at least if the two were symmetrical. 
In this case then we might expect our results to agree very nearly 
with experiment, so far at least as the time of oscillation is con 
cerned. 



ON SOME CASES OF FLUID MOTION. 21 

The forces which act on the fluid are supposed in the following 
investigations to be such that Xdx + Ydy + Zdz is the exact dif 
ferential of a function of x, y and z, where X, Y, Z, are the com 
ponents, parallel to the axes, of the acccelerating force acting on 
the particle whose co-ordinates are x, y, z. The only effect of such 
forces, in the case of a homogeneous, incompressible fluid, being 
to add the quantity pf(Xdx + Ydy +Zdz) to the pressure, the forces, 
as well as the pressure due to them, will for the future be omitted 
for the sake of simplicity. 

2. It is a recognized principle, and one of great importance in 
these investigations, that when a problem is determinate any solu 
tion which satisfies all the requisite conditions, no matter how ob 
tained, is the solution of the problem. In the case of fluid motion, 
when the initial circumstances and the conditions with respect to 
the boundaries of the fluid are given, the problem is determinate. 
If it were required to find what sort of steady motion could take 
place between given surfaces, the problem would not be determi 
nate, since different kinds of steady motion might result from dif 
ferent initial circumstances. 

It may be well here to enumerate the conditions which must 
be satisfied in the case of a homogeneous incompressible fluid 
without a free surface, the case which is considered in this paper. 
We have first the equations, 

1 dp I dp 1 dp 

-p dx = w " pdy = ^ pd~ 2 = -^ ............ ^)J 

... du du du du , 

putting ^fa + U +V + M an(i OT 2 > OT 3> for the cor 

responding quantities for y and , and omitting the forces. 
We have also the equation of continuity, 



.. ,.. 

dx dy dz 

(A) and (B) hold at all times for all points of the fluid mass. 

If a- be the velocity of the point (x, ?/, z) of the* surface of a 
solid in contact with the fluid resolved along the normal, and v 
the velocity, resolved along the same normal, of the fluid particle, 



22 OX SOME CASES OF FLUID MOTION. 

which at the time t is in contact with the above point of the solid, 
we must have 

v = * ........ ............................ (a)*, 

at all times and for all points of the fluid which are in contact with 
a solid. 

If the fluid extend to infinity, and the motion at first be zero 
at an infinite distance, we must have 

u = 0, v = 0, w = 0, at an infinite distance ............. (b). 

An analagous condition is, that the motion shall not become 
infinitely great about a particular point, as the origin. 

Lastly, if u 0) v ot w , be the initial velocities, subject of course 
to satisfy equations (B) and (a), we must have 

u u^ V = V Q , w = w , when = ..................... (c). 

In the most general cases the equations which u, v and w are 
to satisfy at every point of the mass and at every time are (B) and 
the three equations 



. ~, 



~dy~~dx* dz ~~ dy dx ~ dz " 

These equations being satisfied, the quantity 
will be an exact differential, whence p may be determined by inte 
grating the value of dp given by equations (A). Thus the condi 
tion that these latter equations shall be satisfied is equivalent to 
the condition that the equations ( C) shall be satisfied 

In nearly all the cases considered in this paper, and in all those 
of which the complete solution is given, the motion is such that 
udx + vdy + ^udz is an exact differential dty. This being the case, 
the equations (C) are, as it is well known, always satisfied, the 
value of p being given by the equation 



* For greater clearness, those equations which must hold for all values of the 
variables within limits depending on the problem are denoted by capitals, while 
those which hold only for certain values of the variables, or of some of them, are 
denoted by small letters. The latter class serve to determine the forms of the 
arbitrary functions contained in the integrals of the former. 



ON SOME CASES OF FLUID MOTION. 23 

being an arbitrary function of t, which may if we please be 
included in (/>. In this case, therefore, the single condition which 
has to be satisfied at all times, and at every point of the mass is 
(j5), which becomes in this .case 



(E). 



In the case of impulsive motion, if U Q , v OJ w , be the velocities 
just before impact, u, v, w, the velocities just after, and q the im 
pulsive pressure, the equations (A) are replaced by the equations 

1 da 1 da 1 da 

- -/- = U + U Q , ~~ = V + V Q) --j L = w + w 6 ....(F): 

pdx p dy pdz 

and in order that these equations may be satisfied it is necessary 
and sufficient that (u u )dx + (v v ) dy -f (w W Q ] dz be an exact 
differential d<f), which gives 

q = C- pcf). 

The only equation which must be satisfied at every point of the 
mass is (B), which is equivalent to (E), since by hypothesis u , V Q , 
and w satisfy (B}. The conditions (a) and (b) remain the same 
as before. 

One observation however is necessary here. The values of u, 
v and w are always supposed to alter continuously from one point 
in the interior of a fluid mass to another. At the extreme boun 
daries of the fluid they may however alter abruptly. Suppose now 
values of u, v and w to have been assigned, which do not alter 
abruptly, which satisfy equations (5) and ( C) as well as the con 
ditions (a), (b) and (c), or, to take a particular case, values which 
do not alter abruptly, which satisfy the equation (B) and the same 
conditions, and which render udx + vdy + wdz an exact differential. 
Then the values of dp/dx, dp/dy and dpjdz will alter continuously 
from one point to another, but it does not follow that the value of 
p itself cannot alter abruptly. Similarly in impulsive motion the 
value of q may alter abruptly, although those of dq/dx, dq/dy and 
dqjdz alter continuously. Such abrupt alterations are, however, 
inadmissible; whence it follows as an additional condition to be 
satisfied, 

that the value of p or g, obtained by integrating j 

equations (A) or (F), shall not alter abruptly > ........ (d). 

from one point of the fluid to another. J 



24 ON SOME CASES OF FLUID MOTION. 

An example will make this clearer. Suppose a mass of fluid 
to be at rest in a finite cylinder, whose axis coincides with that of 
z t the cylinder being entirely filled, and closed at both ends. Sup 
pose the cylinder to be moved by impact with an initial velocity C 
in the direction of x ; then shall 

u = C, v = 0, 10 = 0. 

For these values render udx + vdy + wdz an exact differential d<f>, 
where </> satisfies (E) ; they also satisfy (a) ; and, lastly, the value 
of q obtained by integrating equations (F), namely, C 1 Cpx, does 
not alter abruptly. But if we had supposed that <f> was equal 
to Cx + C O, where 6 = tan" 1 yjx t the equation (E) and the con 
dition (a) would still be satisfied, but the value of q would be 
C" - p ( Cx + C ff), in which the term pG 6 alters abruptly from 
%7rpC to 0, as 6 passes through the value 2?r. The condition (d) 
then alone shews that the former and not the latter is the true 
solution of the problem. 

The fact that the analytical conditions of a problem in fluid 
motion, as far as those conditions depend on the velocities, may be 
satisfied by values of those velocities, which notwithstanding cor 
respond to a pressure which alters abruptly, may be thus explained. 
Conceive two masses of the same fluid contained in two similar 
and equal close vessels A and B. For more simplicity, suppose 
these vessels and the fluid in them to be at first at rest. Conceive 
the fluid in B to be divided by an infinitely thin lamina which is 
capable of assuming any form, and, at the same time, of sustaining 
pressure. Suppose the vessels A and B to be moved in exactly 
the same manner, the lamina in B being also moved in any arbi 
trary manner. It is clear that, except for one particular motion 
of the lamina, the motion of the fluid in B will be different from 
that of the fluid in A. The velocities u, v, w, will in general be 
different on opposite sides of the lamina in B. For particular 
motions of the lamina however the velocities u, v, w, may be the 
same on opposite sides of it, while the pressures are different. 
The motion which takes place in B in this case might, only for 
the condition (d) t be supposed to take place in A. 

It is true that equations (A) or (F), could not strictly speaking 
be said to hold good at those surfaces where such a discontinuity 
should exist. Still, to avoid the liability to error, it is well to 
state the condition (d) distinctly. 



ON SOME CASES OF FLUID MOTION. 25 

When the motion begins from rest, not only must udx+vdy+wdz 
be an exact differential d(f>, and u, v, w, not alter abruptly, but 
also (j> must not alter abruptly, provided the particles in contact 
with the several surfaces remain in contact with those surfaces ; 
for if this condition be not fulfilled, the surface for which it is not 
fulfilled will as it were cut the fluid into two. For it follows from 
the equation (D) that d<f>/dt must not alter abruptly, since other 
wise p would alter abruptly from one point of the fluid to another; 
and d<p/dt neither altering abruptly nor becoming infinite, it fol 
lows that (/> will not alter abruptly. Should an impact occur at 
any period of the motion, it follows from equations (F) that that 
cannot cause the value of (f> to alter abruptly, since such an abrupt 
alteration would give a corresponding abrupt alteration in the 
value of qr. 

3. A result which follows at once from the principle laid down 
in the beginning of the last article is this, that when the motion 
of a fluid in a close vessel which is at rest, and is completely filled, 
is of such a kind that udx + vdy + wdz is an exact differential, it 
will be steady. For let u, v, w, be the initial velocities, and let 
us see if the velocities at the same point can remain u, v, w. First, 
udx + vdy + wdz being an exact differential, equations (A) will be 
satisfied by a suitable value of p, which value is given by equation 
(D). Also equation (B) is satisfied since it is so at first. The con 
dition (a) becomes v = 0, which is also satisfied since it is satis 
fied at first. Also the value of p given by equation (U) will not 
alter abruptly, for dfyjdt 0, or a function of t, and the velocities 
d(j)/dx &c., are supposed not to alter abruptly. Hence, all the 
requisite conditions are satisfied ; and hence, (Art. 2) the hypo 
thesis of steady motion is correct*. 

4. In the case of an incompressible fluid, either of infinite ex 
tent, or confined, or interrupted in any manner by any solid bodies, 
if the motion begin from rest, and if there be none of the cutting 
motion mentioned in Art. 2, the motion at the time t will be the 

* [N.B. It is only within a space which is at least doubly connected that such a 
motion is possible. Thus in the example given in the preceding article, the axis of 
the cylinder, where the velocity becomes infinite, may be regarded as an infinitely 
slender core which we are forbidden to cross, and which renders the space within 
the cylinder virtually ring- shaped.] 



26 ON SOME CASES OF FLUID MOTION. 

same as if it were produced instantaneously by the impulsive 
motion of the several surfaces which bound the fluid, including 
among these surfaces those of any solids which may be immersed in 
it. For let u t v, w, be the velocities at the time t. Then by a known 
theorem udx + vdy + wdz will be an exact differential d<f>, and </> 
will not alter abruptly (Art. 2). (f> must also satisfy the equation 
(E}> and the conditions (a) and (b). Now if u, v , w , be the velo 
cities on the supposition of an impact, these quantities must be 
determined by precisely the same conditions as u, v and w. But 
the problem of finding u , v and w , being evidently determinate, it 
follows that the identical problem of finding u, v and w is also 
determinate, and therefore the two problems have the same solu 
tion ; so that 

u = u , v v, w = w . 

This principle has been mentioned by M. Cauchy, in a memoir 
entitled Memoir e sur la Theorie des Ondes, in the first volume of 
the Memoires des Savans Etrangers (1827), page 14. It will 
be employed in this paper to simplify the requisite calculations by 
enabling us to dispense with all consideration of the previous motion, 
in finding the motion of the fluid at any time in terms of that of 
the bounding surfaces. One simple deduction from it is that, 
when all the bounding surfaces come to rest, each element of the 
fluid will come to rest. Another is, that if the velocities of the 
bounding surfaces are altered in any ratio the value of < will be 
altered in the same ratio. 

5. Superposition of different motions. 

In calculating the initial motion of a fluid, corresponding to 
given initial motions of the bounding surfaces, we may resolve the 
latter into any number of systems of motions, which when com 
pounded give to each point of each bounding surface a velocity, 
which when resolved along the normal is equal to the given 
velocity resolved along the same normal, provided that, if the 
fluid be enclosed on all sides, each system be such as not to alter 
its volume. For let u , v f , w, v , or , be the values of u, v, &c., corre 
sponding to the first system of motions ; u", v", &c., the values of 
those quantities corresponding to the second system, and so on ; 
so that 



v v 



a cr + a" -f ... . 



ON SOME CASES OF FLUID MOTION. 27 

Then since we have by hypothesis u dx + vdy + wdz an exact 
differential d<j> , u"dx + v dy + w dz an exact differential d<f>" t and 
so on, it follows that udx + vdy + wdz is an exact differential. Again 
by hypothesis v = a, v" = a", &c., whence v = cr. Also, if the fluid 
extend to an infinite distance, u, v, and w must there vanish, since 
that is the case with each of the systems u, v, w\ &c. Lastly, the 
quantities < , <", &c., not altering abruptly, it follows that <, 
which is equal to < + </>"+ ... , will not alter abruptly. Hence the 
compounded motion will satisfy all the requisite conditions, and 
therefore (Art. 2) it is the actual motion. 

It will be observed that the pressure p will not be obtained 
by adding together the pressures due to each of the above systems 
of velocities. To find p we must substitute the complete value of 
(f) in equation (D). If, however, the motion be very small, so that 
the square of the velocity is neglected, it will be sufficient to add 
together the several pressures just mentioned. 

In general the most convenient systems into which to decom 
pose the motion of the bounding surfaces are those formed by 
considering the motion of each surface, or of a certain portion of 
each surface, separately". Such a portion may be either finite or 
infinitesimal. In fact, in some of the cases of motion that will be 
presently given, where (f> is expressed by a double integral with a 
function under the integral sign expressing the motion of the 
bounding surfaces, it will be found that each element of the inte 
gral gives a value of (/> such that, except about the corresponding 
element of the bounding surface, the motion of all particles in 
contact with those surfaces is tangential. 

A result which follows at once from this principle, and which 
appears to admit of comparison with experiment, is the following. 
Conceive an ellipsoid, or any body which is symmetrical with 
respect to three planes at right angles to each other, to be made 
to oscillate in a fluid in the direction of each of its three axes in 
succession, the oscillations being very small. Then, in each case, 
as may be shewn by the same sort of reasoning as that employed 
in Art. 8, in the case of a cylinder, the effect of the inertia of the 
fluid will be to increase the mass of the solid by a mass having a 
certain unknown ratio to that of the fluid displaced. Let the axes 
of co-ordinates be parallel to the axes of the solid; let x, y, z t be 



28 ON SOME CASES OF FLUID MOTION. 

the co-ordinates of the centre of the solid, and let M, M , M", be 
the imaginary masses which we must suppose added to that of the 
solid when it oscillates in the direction of the axes of x, y, z, respec 
tively. Let it now be made to oscillate in the direction of a line 
making angles a, /3, 7, with the axes, and let s be measured along 
this line. Then the motions of the fluid due to the motions of 
the solid in the direction of the three axes will be superimposed. 
The motion being supposed to be small, the resultant of the pres 
sures of the fluid on the solid will be three forces, equal to 
2 <? ,7 2 <? /7 2 <? 

a -.ft f~.lv o -AIT 1 1 lit a 

M cos) M co S 



respectively, in the directions of the three axes. The resultant of 
these in the direction of the motion will be M t d 2 s/df where 

M t = if cos 2 a + M f cos 2 /3 + M" cos 2 7 . 

Each of the quantities M, M , M" and M ft may be determined 
by observation, and we may find whether the above relation holds 
between them. Other relations of the same nature may be de 
duced from the principle explained in this article. 

6. Reflection. 

Conceive two solids, A and B, immersed in a fluid of infinite 
extent, the whole being at rest. Suppose A to be moved in any 
manner by impulsive forces, while B is held at rest. Suppose the 
solids A and B of such forms that, if either were removed, and 
the several points of the surface of the other moved instantaneously 
in any given manner, the motion of the fluid could be determined : 
then the actual motion can be approximated to in the following 
manner. Conceive the place of B to be occupied by fluid, and A 
to receive its given motion ; then by hypothesis the initial motion 
of the fluid can be determined. Let the velocity with which the 
fluid in contact with that which is supposed to occupy B s place 
penetrates into the latter be found, and then suppose that the 
several points of the surface of B are moved with normal velocities 
equal and opposite to those just found, A a place being supposed 
to be occupied by fluid. The motion of the fluid corresponding to 
the velocities of the several points of the surface of B can then be 
found, and A must now be treated as B has been, and so on. The 
system of velocities of the particles of the fluid corresponding to 



ON SOME CASES OF FLUID MOTION. 29 

the first system of velocities of the particles of the surface of B, 
form what may be called the motion of A reflected from B , the 
motion of the fluid arising from the second system of velocities of 
the particles of the surface of A may be called the motion of A 
reflected from B and again from A, and so on. It must be re 
membered that all these motions take place simultaneously. It 
is evident that these reflected motions will rapidly decrease, at 
least if the distance between A and B is considerable compared 
with their diameters, or rather with the diameter of either. In 
this case the calculation of one or two reflections will give the 
motion of the fluid due to that of A with great accuracy. It is 
evident that the principle of reflection will extend to any number 
of solid bodies immersed in a fluid ; or again, the body B may be 
supposed to be hollow, and to contain the fluid and A, or else A 
to contain B. In some cases the series arising from the successive 
reflections can be summed, in which case the motion will be deter 
mined exactly. The principle explained in this article has been 
employed in other subjects, and appears likely to be of great use 
in this. It is the same for instance as that of successive influences 
in Electricity. 

7. If a mass of fluid be at rest or in motion in a close vessel 
which it entirely fills, the vessel being either at rest or moving in 
any manner, any additional motion of translation communicated 
to the vessel will not affect the relative motion of the fluid. For 
it is evident that on the supposition that the relative motion is 
not affected the equation (B) and the condition (a) will still be 
satisfied. Also, if Wj, tn- 2 , v? 3 , be the components of the effective force 
of any particle in the first case, and U, V, W, be the components 
of the velocity of translation, then 

dU dV dW 



will be the components of the effective force of the same particle 
in the second case. Now since by hypothesis vr^dx + vr z dy -f vr s dz 
is an exact differential, as follows from equations ((7), and U, V } W, 
are functions of t only, it follows at once that 

dU\, dV 



30 ON SOME CASES OF FLUID MOTION. 

is an exact differential, where x, y, z, are the co-ordinates of any 
particle referred to the old axes, which are themselves moving in 
space with velocities U, V, W. But if x lt y^z^ be the co-ordinates 
of the same particle referred to parallel axes fixed in space, we 
have 

a^x + fUdt, y^y+fVdt, z^z+fWdt, 

whence, supposing the time constant, dx=dx 1 , dy = dy lt dz = dz 1 , 
and therefore 



dU\ 7 . f . d 



) & > 



is an exact differential. Hence, equations (A) can be satisfied by 
a suitable value of p. Denoting by p the pressure about the par 
ticle whose co-ordinates are x, y, z, in the first case, the pressure 
about the same particle in the second case will be 

{dU dV dW 



It * dt 

none of the terms of which will alter abruptly, since by hypothesis 
p does not. 

Since then the present hypothesis satisfies all the requisite 
conditions, it follows from Art. 2 that that hypothesis is correct. 
If F be the additional effective force of any particle of the vessel 
in consequence of the motion of translation, and we take new axes 
of x, y, z , of which the first is in the direction of F, the additional 
term introduced into the value of the pressure will be pFx, 
omitting the arbitrary function of the time. The resultant of the 
additional pressures on the sides of the vessel will be equal to F 
multiplied by the mass of the fluid, and will pass through the 
centre of gravity of the fluid, and act in the directon of x. 

8. Motion between two cylindrical surfaces having a common 
axis. 

Let us conceive a mass of fluid at rest, bounded by two cylin 
drical surfaces having a common axis, these surfaces being either 
infinite or bounded by two planes perpendicular to their axis. Let 
us suppose the several generating lines of these cylindrical surfaces 
to be moved parallel to themselves in any given manner consistent 
with the condition that the volume of the fluid be not altered : 



ON SOME CASES OF FLUID MOTION. 31 

it is required to determine the initial motion at any point of the 
mass. 

Since the motion will take place in two dimensions, let the 
fluid be referred to polar co-ordinates r, 0, in a plane perpendicular 
to the axis, r being measured from the axis. Let a be the radius 
of the inner surface, 6 that of the outer, f(6) the normal velocity 
of any point of the inner surface, F(6) the corresponding quantity 
for the outer. 

Since for any particular radius vector between a and b the 
value of (j> is a periodic function of 6 which does not become in 
finite, (for the motion at each point of each bounding surface 
is supposed to be finite), and which does not alter abruptly, it 
may be expanded in a converging series of sines and cosines of 
6 and its multiples. Let then 

= P + 2r (P n cos nO + Q n sin nff) (1). 

Substituting the above value in the equation 
d 



which <f) is to satisfy, and equating to zero the coefficients of 
corresponding sines and cosines, which is allowable, since a given 
function can be expanded in only one series of the form (1), we 
find that P must satisfy the equation 

A 

of which the general integral is 



the base being e, and P n and Q n must both satisfy the same 
equation, viz. 



dr dr 



of which the general integral is 

P n = Cr-*+C r*. 
We have then, omitting the arbitrary constant in </>, as will 



32 ON SOME CASES OF FLUID MOTION. 

be done for the future, since we have occasion to use only the 
differential coefficients of (, 

= A \og r + Sr {(A n r~ n + A n r n ] cos W 

+ (J? n r-" + J5>")sin^} ......... (3), 

with the conditions 

when r = a ..................... (4), 

when r=6 ............ (5). 



Let / (&) = + ( n cos n< + n sn w 

j^(6>) = + SrC^cos n0 + & n sin TI 
so that 



with similar expressions for , &c. Then the condition (4) 
gives 

cos TZ^ 



whence, 



Similarly, from the condition (5), we get 



ON SOME CASES OF FLUID MOTION. 33 

It will be observed that aC Q = bC Q , by the condition that the 
volume of fluid remains unchanged, which gives 



o Jo 

From the above equations we easily get 



and, changing the sign of n, 





with similar expressions for B n and B M involving D in place of C. 
We have then 

</> = a(7 log r + Sr - (& 2W - a 2 ") 1 {[(b~ n+l C n - a~ n+1 CJ cos nO 
fi 

+ (b~ n+l D n - a- n+l DJ sin nff] a?" b^r^ 

+ [(b n+1 C n -a n "C n )cosn6 

+ (b n+l D n -a n+1 D n )siun0]r n } ..................... (6), 

which completely determines the motion. 

It will be necessary however, (Art. 2), to shew that this value 
of (f> does not alter abruptly for points within the fluid, as may 
be easily done. For the quantities C n> D n cannot be greater than 

} where each element of the integral is taken posi- 



tively ; and since by hypothesis / (&) is finite for all values of 9 
from to 2?r, it follows that neither G n nor D n can be numerically 
greater than a constant quantity which is independent of n. The 
same will be true of C n and D n . Remembering then that r>a 
and < 6, it can be easily shewn that the series which occur in (6) 
have their terms numerically less than those of eight geometric 
series respectively whose ratios are less than unity; and since 
moreover the terms of the former set of series do not alter abruptly, 
it follows that < cannot alter abruptly. The same may be proved 
in a similar manner of the differential coefficients of (/>. The other 
infinite series expressing the value of < which occur in this paper 
may be treated in the same way : and in Art. 10, where (/> is 
expressed by a definite integral, the value of (/> and its differential 
s. 3 



84 ON SOME CASES OF FLUID MOTION. 

coefficients will alter continuously, since that is the case with each 
element of the integral. It will be unnecessary therefore to 
refer again to the condition (d), 

If the fluid be infinitely extended, we must suppose C n and 
D n to vanish in (6), since the velocity vanishes at an infinite 
distance ; we must then make b infinite, which reduces the above 
equation to 






..... (7). 



This value of (/> may be put under the form of a definite 
integral : for, replacing C Q , C n and D n by their values, it becomes 



(0-ff) dff, 
which becomes on summing the series 

log r*f(ff)W + **loe l - 2 a - cos (6 - ff] + / (P) dff; 



whence 



_^ 
~irr J 



, 



dr ~irr 2 r 2 - 2ar cos (0- ff)+ 

If we suppose r to become equal to a the quantity under the 
integral sign vanishes, except for values of 6 , which are indefinitely 
near to 6. The value of the integral itself becomes irf(0)*. Hence 
it appears, that to the disturbance of each element of the surface, 
there corresponds a normal velocity of the particles in contact 
with the surface, which is zero, except just about the disturbed 
element. The whole disturbance of the fluid will be the aggregate 
of the disturbances due to those of the several elements of the 
surface. The case of the initial motion of fluid within a cylinder, 
and the analogous cases of motion within and without a sphere, 
which will be given in the next article, may be treated in the 
same manner. 

The velocity in the direction of r given by the equation (7), 



= > + 2- {G n cos ne + D n sin nO} t 

* Poisson, TMorie de la Chalcur, Chap. vn. 



ON SOME CASES OF FLUID MOTION. 35 

and that perpendicular to r, and reckoned positive in the same 
direction as 0, (= d(f)/rd0), 

n+l 

{C n smn0-D n cosn6}. 

Conceive a mass of fluid comprised between two infinite 
parallel planes, and suppose that a certain portion of this fluid 
contains solid bodies bounded by cylindrical surfaces perpendicular 
to these planes. The whole being at first at rest, suppose that 
the surfaces of these solids are moved in any manner, the motion 
being in two dimensions. Conceive a circular cylindrical surface 
described perpendicular to the parallel planes, and with a radius so 
large that all the solids are comprised with it. Then, (Art. 4), we 
may suppose the motion of the fluid at any time to have been 
produced directly by impact. On this supposition the initial 
motion of the part of the fluid without the above cylindrical 
surface will be determined in terms of the normal motion of the 
fluid forming that surface, as has just been done. If (7 be different 
from zero, then, at a great distance in the fluid, the velocity will 
be ultimately aCJr, and directed to or from the axis of the 
cylinder, and alike in all directions. Since the rate of increase 
of volume of a length I of the cylinder is equal to 



it appears that the velocity at a great distance is proportional 
to the expansion or contraction of a unit of length of the solids. 
If however there should be no expansion or contraction, or if 
the expansion of some of the solids should make up for the con 
traction of the rest, then in general the most important part of 
the motion at a great distance will consist of a velocity O cos l . /r 2 
directed to or from the centre, and another C sin l . /r z perpen 
dicular to the radius vector, the value of C and the direction from 
which 0j is measured varying from one instant to another. The 
resultant of these velocities will vary inversely as the square of 
the distance. 

Resuming the value of <f> given by equation (6), let us suppose 
that the interior cylindrical surface is rigid, and moved with a 
velocity C in the direction from which 6 is measured, the outer 

32 



36 ON SOME CASES OF FLUID MOTION. 

surface being at rest: then / (6) = G cos 0, F (0) = Q ; whence 
C l = C, and the other coefficients are each zero. We have then 



Suppose now that the inner cylinder has a small oscillatory 
motion about an axis parallel to the axis of the cylinders, the 
cylinders having their axes coincident in the position of equi 
librium. Let ty be the angle which a plane drawn through the 
axis of rotation, and that of the solid cylinder at any time makes 
with a vertical plane drawn through the former. The motion 
of translation of the axis of the cylinder will differ from a recti 
linear motion by quantities depending on iff: the motion of 
rotation about its axis will be of the order -^, but will have no 
effect on the fluid. Therefore in considering the motion of the 
fluid we niay, if we neglect squares of ^, consider the motion 
of the cylinder rectilinear. The expression given for </> by equa 
tion (8) will be accurately true only for the instant when the 
axes of the cylinders coincide ; but since the whole resultant 
pressure on the solid cylinder in consequence of the motion is 
of the order ty, we may, if we neglect higher powers of -vjr than the 
first, employ the approximate value of </> given by equation (8). 
Neglecting the square of the velocity, we have 

d$ 
P~ P dt 

In finding the complete value of d(f>/dt it would be necessary to 
express (f> by co-ordinates referred to axes fixed in space, which 
after differentiation we might suppose to coincide with others 
fixed in the body. But the additional terms so introduced de 
pending on the square of the velocity, which by hypothesis is 
neglected, we may differentiate the value of $ given by equation 
(8) as if the axes were fixed in space. We have then, to the first 
order of approximation, 

ci 

-n = - ?~ 2 \- + r l cos & 
dt b -or (r J 

If I be the length of the cylinder, the pressure on the element 
ladO, resolved parallel to x and reckoned positive when it acts 
in the direction of x, 



ON SOME CASES OF FLUID MOTION. 87 

P la *-^ 



and integrating from = to = 2?r, we have the whole resultant 
pressure parallel to x 

tf + a? 2 dC 

= 72 9 Vpla -j- . 

If or dt 

Since dC/dt is the effective force of the axis, parallel to x, and 
that parallel to y is of the order i|r 2 , we see that the effect of 
the inertia of the fluid is to increase the mass of the cylinder 

by TTT~ 2 A 6 * where p is the mass of the fluid displaced. This 

imaginary additional mass must be supposed to be collected at the 
axis of the cylinder. 

If the cylinder oscillate in an infinitely extended fluid b = & , 
and the additional mass becomes equal to that of the fluid dis 
placed. This appears to be a result capable of being compared 
with experiment, though not with very great accuracy. Two 
cylinders of the same material, and of the same radius, but whose 
lengths differ by several radii, might be made to oscillate in 
succession in a fluid, at a depth sufficiently great to allow us 
to neglect the motion of the surface of the fluid. The time of 
oscillation of each might then be calculated as if the cylinder 
oscillated in vacuum, acted on by a moving force equal to its 
weight minus that of the fluid displaced, acting downwards 
through its centre of gravity, and having its mass increased by an 
unknown mass collected in the axis. Equating the time of oscil 
lation so calculated to that given by observation, we should 
determine the unknown mass. The difference of these masses 
would be very nearly equal to the mass which must be added 
to that of a cylinder whose length is equal to the difference of 
the lengths of the first two, when the motion is in two dimensions. 
This evidently comes to supposing that, at a distance from the 
middle of the longer cylinder not greater than half the difference 
of the lengths of the two, the motion may be taken as in two 
dimensions. The ends of the cylinders may be of any form, 
provided that they are all of the same. They may be suspended 
by fine equal wires, in which case we should have a compound 



38 ON SOME CASES OF FLUID MOTION. 

pendulum, or attached to a rigid body oscillating above the fluid 
by means of thin flat bars of metal, whose plane is in the plane of 
motion. Another way of getting rid of the motion in three 
dimensions about the ends would be, to make those ends plane, 
and to fix two rigid planes parallel to the plane of motion, which 
should be almost in contact with the ends of the cylinder. 

9. Motion between two concentric spherical surfaces. Motion 
of a ball pendulum enclosed in a spherical case. 

Let a mass of fluid be at rest, comprised between two con 
centric spherical surfaces. Let the several points of these surfaces 
be moved in any manner consistent with the condition that the 
volume of the fluid be not changed : it is required to determine 
the initial motion at any point of the mass. 

Let a, b, be the radii of the inner and outer spherical surfaces 
respectively ; then employing the co-ordinates r } 0, o>, where r 
is the distance from the centre, 6 the angle which r makes with 
a fixed line passing through the centre, G> the angle which a plane 
passing through these two lines makes with a fixed plane through 
the latter, the value of </> corresponding to any radius vector 
comprised between a and b can be expanded in a converging 
series of Laplace s coefficients. Let then 



V n being a Laplace s coefficient of the n th order. 
Substituting in the equation, 

dV0 1 d ( . a d$\ , 1 

r TT + - Q -jh sm JQ + -=-T 
dr sin 6 dv \ dvj sin 

which </> is to satisfy, employing the equation 



and then equating to zero the Laplace s coefficients of the several 
orders, we find 



The general integral of this equation is 



ON SOME CASES OF FLUID MOTION. 39 

where C and are functions of 6 and G>. Substituting in the 
equation (9), and equating coefficients of the two powers of r 
which enter into it separately to zero, we find that both C and G 
satisfy it, and therefore are both Laplace s coefficients of the n ih 
order. We have then 



where Y n and Z n are each Laplace s coefficients of the n ih order, 
and do not contain r. Let f(0, w) be the normal velocity of the 
point of the inner surface corresponding to 6 and co, F(6, o>) the 
corresponding quantity for the outer ; then the conditions which 
<j) is to satisfy are that 



-~ = f(Q } G)) when r = a, 
dr 

^0 T^/a \ T_ 7 

-r- = if (u, &)) when r b. 
Let /(#, ft)), expanded in a series of Laplace s coefficients, be 

P.+P....+P.+... 

which expansion may be performed by the usual formula,, if not 
by inspection: then the first condition gives 

STw Y n n ~ l (n -1- 1\ ^ /7-( n + 2U ^P 
t 7i \ * / n ** j ^o 71 > 

and equating Laplace s coefficients of the same order, we get 

V ^n-l Inn i "T \ ^ ^-(n+2) P f~\T\ 

71 2 n d \n-\- i)Zj n d v = JL n \^-^/ 

Let F(0, ft)), expanded in a series of Laplace s coefficients, be 

F t +P t ...P,+ ...! 

then from the second condition, we get 

7 iF w ^-l_( w+ l)^-(n+2) == p n (12). 

From (11) and (12) we easily get 

P f lkn+2 P s*n+2 
T-r -JL u _/: (jj 



provided n be greater than 0. If n 0, we have 



40 ON SOME CASES OF FLUID MOTION. 

But the condition that the volume of the fluid be not altered, 
gives 

tf I I *f(0, ) sin OdBdto = &*[* f F(0, to) sin 0d0da, 

J OJQ JO ./O 

or 47ra 2 P =47r& 2 P , 

which reduces the two equations just given to one. 
We have then, omitting the constant Y QJ 

i (P n 6J;2 - P w a + 2 ) r n 



which determines the motion. 

When the fluid is infinitely extended, we have P n = since 
the velocity vanishes at an infinite distance, and b = GO , whence 



It may be proved, precisely as was done, (Art. 8), for motion 
in two dimensions, that if any portion of an infinitely extended 
fluid be disturbed by the motion of solid bodies, or otherwise, 
if all the fluid beyond a certain distance from the part disturbed 
were at first at rest, the velocity at a great distance will ultimately 
be directed to or from the disturbed part, and will be the same 
in all directions, and will vary as r~ 2 . The coefficient of r~ 2 will 
be proportional to the rate of gain or loss of volume of the part 
disturbed. If however this rate should be zero, then the most 
important part of the velocity at a great distance will in general 
be that depending on the term -^a s P l .r~ z in <, Since the 
general form of P l is 

J.cos 0+ J9sin0cosa> + sin0sin, 

we easily find, by making use of rectangular co-ordinates, changing 
the direction of the axes, and then again adopting polar co 
ordinates, that the above term in <f> takes the form D cos 6^ . r~ 2 , 
0j being measured from some line passing through the origin. 
The motion will therefore be the same as that round a ball 
pendulum in an incompressible fluid, the centre of the ball being 
in the origin; a case of motion which will be considered im 
mediately. In order to represent the motion at different times, 



ON SOME CASES OF FMJID MOTION. 41 

we must suppose the velocity and direction of motion of the 
ball to change with the time. 

The value of </> given by equation (13) is applicable to the 
determination of the motion of a ball pendulum enclosed in a 
spherical case which is concentric with the ball in its position of 
equilibrium. If G be the velocity of the centre of the ball at 
the instant when the centres of the ball and case coincide, and 
if 6 be measured from the direction in which it is moving, we 
shall have 



/. P = 0, P^C cos 0, P 2 = 0, &c., P = 0, &c., 
and the value of (/> for this instant is accurately 
Co 3 b* 



which, when b = oo , becomes 

Co? cos 
2r z 

which is the known expression for the value of </> for a sphere 
oscillating in an infinitely extended, incompressible fluid. 

It may be shewn, by precisely the same reasoning as was 
employed in the case of the cylinder, that in calculating the 
small oscillations of the sphere the value of d(f>/dt to be employed is 




and from the equation p = p d<f>/dt, we easily find that the whole 
resultant pressure on the sphere in the direction of its centre, and 
tending to retard it is 

4 Trpa 8 t V_\dCL 

and that perpendicular to this direction is zero. Since dC/dt is 
the effective force of the centre in the direction of the motion, and 
that perpendicular to this direction is of the second order, the 
effect of the inertia of the fluid will be to increase the mass of the 
sphere by a mass 



42 ON SOME CASES OF FLUID MOTION. 

IJL being the mass of the fluid displaced ; so that the effect of the 
case is, to increase the mass which we must suppose added to 
that of the ball in the ratio of b 3 + 2a 3 to V - a\ 

Poisson, in his solution of the problem of the oscillating sphere 
given in the Memoir es de I Acade mie, Tome XL arrives at a different 
conclusion, viz. that the case does not at all affect the motion of 
the sphere. When the elimination which he proposes at p. 563 
is made, the last term of equation (/), p. 550, becomes 



where a is the velocity of propagation of sound, and 8 the ratio 
of the density of air to that of the ball, f and " being functions 
derived from others which enter into the value of <f> by putting 
r = c, where c is the radius of the ball. He then argues that 
this term may be neglected as insensible, since it involves 8 in 
the numerator and a 2 in the denominator, tacitly assuming that 



jy + is not large since is not large. Now for the disturb 
ed dt 

ances of the air which have the same period as those of the 
pendulum d$/dt is not large compared with <, as it is for those on 
which sound depends. Let then Poisson s solution of equation (a), 
p. 547 of the volume already mentioned, be put under the form 



/ and F denoting the derived functions, and all the Laplace s 
coefficients except those of the first order being omitted, the value 
of </> just given being supposed to be a Laplace s coefficient of that 
order. Then if we expand the above functions in series ascending 
according to powers of r/a, we find 



and in order that when a = oo this equation may coincide with 
(10), when all the Laplace s coefficients except those of the first 
order are omitted in that equation, it will be seen that it is 



ON SOME CASES OF FLUID MOTION. 43 

necessary to suppose f"(t)-F "(t) t and therefore f(t)-F(), 
to be of the order a/ 5 , while f(t) + F (t) is not large. Putting then 



=x (0 

l?(*)-X() 

we shall have 



. 
so that -^7-3-^ will contain a term of the order a 2 , and the 

Cut 

term which Poisson proposes to leave out will be of the same 
order of magnitude as those retained. 

In making the experiment of determining the resistance of 
the air to an oscillating sphere, it would appear to be desirable 
to enclose the sphere in a concentric spherical case, which would 
at the same time exclude currents of air, and facilitate in some 
measure the experiment by increasing the small quantity which is 
the subject of observation. The radius of the case however ought 
not to be nearly as small as that of the ball, for if it were, in 
the first place a small error in the position of the centre of the 
ball when at rest might not be insensible, and in the second place 
the oscillations would have to be inconveniently small, in order 
that the value of <f> which has been given might be sufficiently 
approximate. The effect of a small slit in the upper part of the 
case, sufficient to allow the wire by which the ball is supported 
to oscillate, would evidently be insensible, for the condensation 
being insensible in a vertical plane passing through the axis of 
rotation, since the alteration of pressure in that plane is insensible, 
the air would not have a tendency alternately to rush in and out 
at the slit. 

10. Effect of a distant rigid plane on the motion of a ball 
pendulum. 

Although this problem may be more easily solved by an arti 
fice, it may be well to give the direct solution of it by the method 
mentioned in Article 6. In order to calculate the motion re 
flected from the plane, it will be necessary to solve the following 
problem : 



44 ON SOME CASES OF FLUID MOTION. 

To find the initial motion at any point of a mass of fluid in 
finitely extended, except where it is bounded by an infinite solid but 
not rigid plane, the initial motion of each point of the solid plane 
being given. 

It is evident that motion directed to or from a centre situated 
in the plane, the velocity being the same in all directions, and 
varying inversely as the square of the distance from that centre, 
would satisfy the condition that udx + vdy + wdz is an exact 
differential, and would give to the particles in contact with the 
plane a velocity directed along the plane, except just about the 
centre. Let us see if the required motion can be made up of an 
infinite number of such motions directed to or from an infinite 
number of such centres. 

Let x, y, z, be the co-ordinates of any particle of fluid, the 
plane xy coinciding with the solid plane, and the axis of z being 
directed into the fluid. Let x, y y be the co-ordinates of any point 
in the solid plane : then the part of < corresponding to the motion 
of the element dxdy of the plane will be 

ty(x, y )dx dy 



and therefore the complete value of </> will be given by the equa 
tion 

4 = f f _ *fr.yWfr ( H). 

* /(/ \a i /. ^. \* i -.2) V X 



The velocity parallel to z at any point = dcf>/dz 



Now when z vanishes the quantity under the integral signs 
vanishes, except for values of x and y indefinitely near to x arid y 
respectively, the function ty(x t y ) being supposed to vanish when 
x or y is infinite. Let then x = x + f , y = y + 77, then, and 77, 
being as small as we please, the value of the above expression 
when z = becomes 

-the limit off 

/- 

Now if ^r(x, y } does not alter abruptly between the limits x- 



ON SOME CASES OF FLUID MOTION. 45 

and x + % , of x, and y 77, and ;y + 77, of y , the above expression 
may be replaced by 

- x the limit 



which is = 27Ti/r(a?, y}. 

If now/(# , y") be the given normal velocity of any point (a? , y} 
of the solid plane, the expression for < given by equation (14) may 
be made to give the required normal velocity of the fluid particles 
in contact with the solid plane by assuming 



whence 

A = IT f f( 

2vr J _ J .. {(aj- aj ) a + (y- 

This expression will be true for any point at a finite distance from 
the plane xy even when / (x, y } does alter abruptly; for we may 
first suppose it to alter continuously, but rapidly, and may then 
suppose the rapidity of alteration indefinitely increased : this will 
not cause the value of just given to become illusory for points 
situated without the plane xy. 

If it be convenient to use polar co-ordinates in the plane xy, 
putting x = q cos co, y = q sin co, x = q cos co , y q sin &/, and re 
placing/^ , y } by/(/, < ), the equation just given becomes 



2?r o o {q 2 + <f- Zqq cos (co - co ) + 

To apply this to the case of a sphere oscillating in a fluid per 
pendicularly to a fixed rigid plane, let a be the radius of the sphere, 
and let its centre be moving towards the plane with a velocity C 
at the time t. Then, (Art. 4), we may calculate the motion as if 
it were produced directly by impact. Let h be the distance of the 
centre of the sphere from the fixed plane at the time t, and let 
the line h be taken for the axis of z, and let r, 0, be the polar co 
ordinates of any point of the fluid, r being the distance from the 
centre of the sphere, and 6 the angle between the lines r and h. 
Then if the fluid were infinitely extended around the sphere we 
should have 

(7a s cos 



40 ON SOME CASES OF FLUID MOTION. 

The velocity of any particle, resolved in a direction towards the 
plane, = d$/dr . cos d$/rd& . sin 6 

LsCL r n * 






For a particle in the plane xy we have 

r cos 6 h, rsm6 = q> 
and the above velocity becomes 



We must now, according to the method explained in (Art. 6), sup 
pose the several points of the plane xy moved with the a,bove 
velocity parallel to z. We have then 



whence, for the motion of the sphere reflected from the plane, 

*- a* a rf (w-Mw* 

47r J o J tf+ < + - 2 cos a, - + z 2 * " 



- 2 qq cos (a, - 

We must next find the velocity, corresponding to this value of 
(f>, with which the fluid penetrates the surface of the sphere. We 
have in general 

z h r cos 0, q r sin 0, 
whence 

[f + f - 2qq cos (o> - < ) + z 2 }^ 

= {h 2 + r 2 + # 2 - 2hr cos - 2q r sin cos (co - a> )}~*. 
Now supposing the ratio of a to h to be very small, and retaining 
the most important term, the value of d(f>/dr when r = a will be 
equal to the coefficient of r when is expanded in a series ascend 
ing according to powers of r, 

_Ca?_ r ^ (2A 2 - q z ) {h COS + tf sin 6 cos (a - a) )} q dq da) 
47rJ J " (h* + q *)* 

Ca 5 cos 9 



In order now to determine the motion reflected from the 
plane and again from the sphere, we must suppose the several 
points of the sphere to be moved with a normal velocity 



ON SOME CASES OF FLUID MOTION. 47 

Ca 3 cos 6 . /8h 3 , or, which is the same, we must suppose the whole 
sphere to be moved towards the plane with a velocity Ca s /8h 3 . 
Hence the value of <f> corresponding to this motion will be given 
by the equation 

Ca 6 cos 



For points at a great distance from the centre of the sphere, 
the motion which is twice reflected will be very small compared 
with that which is but once reflected. For points close to the 
sphere however, with which alone we are concerned, those motions 
will be of the same order of magnitude, and if we take account 
of the one we must take account of the other. 

Putting 2=rsin#, z = h-rcos0 in (16), expanding, and 
retaining the two most important terms, we have 



K being a constant, the value of which is not required, and the 
second term being evidently found by multiplying the quantity 
at the second side of (17) by r. Adding together the parts of $ 
given by equations (15), (18) and (19), putting r = a t replacing 
G by dC/dt, and taking for h the value which it has in equili 
brium, just as in the case of the oscillating cylinder in Article 8, 
we have for the small motion of the sphere 

cty j-rdC a/- Sa 3 \dC 



The resultant of the part of the pressure due to the first term 
is zero : that due to the second term is greater than if the plane 
were removed in the ratio of l + Sa 3 /8h 3 to 1. Consequently, if 
we neglect quantities of the order a 4 //*, 4 , the effect of the inertia 
of the fluid is, to add a mass equal to (1 + 3a 3 /8h?) . \p to that of 
the sphere, without increasing the moment of inertia of the latter 
about its diameter. The effect therefore of a large spherical case 
is eight times as great as that of a tangent plane to the case, 
perpendicular to the direction of the motion of the ball. 

The effect of a distant rigid plane parallel to the direction 
of motion of an oscillating sphere might be calculated in the 
same manner, but as the method is sufficiently explained by the 



48 ON SOME CASES OF FLUID MOTION. 

first case, it will be well to employ the artifice before alluded to, 
an artifice which is frequently employed in this subject. It con 
sists in supposing an exactly symmetrical motion to take place 
on the opposite side of a rigid plane, by which means we may 
evidently conceive the plane removed. 

Let the sphere be oscillating in the direction of the axis of oc, 
the oscillations in this case, as in the last, being so small that 
they may be taken as rectilinear in calculating the motion of the 
fluid ; and instead of a rigid plane conceive an equal sphere to exist 
at an equal distance on the opposite side of the plane xy, moving 
in the same direction and with the same velocity as the actual 
sphere. Let r, 0, ew, be the polar co-ordinates of any particle 
measured from the centre of the sphere, 6 being the angle between 
r and a line drawn through the centre parallel to the axis of x, 
and w the angle which the plane passing through these lines makes 
with the plane ocz. Let r t , o> 3 be the corresponding quantities 
symmetrically measured from the centre of the imaginary sphere. 

If the fluid were infinite we should have for the motion cor 
responding to that of the given sphere 



The motion reflected from the plane is evidently the same as 
that corresponding to the motion of the imaginary sphere in an 
infinite mass of fluid, for which we have 

Co? cos & 



Now r cos = r cos 6, r sin & sin to = r sin 6 sin to, 

/ sin & cos o> + r sin 6 cos o> = 2h ; 
whence r 2 = r 2 + 4A 2 4<hr sin cos co, 

and equation (21) is reduced to 

<7aV cos 6 

2 {r 2 + 4 2 - 4/tr sin cos o>} f 

Retaining only the terms of the order a*r/h a or r 4 /^ 3 , so as to get 
the value of d<j>/dr to the order a 3 /h 3 , the above equation is re 

duced to 

Ca*r cos , 

...................... (22) 



ON SOME CASES OF FLUID MOTION. 49 

and the value of dfy/dr when r = a is, to the required degree of 
approximation, 

Ca 3 cos 6 



For the value of </> corresponding to the motion of the imaginary 
sphere reflected from the real sphere, we shall therefore have 
, Co? cos 



Adding together the values of $ given by (20), (22) and (23), 
putting r = a, and replacing C by dC/dt, we have, to the requisite 
degree of approximation, 

3 a s \ dO 



Hence in this case the motion of the sphere will be the same as 
if an additional mass equal to (l + 3a 8 /16& 8 ) . \p were collected 
at its centre. The effect therefore of a distant rigid plane which 
is parallel to the direction of the motion of a ball pendulum will 
be half that of a plane at the same distance, and perpendicular 
to that direction. It would seem from Poisson s words at page 562 
of the eleventh volume of the Memoir es de VAcademie, that he 
supposed the effect in the former case to depend on a higher 
order of small quantities than that in the latter. 

If the ball oscillate in a direction inclined to the plane, the 
motion may be easily deduced from that in the two cases just 
given, by means of the principle of superposition. 

11. The values of </> which have been given for the motion 
of translation of a sphere and cylinder do not require us to 
suppose that either the velocity, or the distance to which the 
centre of the sphere or axis of the cylinder has been moved, is 
small, provided the same particles remain in contact with the 
surface. The same indeed is true of the values corresponding to 
a motion of translation combined with a motion of contraction 
or expansion which is the same in all directions, but varies in any 
manner with the time. The value of </> corresponding to a motion 
of translation of the cylinder is - Ca 2 cos 9 . r~\ C being the velo 
city of the axis, and 6 being measured from a line drawn in the 
direction of its motion. The whole resultant of the part of the 
pressure due to the square of the velocity is zero, since the velocity 
at the point whose co-ordinates are r, 0, is the same as that at 
S. 4 



50 ON SOME CASES OF FLUID MOTION. 

the point whose co-ordinates are r and -rr-6. To find the re 
sultant of the part depending on d(j>/dt, it will be necessary to 
express ^ by means of co-ordinates referred to axes fixed in space. 
Let Ox, Oy, be rectangular axes passing through the centre of 
any section of the cylinder, OT the angle which the direction of 
motion of the axis makes with Ox, & the inclination of any radius 
vector to Ox \ then 

Co? 
<f>= -- jj- (r cos 6 cos w + r sin 6 sin -BT) 

a*(C x+C"y) 

x* + y* 

putting Q and C" for the resolved parts of the velocity G along 
the axes of x and y respectively. Taking now axes Ax, Ay, 
parallel to the former and fixed in space, putting a and /3 for the 
co-ordinates of 0, differentiating <j> with respect to t, and replacing 
da/dt by C , and d/3/dt by C", and then supposing a and /3 to 
vanish, we have 

,/ dC?_ dC"\ 

d = a 2 2 _ 2o^( * + (Ty)! _ * V dt V dt 
dt ~ 



The resultant of the part of the pressure due to the first two 
terms is zero, since the pressure at the point (x, y) depending on 
these terms is the same as that at the point ( x, y). It will 
be easily found that the resultant of the whole pressure parallel 
to x, and acting in the negative direction, on a length I of the 
cylinder, is equal to irpla? . d C /dt, and that parallel to y equal to 
Trpla 2 . dC"ldt. The resultant of these two will be TrplcfF, where 
F is the effective force of a point in the axis of the cylinder, and 
will act in a direction opposite to that of F. Hence the only 
effect of the motion of the fluid will be, to increase the mass of 
the cylinder by that of the fluid displaced. In a similar manner 
it may be proved that, when a solid sphere moves in any manner 
in an infinite fluid, the only effect of the motion of the fluid is to 
increase the mass of the sphere by half that of the fluid displaced. 
A similar result may be proved to be true for any solid sym 
metrical with respect to two planes at right angles to each other, 
and moving in the direction of the line of their intersection in 
an infinitely extended fluid, the solid and fluid having been at 
first at rest. Let the planes of symmetry be taken for the planes 
of xy and xz, the origin being fixed in the body : then it is evident 



ON SOME CASES OF FLUID MOTION. 51 

that the resultant of the pressure on the solid due to the motion 
will be in the direction of the axis of x, and that there will be 
no resultant couple. Let C be the velocity of the solid at any 
time ; then the value of < at that time will be of the form 
Cifr (x, y, 2), where G alone contains t (Art. 4), and the velocity 
of the particle whose co-ordinates are #, y, z, being proportional 
to (7, the vis viva of the solid and the fluid together will be 
proportional to C*. Now if no forces act on the fluid and solid, 
except the pressure of the fluid, this vis viva must be constant * ; 
therefore G must be constant ; therefore the resultant of the fluid 
pressure on the solid must be zero. If now G be a function of t 
we shall have 

p = _p^ (a-, #*) +/, 

p being the pressure when G is constant. Since therefore the 
resultant of the fluid pressure varies for the same solid and fluid 
as dC/dt the effective force, and for different fluids varies as p, 
the effect of the inertia of the fluid will be, to increase the mass 
of the solid by n times that of the fluid displaced, n depending 
only on the particular solid considered. 

Let us consider two such solids, similar to each other, and 
having the co-ordinate planes similarly situated, and moving with 
the same velocities. Let the linear dimensions of the second 
be greater than those of the first in the ratio of m to 1. Let 

* If an incompressible fluid which is homogeneous or heterogeneous, and con 
tains in it any number of rigid bodies, be in motion, the rigid bodies being also 
in motion, if the rigid bodies are perfectly smooth, and no contacts are formed or 
broken among them, and if no forces act except the pressure of the fluid, the 
principle of vis viva gives 



where v is the whole velocity of the mass m, and the sign 2 extends over the whole 
fluid and the rigid bodies spoken of, and where dS is an element of the surface 
which bounds the whole, p / the pressure about the element dS, and v the normal 
velocity of the particles in that element, reckoned positive when tending into the 
fluid, and where the sign ff extends to all points of the bounding surface. To apply 
equation (a) to the case of motion at present considered, let us first confine our 
selves to a spherical portion of the fluid, whose radius is r, and whose centre is near 
the solid, so that dS refers to the surface of this portion. Let us now suppose r to 
become infinite : then the second side of (a) will vanish, provided^ remain finite, 
and v decrease in a higher ratio than r~ 2 . Both of these will be true, (Art. 9) ; for 
v will vary ultimately as r~ 3 , since there is no alteration of volume. Hence if the 
sign S extend to infinity, we shall have 2/mv 2 constant. 

. 42 



52 ON SOME CASES OF FLUID MOTION. 

u, v, w, be the velocities, parallel to the axes, of the particle (x, y, z] 
in the fluid about the first ; then shall the corresponding velocities 
at the point (mx, my, mz) in the fluid about the second be also 
u, v, to. For 

udmx + vdmy + wdmz = m (udx + vdy -f wdz) (24), 

and is therefore an exact differential, since udx + vdy+wdz is 
one : also the normal at the point (x, y, z) in the first surface will 
be inclined to the axes at the same angles as the normal at the 
point (mx, my, mz) of the second surface is inclined to its axes, 
and therefore the normal velocities of the two surfaces at these 
points are the same ; and the velocities of the fluid at these two 
points parallel to the axes being also the same, it follows that the 
normal velocity of each point of the second surface is equal to 
that of the fluid in contact with it. Lastly, the motion about 
the first solid being supposed to vanish at an infinite distance 
from it, that about the second will vanish alsU Hence the sup 
position made with respect to the motion of the fluid about the 
second surface is correct. Now putting for $(udx + vdy + wda}} 
for the fluid in the first case, the corresponding integral for the 
fluid in the second case will be ???<, if the constant be properly 
chosen, as follows from equation (24). Consequently the value of 
that part of the expression for the pressure, on which the resist 
ance depends, will be m times as great for any point in the second 
case as it is for the corresponding point in the first. Also, each 
element of the surface of the second solid will be m 2 times as 
great as the corresponding element of the surface of the first. 
Hence the whole resistance on the second solid will be m 3 times 
as great as that on the first, and therefore the quantity n depends 
only on the form, and not on the size of the solid. 

When forces act on the fluid, it will only be necessary to add 
the corresponding pressure. Hence when a sphere descends from 
rest in a fluid by the action of gravity, the motion will be the same 
as if a moving force equal to that of the sphere minus that of 
the fluid displaced acted on a mass equal to that of the sphere 
plus half that of the fluid displaced. For a cylinder which is 
so long that we may suppose the length infinite, descending hori 
zontally, every thing will be the same, except that the mass to be 
moved will be equal to that of the cylinder plus the whole of the 
fluid displaced. In these cases, as well as in that of any solid 



ON SOME CASES OF FLUID MOTION. 53 

which is symmetrical with respect to two vertical planes at right 
angles to each other, the motion will be uniformly accelerated, 
and similar solids of the same material will descend with equal 
velocities. These results are utterly opposed even to the com 
monest observation, which shews that large solids descend much 
more rapidly than small ones of the same shape and material, 
and that the velocity of a body falling in a fluid (such as water), 
does not sensibly increase after a little time. It becomes then 
of importance in the theory of resistances to enquire what may be 
ths cause of this discrepancy between theory and observation. 
The following are the only ways of accounting for it which suggest 
themselves to me. 

First. It has been supposed that the same particles remain in 
contact with the solid throughout the motion. It must be re 
membered that we suppose the ultimate molecules of fluids (if 
such exist), to be so close that their distance is quite insensible, a 
supposition of the truth of which there can be hardly any doubt. 
Consequently we reason on a fluid as if it were infinitely divisible. 
Now if the motion which takes place in the cases of the sphere 
and cylinder be examined, supposing for simplicity their motions 
to be rectilinear, it will be found that a particle in contact with 
the surface of either moves along that surface with a velocity which 
at last becomes infinitely small, and that it does not reach the 
end of the sphere or cylinder from which the whole is moving 
until after an infinite time, while any particle not in contact with 
the surface is at last left behind. It seems difficult to conceive of 
what other kind the motion can be, without supposing a line 
(or rather surface) of particles to make an abrupt turn. If it 
should be said that the particles may come off in tangents, it must 
be remembered that this sort of motion is included in the con 
dition which has been assumed with respect to the surface. 

Secondly. The discrepancy alluded to might be supposed to 
arise from the friction of the fluid against the surface of the solid. 
But, for the reason mentioned in the beginning of this paper, this 
explanation does not appear to me satisfactory. 

Thirdly. It appears to me very probable that the spreading 
out motion of the fluid, which is supposed to take place behind 
the middle of the sphere or cylinder, though dynamically possible, 
nay, the only motion dynamically possible when the conditions 



54 ON SOME CASES OF FLUID MOTION. 

which have been supposed are accurately satisfied, is unstable ; 
so that the slightest cause produces a disturbance in the fluid, 
which accumulates as the solid moves on, till the motion is quite 
changed. Common observation seems to shew that, when a solid 
moves rapidly through a fluid at some distance below the surface, 
it leaves behind it a succession of eddies in the fluid. When the 
solid has attained its terminal velocity, the product of the resist 
ance, or rather the mean resistance, and any space through which the 
solid moves, will be equal to half the via viva of the corresponding 
portion of its tail of eddies, so that the resistance will be measured 
by the vis viva in the length of two units of that tail. So far 
therefore as the resistance which a ship experiences depends 
on the disturbance of the water which is independent of its 
elevation or depression, that ship which leaves the least wake 
ought, according to this view, to be cceteris paribus the best sailer. 
The resistance on a ship differs from that on a solid in motion 
immersed in a fluid in the circumstance, that part of the resist 
ance is employed in producing a wave. 

Fourthly. The discrepancy alluded to may be due to the 
mutual friction, or imperfect fluidity of the fluid. 

12. Motion alout an elliptic cylinder of small eccentricity*. 

The value of <, which has been deduced (Art. 8), for the 
motion of the fluid about a circular cylinder, is found on the 
supposition that for each value of r there exists, or may be 

[* This particular problem, so far at least as concerns motion of translation, 
is of little interest in itself, because Green (see Transactions of, the Eoijal Society 
of Edinburgh, Vol. xm. p. 5.4, or p. 315 of his collected works) has determined the 
motion of a fluid about an ellipsoid moving in any manner with a motion of trans 
lation only; and the ellipsoid includes of course as a particular case an elliptic 
cylinder of any eccentricity. The problem in the text will however serve as an 
example of the mode of proceeding in the case of a cylinder of any kind differing 
little from a circular cylinder. 

In the case of such a cylinder, supposed to be free from abrupt changes of form, 
it might safely be assumed that the expression for which applies to the fluid 
beyond the greatest radius vector of any point of the surface might also be used 
for some distance within, as explained in the text. By starting with this assumption, 
which would be verified in the end, the process of solution would of course be 
shortened. We should simply have to take the expression (31 ), form the expression 
(26 ) for the velocity normal to the surface, putting r = c (1 + e cos 20), and expand 
ing as far as the first power of e, and equate the result to the expression (26). We 
should thus determine the arbitrary constants in (31 ), which would complete the 
solution of the problem.] 



ON SOME CASES OF FLUID MOTION. 55 

supposed to exist, a real and finite value of <. This will be true, 
in any case of motion in two dimensions where udx + vdy is an 
exact differential, for those values of r for which the fluid is not 
interrupted, but will be true for values of r for which it is in 
terrupted by solids only when it is possible to replace those solids 
at any instant by masses of fluid, without affecting the motion 
of the fluid exterior to them, those masses moving in such a 
manner that the motion of the whole fluid might have been 
produced instantaneously by impact. In some cases such a 
substitution could be made, while in others it probably could not. 
In any case however we may try whether the expansion given 
by equation (3) will enable us to get a result, and if it will, we 
need be in no fear that it is wrong (Art. 2). The same remarks 
will apply to the question of the possibility of the expansion of < 
in the series of Laplace s coefficients given in equation (10), for 
values of r for which the fluid is interrupted. They will also 
apply to such a question as that of finding the permanent tempe 
rature of the earth due to the solar heat, the earth being supposed 
to be a homogeneous oblate spheroid, and the points of the 
surface being supposed to be kept up to constant temperatures, 
given by observation, depending on the latitude. 

In cases of fluid motion such as those mentioned, the motion 
may be determined by conceiving the whole mass of fluid divided 
into two or more portions, taking the most general value of </> for 
each portion, this value being in general expressed in a different 
manner for the different portions, then limiting the general value 
of (f> for each portion so as to satisfy the conditions with respect to 
the surfaces of solids belonging to that portion, and lastly in 
troducing the condition that the velocity arid direction of motion 
of each pair of contiguous particles in any two of the portions are 
the same. The question first proposed will afford an example 
of this method of solution. 

Let an elliptic cylinder be moving with a velocity (7, in the 
direction of the major axis of a section of it made by a plane 
perpendicular to its axis. The motion being supposed to be in 
two dimensions, it will be sufficient to consider only this section. 
Let 

r = c (1 + e cos 20) 

be the approximate equation to the ellipse so formed, the centre 



56 ON SOME CASES OF FLUID MOTION. 

being the pole, and powers of e above the first being neglected. 
Let a circle be described about the same centre, and having a 
radius 7 equal to (1 + k) c, k being ^ e, and being a small quantity 
of the order e. Let the portions of fluid within and without the 
radius 7 be considered separately, and putting 

r = c + z t 
let the value of (f> -corresponding to the former portion be 



P, Q and R being functions of 0, and the term in 2 being retained, 
in order to get the value of dfyjdr true to the order e, while the 
terms in z s , &c. are omitted. Substituting this value of <f> in 
equation (2), and equating to zero coefficients of different powers 
of z, we have 



_ 

2c 2c 2 d6* 

which is the only condition to be satisfied, since the other equations 
would only determine the coefficients of z 3 , &c. in terms of the 
preceding ones. We have then 



Now if be the angle between the normal at any point of the 
ellipse, and the major axis, we have 



and the velocity of the ellipse resolved along the normal 

= (7 cos f = G (I - e) cos + Ce cos 30 ......... (26). 

The velocity of the fluid at the same point resolved along the 
normal is 



Let P and Q be expanded in series of cosines of 6 and its mul 
tiples, so that 

P = 2 " P H cos n9, Q = ^ Q n cos w0, 



ON SOME CASES OF FLUID MOTION. 57 

there being no sines in the expansions of P and Q, since the 
motion is symmetrical with respect to the major axis ; then 

j - ~ (Q,- J P.) cos n6 . . ..(28) ; 

(29); 



For a point in the ellipse, z cecos 20, whence from (27), (29) and 
(30), we find that the normal velocity of the fluid 

= 2 " Q n cos 7i0 + I n (n - 2) * - Q ;i cos (n - 2) 



which is the same thing as 
}[<- 2) *-L 

+ n (n + 2) -Q,, +2 cos0....(31), 



if we suppose P and Q to be zero when affected with a negative 
suffix. This expression will have to be equated to the value of 
C cos given by equation (26). 

For the part of the fluid without the radius 7 we have 

<=.^ logr + 2r cosn0* ............. (31 ), 

since there will be no sines in the expression for <, because the 
motion is symmetrical with respect to the major axis, and no 
positive powers of r, because the velocity vanishes at an infinite 
distance. 

From the above value of </> we have, for the points at a distance 
7 from the centre, 

* The first term of this expression is accurately equal to zero, since there is 
no expansion or contraction of the solid (Art. 8). I have however retained it, in 
order to render the solution of the problem in the present article independent of 
the proposition referred to. 



58 ON SOME CASES OF FLUID MOTION. 

d<f> A Q ^<*nA n 

-y- = ^ Zj -j&i cos rc#, 
dr 7 1 7" 

e_ S w4, - 

~~* S 



Equating the above expressions to the velocities along and per 
pendicular to the radius vector given by equations (29) and (30), 
when z is put = kc, and then equating coefficients of corresponding 
sines and cosines, we have 

(!-&)&+* S = _^ .............. (32), 

(33), 






when n > 0, and equating constant terms we have 



from which equation with (32) and (33) we have, putting 



T) A M A A 

^ = -t?, <3 = -^?- when>0, and , = =*. 

p C v t/ 

Substituting these values in the expression (31), it becomes 
2: (n + !)(- 2) -- + 1 ( + 1) ( + 2) g? cos 



p c 

In the case of a circular cylinder the quantities A , A, 2 , A a , &c. are 
each zero. In the present case therefore they are small quantities 
depending on e. Hence, neglecting quantities of the order e 2 
in the above expression, it becomes 



which must be equal to <7{(1 -e) cos ^ + e cos- 3^}. Equating 
coefficients of corresponding cosines, we have 



and the other quantities A , A a , &c. are of an order higher than e. 



ON SOME CASES OF FLUID MOTION. 59 

Hence, for the part of the fluid which lies without the radius 7, 
we have 



(34), 



and for the part which lies between that radius and the ellipse we 
have from (28) 
< = _ Cc {(I - e) cos + cos 30} + C {(I - e) cos + 3e cos 30} z 

- -cos 6z z .. ..(35). 
c v 

The value of <f> given by equation (So) may be deduced from 
that given by equation (3 4) by putting r c + z, and expanding as 
far as to 2 2 . In the case of the elliptic cylinder then it appears 
that the same value of <f> serves for the part of the fluid without, 
and the part within the radius jy. If the cylinder i be moving with 
a velocity C in the direction of the minor axis of a section, the 
value of (f> will be found from that given by equation (34) by 
changing the sign of e, putting C for C, and supposing 6 to be 
measured from the minor axis. 

If the cylinder revolve round its axis with an angular velocity 
&), the normal velocity of the surface at any point will be 2coec sin 20. 
Since e 2 is neglected, we may suppose this normal velocity to 
take place on the surface of a circular cylinder whose radius is c ; 
whence (Art. 8) the corresponding value of will be 

- ~ sin 20. 



If we suppose all these motions to take place together, we have 
only (Art. 5) to add together the values of < corresponding to 
each. If we suppose the motion very small, so as to neglect 
the square of the velocity, we need only retain the terms depend 
ing on dw/dt, dC/dt and dC /dt, in the value of d(f>/dt, and we 
may calculate the pressure due to each separately. The resultant 
of the pressure due to the term dco/dt will evidently be zero, on 
account of the symmetry of the corresponding motion, while the 
resultant couple will be of the order e 2 , since the pressure on 
any point of the surface, and the perpendicular from the centre on 
the normal at that point, are each of the order e. The pressure 
due to the term dC/dt will evidently have a resultant in the 
direction of the major axis of a section of the cylinder ; and it will 



60 ON SOME CASES OF FLUID MOTION. 

be easily proved that the resultant pressure on a length I of the 
cylinder is TrpcH (1 - 2e) dC/dt. That due to the term dC /dt will 
be 7rpc z l (1 -f 2e) dC /dt, acting along the minor axis. If the 
cylinder be constrained to oscillate so that its axis oscillates in a 
direction making an angle a with the major axis, and if C" be 
its velocity, which is supposed to be very small, the resultant 
pressures along the major and minor axes will be 

a<^ \ cL\j i ,., /- \ a(-j 

2e) cos a , and yu, (1 + 2e) sin V-TT 

respectively, where ft is the mass of the fluid displaced. Resolving 
these pressures in the direction of the motion, the resolved part 
will be p(I-2ecoa2y)dC"/dt 9 or p (1 - Je* cos 2a) dC"/dt, e 
being the eccentricity ; so that the effect of the inertia of the fluid 
will be, to increase the mass of the solid by a mass equal to 
//,(! Je 2 cos 2 a), which must be supposed to be collected at the 
axis. 

A similar method of calculation would apply to any given solid 
differing little either from a circular cylinder or from a sphere. 
In the latter case it would be necessary to use expansions in series 
of Laplace s coefficients, instead of expansions in series of sines 
and cosines. 

13. Motion of fluid in a closed box whose interior is of the form 
of a rectangular parallelepiped. 

The motion being supposed to begin from rest, the motion 
at any time may be supposed to have been produced by impact 
(Art. 4). The motion of the box at any instant f may be resolved 
into a motion of translation and three motions of rotation about 
three axes parallel to the edges, and passing through the centre 
of gravity of the fluid, and the part of </> due to each of these 
motions may be calculated separately. Considering any one 
of the motions of rotation, we shall see that the normal velocity 
of each face in consequence of it will ultimately be the same 
as if that face revolved round an axis passing through its centre, 
and that the latter motion would not alter the volume of the 
fluid. Consequently, in calculating the part of $ due to any one 
of the angular velocities, we may calculate separately the part 
due to the motion of each face. 

Let the origin be in a corner of the box, the axes coinciding 



ON SOME CASES OF FLUID MOTION. 61 

with its edges. Let a, b, c, be these edges, U, V, W, the velocities, 
parallel to the axes, of the centre of gravity of the interior of the 
box, w , CD", &) ", the angular velocities of the box about axes 
through this point parallel to those of #, y, z. Let us first con 
sider the part of < due to the motion of the face xz in conse 
quence of the angular velocity ft) ". 

The value of $ corresponding to this motion must satisfy the 
equation 



with the conditions 

y == 0> when x = Q or a ................. (.37), 

Cv JC 

g=0, when y = b ........................ (33), 

^ = o> "(a;-ia), wheny=0 ............ (39), 

within limits corresponding to those of the box. 

Now, for a given value of y t the value of (/> between x = and 
x = a can be expanded in a convergent series of cosines of irx/a 
and its multiples ; and, since (37) is satisfied, the series by which 
d(j>/dx will be expressed will also hold good for the limiting values 
of x, and will be convergent. The general value of </> then will be 
of the form 2" Y n cos mrx/a. Substituting in (36), and equating 
coefficients of corresponding cosines, which may be done, since any 
function of x can be expanded in but one such series of cosines 
between the limits and a, we find that the general value of 
Y n is Ce n *ul a + C e- n7r ^ a , or, changing the constants, 

Y n =A n (e n * J>-yV* + e -mr(b-y)l 

when n > 0, and for n = Q, 



From the condition (38) we have 

A + TroT 1 2"nB n (e 6 /* - e -V a ) cos mrx/a = : 
whence A Q = 0, B n 0, and, omitting B , 



62 ON SOME CASES OF FLUID MOTION. 

From the condition (39), we have 

TToT 1 ^nA n (e nirb/a e~ nirb/a ) cos nirxja = &/"(# 2 a ) 
Determining the coefficients in the usual manner, we have 



whence 

e -nir(b-y)/a, 



COS 



putting 2 , for shortness, to denote the sum corresponding to odd 
integral values of n from 1 to oo . 

It is evident that the value of corresponding to the motion of 
the opposite face in consequence of the angular velocity a/" will be 
found from that just given by putting b y for y, and changing 
the sign of a/"; whence the value corresponding to the motion 
of these two faces in consequence of &> " will be 



4o/V ^ - - , 

. _ y i - - - - - - =-. - - - cos nirx a. 

^ 72, 3 e mrb/a _ e -#irb/a 

Let this expression be denoted by &>" ^(#, a, y, b). It is 
evident that the part of $ due to the motion of the two faces 
parallel to the plane yz will be got by interchanging x and y, 
a and b, and changing the sign of " in the last expression, and 
will therefore be - w"^r (y, b, x, a). The parts of < corresponding 
to the angular velocities a/, a/ , will be got by interchanging the 
requisite quantities. Also the part of </> due to the velocities 
U, V, W, will be Ux + Vy + Wz (Art. 7), and therefore we have 
for the complete value of </> 



Ux + Vy + Wz + G) "{^(aj, a, y, 6) -^(y, b, x, a)} + a/ ty(y, b, z, c) 
- ^ (si, c, y, 6)) + a>" {^ (z, c, x,a)-ir (x, a, , c)). 



According to Art. 7 we may consider separately the motion of 
translation of the box and fluid, and the motion of rotation about 
the centre of gravity of the latter ; and the whole pressure will be 
compounded of the pressures due to each. The pressures at the 
several points of the box due to the motion of translation will have 
a single resultant, which will be the same as if the mass of the 
fluid were collected at its centre of gravity. Those due to the 



ON SOME CASES OF FLUID MOTION. 63 

motion of rotation will have a single resultant couple, to calculate 
which we have 

= to " [^ (x, a, y, b) - ^ (y, b, x, a)) + &c. 
Since for the motion of rotation there is no resultant force, 
we may find the resultant couple of the pressures round any 
origin, that for instance which has been chosen. If now we 
suppose the motion very small, so as to neglect the square of 
the velocity, we may find d(f)/dt as if the axes were fixed in space. 
We have then for the motion of rotation 

j rrt 

a >y> &)-^(y> > # a)}-&c. 



Hence we may calculate separately the couples due to each of 
the quantities da> "/dt, dco /dt and dco"/dt. It is evident from the 
symmetry of the motion that that due to dco "/dt will act round 
the axis of z, and that the pressures on the two faces perpendicular 
to that axis will have resultants which are equal and opposite. 
Also, since ^ (a, a, y, 6) = - ty (0, a, y,. b) and ^ (x, a, 6, &,) = - ^ 
(x, a, 0, 5), it will be seen that the couples due to the pressures 
on the faces perpendicular to the axes of x and y will be twice 
as great respectively as those due to the pressures on the planes 
yz and xz. The pressure on the element dydz of the plane yz will 
be p x==Q dydz, and the moment of this pressure round the axis of z, 
reckoned positive when it tends to turn the box from x to y, 
will be 

- P -" y W" (0, a, V, 6) - f (y, &, O, a)} dydz. 

Substituting the values of the functions, integrating from y to 
y = b, and from z = to z c, replacing 2 l/n 5 by its value 7r 4 /96, 
and reducing the other terms, it will be found that the couple 
due to the pressure on the plane yz is 
ptfbcda" _ S/oaWcTg 1 l- 
24 dt 7T 5 dt *l + 



_ 

7T 5 dt t l + 

We shall get the couple due to the pressure on the plane xz 
by interchanging a and b, changing the sign of to ", and measuring 
the couple in the opposite direction, or, which is the same, by 
merely interchanging a and b. Adding together these two couples 



64 ON SOME CASES OF FLUID MOTION. 

and doubling their sum we shall find that the couple due to 
do) "/dt is - Cda>"ldt, where 

~ ~6~ ^? ) ^ 1 J. f-nvb/a + 1 _i_ e -mra/l>\ 



a +j) ............ (40). 

Similarly, the couple due to dw -/dt will be J. dco /dt, tending 
to turn the box from y to z, and that due to dco"/dt will be 
Bda)"/dt, tending to turn the box from z to x, where A and B 
are derived from C by interchanging the requisite quantities. 
Hence, considering the motions both of translation and rotation of 
the box, we see that the small motions of the box will take place 
as if the fluid were replaced by a solid having the same mass, 
centre of gravity, and principal axes, and having A, B and G 
for its principal moments. This will be true whether forces act 
on the fluid or not, provided that if there are any they are of 
the kind mentioned in Art. 1. 

Patting A tt B f > C /t for the principal moments of inertia of the 
solidified fluid, we have 



Taking the ratio of C to (7,, replacing each term such as 

2 
] - 



its approximate value 1 "260497, and for 384/Tr 5 its approximate 
value 1-254821, and employing subsidiary angles, we have 



where tan e n = 

so that 

L tan O n = 10 - k nl/a, L tan 6 . n = 10 - k na/b, 

where ^ = 0821882. 

* [It will be shewn further on, in a supplement to this paper, that either of 
these two infinite series may be expressed by means of the other, so that we shall 
have only one of the infinite series to calculate in any case, for which we may 
choose the more rapidly convergent.] 



ON SOME CASES OF FLUID MOTION. G5 

The numerical calculation of this ratio is very easy, on account 
of the great rapidity with which the series contained in it con 
verge, both on account of the coefficients, and on account of the 
rapid diminution of the angles 6 n and n . The values of A/A, 
and B/B t will be derived from that of C/C, by putting c for a in 
the first case, and c for b in the second. The calculation of the 
small motions of the box will thus be reduced to a question of 
ordinary rigid dynamics*. 

When one of the quantities a, 6, becomes infinitely great com 
pared with the other, the ratio C/C t becomes 1, as will be seen 
from equation (40). This result might have been expected. When 
a = 6 the value of C/C, is -156537t- 

The experiment of the box appears capable of great variety 
as well as accuracy. We may take boxes in which the edges have 

* [Corresponding to the two simple cases of steady motion referred to in the 
foot-note to p. 7, are two in which the motion of the fluid within a box of simple 
form can be expressed in finite terms, the box and the fluid being initially at rest, 
and the box being then moved about its axis. 

The first is that in which the box is of the form of a right prism, having for 
its base an equilateral triangle. If as before a be the perpendicular from the 
centre of the triangle on one side, and 6 be measured from this perpendicular, 
we shall have 

0=-7^-r 3 sin 30; 
btt 

and by performing the integrations we shall find that if fc be the radius of gyration 
of what we may call the equivalent solid, that is, the solid, of the same mass as 
the fluid, by which the fluid may be replaced without affecting the motion of the 
box under given forces, 

fc 2 = fa 2 ; 

and as a is the radius of gyration for the fluid supposed solidified, the moment of 
inertia of the equivalent solid is two-fifths of that of the solidified fluid. 

The other is that of a box of the form of a right elliptic prism. In this case < 
is of the form cr 2 sin 20, 6 being measured from the major axis ; and determining c 
so as to suit an ellipse of which a and 6 are the semiaxes, we find 



k having the same meaning as before, it will be found that 



so that the ratio of the moment of inertia of the equivalent solid to that of the 
solidified fluid is that of (a 2 - b 2 ) 2 to (a 2 + b 2 ) 2 .] 

f [A passage containing a proposal to compare this result with experiment is 
here omitted, as the experiment is described, in the form in which it was actually 
carried out, in the supplement before referred to.] 

s. 5 



66 ON SOME CASES OF FLUID MOTION. 

various ratios to each other, and may make the same box oscillate 
in various positions. 

14. Initial motion in a rectangular box, the several points of 
the surface of which are moved with given velocities, consistent with 
the condition that the volume of the fluid is not altered. 

Employing the same notation as in the last case, let F (x, y} 
be the given normal velocity at any point of the face in the plane 

xy. Let I I F(x, y) dxdy = Wab, and let 
Jo Jo 

then, since the normal motion of the above face due to the function 
f(x,y) does not alter the volume of the fluid, we may consider 
separately the part of ^ due to this quantity. For this part we have 



, _ 

-z9 H r~2 n 7~a 
dx* d d*r 



with the conditions 



= 0, when x = or a ............... (42), 

dx 

^ =0, when y = or I ............... (43), 

dy 

^=0, whence ...................... (44), 

dz 

^ =/K 2/)> when ^ = ., ............ (45), 

within limits corresponding to those of the box. 

For a given value of z the value of $ from x = to x = a and 
from y = to y = b may be expanded in a series of the form 

. cosmry/b, 



the sign X referring to m, and S to n : and since the values of 
<, d<j>/dx and d$/dy do not alter abruptly, and equations (42) and 
(43) are satisfied, it follows that the series by which <, d(j>/d^ and 
dfyjdy are expressed are convergent, and hold good for the limiting 
values of x and y. Substituting the value of (/> just given in (41), 
equating to zero coefficients of corresponding cosines, and intro- 



ON SOME CASES OF FLUID MOTION. 67 

ducing the condition (44), we have, omitting the constant, or 
supposing A 0,0=0, 

(f) = 2 2 *-4 m- ^e pw( - c ~ z ^ c -{- e~ pir ^ c ~ z ^ c } cos mirxja . cos niry/b, 

9 n Q 

i p m n 

where % = + y^ . 

c a o 

Determining the coefficients such as A m>n from the condition 
(45) in the usual manner we have, m and n being > 0, 

P _ Q -PTT^ - 1 I I f^ y^ cos m7r jr/a . cos mry/b . c?^ (Zy , 
Jo Jo 



A = - 



Trpab 

2 

_. (atmro/o pmrc/o\ - 1 i 

\ / 

o Jo 



nTc/6_ e -mrc/6)-i f" f y^ ^) cosniry/b . dxdy*, 

Jo Jo 



with a similar expression for J. m , whence the value of <f> corre 
sponding to / (a?, y) is known. In a similar manner we may find 
the values corresponding to the similar functions belonging to 
each of the other faces. If W be the quantity corresponding to 
W for the face opposite to the plane xy, and U, U , correspond to 
W t W, for the faces perpendicular to the axis of x, and if V, V, 
be the corresponding quantities for y, there remains only to be 
found the part of </> due to these six quantities. Since U, U , are 
the velocities parallel to the axis of x of the faces perpendicular 
to that axis, and so for V, V, &c., the motion corresponding to 
these six quantities may be resolved into three motions of trans 
lation parallel to the three axes, the velocities being U, V and W, 
and that motion which is due to the motions of the faces opposite 
to the planes yz, zoo, ocy, moving with velocities U U, V V, 
W W, parallel to the axes of x, y, z, respectively. The condition 
that the volume of the fluid remains the same requires that 



It will be found that the velocities 



a^ b^ c v 

satisfy all the requisite conditions. Hence the part of < due to 

* The function f(x,y) in these integrals may be replaced by F(x,y), since 
P cos mry/b . cosmrx/a . dxdy~Q, unless m = ?j = 0. 

52 



68 ON SOME CASES OF FLUID MOTION. 

the six quantities U, U , V, V, W, W, is 



- . 

This quantity, added to the six others which have already been 
given, gives the value of </> which contains the complete solution 
of the problem. 

The case of motion which has just been given seems at first 
sight to be an imaginary one, capable of no practical application. 
It may however be applied to the determination of the small 
motion of a ball pendulum oscillating in a case in the form of 
a rectangular parallelepiped, the dimensions of the case being 
great compared with the radius of the ball. For this purpose it 
will be necessary to calculate the motion of the ball reflected from 
the case, by means of the formulae just given, and then the motion 
again reflected from the sphere, exactly as has been done iu the 
case of a rigid plane, Art. 10. In the present instance however 
the result contains definite integrals, the numerical calculation of 
which would be very troublesome. 



[From the Cambridge Mathematical Journal, Vol. iv. p. 28. (Nov. 1843).] 
ON THE MOTION OF A PlSTON AND OF THE AlR IN A CYLINDER. 

WHEN a piston is in motion in a cylinder which also contains 
air, if the motion of the piston be not very rapid, so that its 
velocity is inconsiderable compared with the velocity of pro 
pagation of sound, the motions of the air may be divided into 
two classes, the one consisting of those which depend directly on 
the motion of the piston, the other, of those which are propagated 
with the velocity of sound, and depend on the initial state of the 
air, or on a breach of continuity in the motion of the piston. 
If we suppose the initial velocity and condensation of the air in 
each section of the cylinder to be given, and also the initial 
velocity of the piston, both kinds of motion will in general take 
place, and the solution of the problem will be complicated. If, 
however, we restrict ourselves to motions of the first class, the 
approximate solution, though rather long, will be simple. In this 
case we must suppose the inital velocity and condensation of the 
air not to be given arbitrarily, but to be connected, according to 
a certain law which is yet to be found, with the motion of the 
piston. The problem as so simplified may perhaps be of some 
interest, as affording an example of the application of the partial 
differential equations of fluid motion, without requiring the em 
ployment of that kind of analysis which is necessary in most 
questions of that sort. It is, moreover, that motion of the air 
which it is proposed to consider, which principally affects the 
motion of the piston. 

Conceive an air-tight piston to move in a cylinder which is 
closed at one end, and contains a mass of air between the closed 
end and the piston. For more simplicity, suppose the rest of the 



70 ON THE MOTION OF A PISTON AND OF 

cylinder to contain no air. Let a point in the closed end be 
taken for origin, and let x be measured along the cylinder. Let 
% l be the abscissa of the piston ; a the initial value of x^ ; u the 
velocity parallel to x of any particle of air whose abscissa is x ; 
p the pressure, p the density about that particle; II the initial 
mean pressure ; p l the value of p when x = x^\ X, a function of x, 
the accelerating force acting on the air ; then for the motion of the 
air we have 

1 dp ^ r du du 1 
- - = X-j7 u-j- t 

p dx dt dx 



dp dpu _ I (1), 

~T7 "I T ^j 

dt dx 

and p kp, j 

neglecting the variation of temperature. 
We have also the conditions 

u = when x = Q (2); 

dx 

u= ~dl when x=x < (3)> 

for positive values of t, and 

ITa when ^ = (4). 



ra 

pdx = 
Jo 



o 
Eliminating p from equations (1), we have 

1 dp __! ( x _du du 

p dx~k\* dt U dx 

t + ^ = ........................... (6). 

dt dx 

Now, k being very large, for a first approximation let y be 
neglected ; then, integrating (5), 

j->0). 
Substituting in (6), and integrating, 



THE AIR IN A CYLINDER. 71 

The conditions (2) and (3) give 



C 

whence <f> (t) = 

x i 

Substituting in (4) the value ofp when = 0, we have 

dx = C Ha ; 



o ** 

a 



whence J 

Let now, for a second approximation, 

,. a % dec* 

p = U-+p U= -- 37 

v x l ^ #! dt 

so that y and u are small quantities of the order \jk ; then, sub 
stituting these values in (5) and (6), remembering that the quan 
tities which are not small must destroy each other, and retaining 
only small quantities of the first order, we have 



dp 

-* = 7 - 

dx kx 



/ v x d*x\ 

I A -- y, 2 I 

l \ ^ dr J 



dp 1 dx, dp^x a ^ =0 (8) 

~dt + x t dt dx x, dx 

and the conditions (2), (3) and (4) give 

w = when aj = 0, or x = x^ t and t is positive ...(9); 

( a pdx = when = ......... . ........ (10). 

Jo 

Integrating (7), we have 

2 , . n , v 

" ......... 



Substituting the values of p and of its differential coemcients 
in (8), and integrating, we obtain 

x 3 d d z x\ 1 dx. x -, x d, \ 



(12). 



72 ON THE MOTION OF A PISTON 

The conditions (9) give f (t) ; 

1 d ( d*x\ 1 dx^ (** , 1 d 

7TT TL #1 ~TT 1 - vs 7^ 2&W ==: j-r 

6& cfa V d^ / &i ft J o Ha dt 

and integrating, we get 

. tfx. Ila r*i f [ Xi 

-af-TJ. (Jo 

Putting/ for the initial value of tfxjdf we have, from (10) and 
(11), 



o o 

and substituting the value of &> (0) given by this equation in (13), 
after having made t = 0, x l = a, dfxJd? =/in the latter, we have 

TT C a C x 
C=-T dx Xdx. 

K Jo Jo 

Substituting this value of C in that of a (t), and substituting in 
(11) and (12), and then substituting the values of p and u in 
those of p and u t we have 



= n^+S?(f~ w -- 

c&j tf^j VJo 2#, dt 

d*x l 



O/C ttt KX^Ja \J / ^ x K^JQ \J 

(i*); 

^t 



5? dx v x_(- __ x?\ d_ f d*x\ 
x,~drt~Qk\ xfjdt T 1 df) 



Let A be the area of a section of the cylinder, and let TIAa/k=iJL, 
so that fjL is the mass of the air ; then we have 



[* It is best at once to get rid of the double integrals by integration by parts, 
^Yhich simplifies the expression, converting the last two terms into 



AND OF THE AIR IN A CYLINDER. 73 

If there were no motion, the term J//, d*xjdt* would disappear. 
But in that case the value of p^^ the pressure on the piston, 
might be deduced immediately from the equation of equilibrium 
of an elastic fluid 

1 dp^X 

p dx~~ k 

Integrating this equation, determining the constant by the con- 

r*i 

dition that I pdx=Tla, multiplying by -4, and putting x = x v 

we have, neglecting 1/& 2 , 



A = 



- 4 r ( r 

&i Jo VJo 



Comparing this expression with the above, when the second term 
of the latter is left out, we have 

Xdx, 



f^i / f#i \ J 1 ra rx l rx L rx 

( J&teJ =? +=/ dx\ Xdx = \ dx\ 

J a \JO J i UJO JO ^i^O JO 

a formula which may also be proved directly. We have then 

. T-T . a IJL d 2 .^ d ( 1 [* , [* v , 
p t A = UA -- K -srH-Mj I I dx.\ Xdx 
x^ 3 dr dx t \xj^ I JQ 

The first term would be the value of the pressure on the piston 
if the air had no inertia and were acted on by no external forces ; 
the second term is that due to the inertia of the air; the last 
term is that due to the external forces, and in the case of gravity 
expresses the effect of the weight of the air. If M be the mass 
of the piston, P the accelerating force parallel to x acting on it, 
not including the pressure of the air, its equation of motion is 



d f 1 r* 1 7 [ Xl v 7 \ /i/>\ 
-- dx. Xdx). ..(16) 
x l \ajjJo Jo / 



dx l 

Hence the effect of the inertia of the air is to increase the mass 
of the piston by one third of that of the air, without increasing 
the moving force acting on it. If we could integrate equation (16) 
twice, we should determine the arbitrary constants by means of 
the initial values of ^ and dxjdt, and thus get ^ in terms of t : 
then, substituting in (14) and (15), we should obtain p and u as 
functions of x and t. 



74 MOTION OF A PISTON AND OF THE AIR IN A CYLINDER. 

If the cylinder be vertical and smooth and turned upwards, 
we have P X = g ; and if, moreover, the motion be very small, 
putting oc l a + y, and neglecting ?/ 2 , we have 



The term at the second side of this equation is by hypothesis 
small, and if we suppose the mean value of x to be taken for a, 

it is zero. On this supposition II^l = \M+ ^J g, and the time 



/ 

of a small oscillation will be 2?r y . - , which becomes, 

If +2 

since yu, 2 is neglected throughout, 2?r f 1 ^ . J \J - . 

The reader who wishes to see the complete solution of the 
problem, in the case where no forces act on the air, and the air 
and piston are at first at rest, may consult a paper of Lagrange s 
with additions made by Poisson in the Journal de VEcole Poly- 
technique. T. xin. (21 e Cah.) p. 187. 



[From the Transactions of the Cambridge Philosophical Society, 
Vol. VIIL p. 287.] 

ON THE THEORIES OF THE INTERNAL FRICTION OF FLUIDS 

IN MOTION, AND OF THE EQUILIBRIUM AND MOTION OF 

ELASTIC SOLIDS. 

[Eead April 14, 1845.] 

THE equations of Fluid Motion commonly employed depend 
upon the fundamental hypothesis that the mutual action of two 
adjacent elements of the fluid is normal to the surface which 
separates them. From this assumption the equality of pressure 
in all directions is easily deduced, and then the equations of 
motion are formed according to D Alembert s principle. This 
appears to me the most natural light in which to view the sub 
ject ; for the two principles of the absence of tangential action, 
and of the equality of pressure in all directions ought not to be 
assumed as independent hypotheses, as is sometimes done, inas 
much as the latter is a necessary consequence of the former*. 
The equations of motion so formed are very complicated, but yet 
they admit of solution in some instances, especially in the case 
of small oscillations. The results of the theory agree on the 
whole with observation, so far as the time of oscillation is con 
cerned. But there is a whole class of motions of which the 
common theory takes no cognizance whatever, namely, those 
which depend on the tangential action called into play by the 
sliding of one portion of a fluid along another, or of a fluid along 
the surface of a solid, or of a different fluid, that action in fact 
which performs the same part with fluids that friction does with 
solids. 

* This may be easily shewn by the consideration of a tetrahedron of the fluid, 
as in Art. 4. 



76 ON THE FRICTION OF FLUIDS IN MOTION, 

Thus, when a ball pendulum oscillates in an indefinitely ex 
tended fluid, the common theory gives the arc of oscillation 
constant. Observation however shews that it diminishes very 
rapidly in the case of a liquid, and diminishes, but less rapidly, 
in the case of an elastic fluid. It has indeed been attempted to 
explain this diminution by supposing a friction to act on the ball, 
and this hypothesis may be approximately true, but the imper 
fection of the theory is shewn from the circumstance that no 
account is taken of the equal and opposite friction of the ball on 
the fluid. 

Again, suppose that water is flowing down a straight aqueduct 
of uniform slope, what will be the discharge corresponding to 
a given slope, and a given form of the bed ? Of what magnitude 
must an aqueduct be, in order to supply a given place with 
a given quantity of water ? Of what form must it be, in order 
to ensure a given supply of water with the least expense of 
materials in the construction ? These, and similar questions are 
wholly out of the reach of the common theory of Fluid Motion, 
since they entirely depend on the laws of the transmission of that 
tangential action which in it is wholly neglected. In fact, accord 
ing to the common theory the water ought to flow on with 
uniformly accelerated velocity ; for even the supposition of a 
certain friction against the bed would be of no avail, for such 
friction could not be transmitted through the mass. The practical 
importance of such questions as those above mentioned has made 
them the object of numerous experiments, from which empirical 
formulae have been constructed. But such formulas, although 
fulfilling well enough the purposes for which they were con 
structed, can hardly be considered as affording us any material 
insight into the laws of nature; nor will they enable us to pass 
from the consideration of the phenomena from which they were 
derived to that of others of a different class, although depending 
on the same causes. 

In reflecting on the principles according to which the motion 
of a fluid ought to be calculated when account is taken of the 
tangential force, and consequently the pressure not supposed the 
same in all directions, I was led to construct the theory explained 
in the first section of this paper, or at least the main part of it, 
which consists of equations (13), and of the principles on which 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 77 

they are formed. I afterwards found that Poisson had written 
a memoir on the same subject, and on referring to it I found that 
he had arrived at the same equations. The method which he em 
ployed was however so different from mine that I feel justified in 
laying the latter before this Society*. The leading principles of my 
theory will be found in the hypotheses of Art. 1, and in Art. 3. 

The second section forms a digression from the main object of 
this paper, and at first sight may appear to have little connexion 
with it. In this section I have, I think, succeeded in shewing 
that Lagrange s proof of an important theorem in the ordinary 
theory of Hydrodynamics is untenable. The theorem to which I 
refer is the one of which the object is to shew that udx+vdy+wdz, 
(using the common notation,) is always an exact differential when 
it is so at one instant. I have mentioned the principles of 
M. Cauchy s proof, a proof, I think, liable to no sort of objection. 
I have also given a new proof of the theorem, which would have 
served to establish it had M. Cauchy not been so fortunate as to 
obtain three first integrals of the general equations of motion. 
As it is, this proof may possibly be not altogether useless. 

Poisson, in the memoir to which I have referred, begins with 
establishing, according to his theory, the equations of equilibrium 
and motion of elastic solids, and makes the equations of motion 
of fluids depend on this theory. On reading his memoir, I was 
led to apply to the theory of elastic solids principles precisely 
analogous to those which I have employed in the case of fluids. 
The formation of the equations, according to these principles, 
forms the subject of Sect. III. 

The equations at which I have thus arrived contain two arbi 
trary constants, whereas Poisson s equations contain but one. In 
Sect. IV. I have explained the principles of Poisson s theories of 
elastic solids, and of the motion of fluids, and pointed out what 
appear to me serious objections against the truth of one of the 
hypotheses which he employs in the former. This theory seems 
to be very generally received, and in consequence it is usual to 
deduce the measure of the cubical compressibility of elastic solids 
from that of their extensibility, when formed into rods or wires, 

* The same equations have also been obtained by Navier in the case of an in 
compressible fluid (Mem. de V Academic, t. vi. p. 389), but his principles differ from 
mine still more than do Poisson s. 



78 ON THE FEICTION OF FLUIDS IN MOTION, 

or from some quantity of the same nature. If the views which 
I have explained in this section be correct, the cubical compres 
sibility deduced in this manner is too great, much too great in 
the case of the softer substances, and even the softer metals. 
The equations of Sect. III. have, I find, been already obtained by 
M. Cauchy in his Exercises Mathematiques, except that he has not 
considered the effect of the heat developed by sudden compression. 
The method which I have employed is different from his, although 
in some respects it much resembles it. 

The equations of motion of elastic solids given in Sect. in. 
are the same as those to which different authors have been led, 
as being the equations of motion of the luminiferous ether in 
vacuum. It may seem strange that the same equations should 
have been arrived at for cases so different ; and I believe this has 
appeared to some a serious objection to the employment of those 
equations in the case of light. I think the reflections which 
I have made at the end of Sect. IV., where I have examined the 
consequences of the law of continuity, a law which seems to per 
vade nature, may tend to remove the difficulty. 

SECTION I. 

Explanation of the Theory of Fluid Motion proposed. Formation 
of the Differential Equations, Application of these Equations 
to a few simple cases. 
1. Before entering on the explanation of this theory, it will 

be necessary to define, or fix the precise meaning of a few terms 

which I shall have occasion to employ. 

In the first place, the expression " the velocity of a fluid at 
any particular point" will require some notice. If we suppose 
a fluid to be made up of ultimate molecules, it is easy to see that 
these molecules must, in general, move among one another in an 
irregular manner, through spaces comparable with the distances 
between them, when the fluid is in motion. But since there is 
no doubt that the distance between two adjacent molecules is 
quite insensible, we may neglect the irregular part of the velocity, 
compared with the common velocity with which all the molecules 
in the neighbourhood of the one considered are moving. Or, we 
may consider the mean velocity of the molecules in the neigh 
bourhood of the one considered, apart from the velocity due to 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 79 

the irregular motion. It is this regular velocity which I shall 
understand by the velocity of a fluid at any point, and I shall 
accordingly regard it as varying continuously with the co-ordinates 
of the point. 

Let P be any material point in the fluid, and consider the 
instantaneous motion of a very small element E of the fluid 
about P. This motion is compounded of a motion of translation, 
the same as that of P, and of the motion of the several points of 
E relatively to P. If we conceive a velocity equal and opposite 
to that of P impressed on the whole element, the remaining 
velocities form what I shall call the relative velocities of the points 
of the fluid about P ; and the motion expressed by these velocities 
is what I shall call the relative motion in the neighbourhood of P. 

It is an undoubted result of observation that the molecular 
forces, whether in solids, liquids, or gases, are forces of enormous 
intensity, but which are sensible at only insensible distances. 
Let E be a very small element of the fluid circumscribing E, and 
of a thickness greater than the distance to which the molecular 
forces are sensible. The forces acting on the element E are the 
external forces, and the pressures arising from the molecular 
action of E 1 . If the molecules of E were in positions in which 
they could remain at rest if E were acted on by no external force 
and the molecules of E f were held in their actual positions, they 
would be in what I shall call a state of relative equilibrium. Of 
course they may be far from being in a state of actual equilibrium. 
Thus, an element of fluid at the top of a wave may be sensibly 
in a state of relative equilibrium, although far removed from its 
position of equilibrium. Now, in consequence of the intensity of 
the molecular forces, the pressures arising from the molecular 
action on E will be very great compared with the external 
moving forces acting on E. Consequently the state of relative 
equilibrium, or of relative motion, of the molecules of E will not 
be sensibly affected by the external forces acting on E. But the 
pressures in different directions about the point P depend on that 
state of relative equilibrium or motion, and consequently will not 
be sensibly affected by the external moving forces acting on E. 
For the same reason they will not be sensibly affected by any 
motion of rotation common to all the points of E\ and it is 
a direct consequence of the second law of motion, that they will 



80 ON THE FRICTION OF FLUIDS IN MOTION, 

not be affected by any motion of translation common to the whole 
element. If the molecules of E were in a state of relative equi 
librium, the pressure would be equal in all directions about P, 
as in the case of fluids at rest. Hence I shall assume the follow 
ing principle : 

That the difference between the pressure on a plane in a given 
direction passing through any point P of a fluid in motion and the 
pressure which would exist in all directions about P if the fluid in 
its neighbourhood were in a state of relative equilibrium depends 
only on the relative motion of the fluid immediately about P ; and 
that the relative motion due to any motion of rotation may be elimi 
nated without affecting the differences of the pressures above men 
tioned. 

Let us see how far this principle will lead us when it is 
carried out. 

2. It will be necessary now to examine the nature of the 
most general instantaneous motion of an element of a fluid. 
The proposition in this article is however purely geometrical, and 
may be thus enunciated : " Supposing space, or any portion of 
space, to be filled with an infinite number of points which move 
in any continuous manner, retaining their identity, to examine 
the nature of the instantaneous motion of any elementary portion 
of these points." 

Let u, v, w be the resolved parts, parallel to the rectangular 
axes, Ox, Oy, Oz, of the velocity of the point P, whose co-ordinates 
at the instant considered are x, y, z. Then the relative velocities 
at the point P , whose co-ordinates are x + x, y + y , z + z, will be 



dw , dw , dw ^ ^ 

dx dy dz 

neglecting squares and products of SB , y, z. Let these velocities 
be compounded of those due to the angular velocities w , o>", to" 
about the axes of x, y, z, and of the velocities U, F, W parallel 



AND THE EQUILIBR\UM AND MOTION OF ELASTIC SOLIDS. 81 

to x, y, z. The linear velocities due to the angular velocities 
being w"z <*>" y , G> "X wz, wy f a>"x parallel to the axes of 
-r, ?/, z, we shall therefore have 

du , fdu , f \ , fdu 

dx \dy J \dz 

dv ,,\ . dv . fdv . 



- 

dy ) * dz 

Since &> , <w", &> " are arbitrary, let them be so assumed that 
dU _dV dV _dW dW_dU 
dy ~ dx ~dz ~ dy W ~ dz 7 
which gives 

dw dv\ ,, - du dw\ , fdv du 



du , 

=&* + 

= fdv u\ , + v / ...... (2). 

* \dx dy) dy * * \dz dy ) 

TT7 . (dw du\ , . fdw dv\ , dw , 

W= -, - + -- I a? + - -- +J-) y + j- * , 





The quantities &> , &>", w" are what I shall call the angular 
velocities of the fluid at the point considered. This is evidently 
an allowable definition, since, in the particular case in which the 
element considered moves as a solid might do, these quantities 
coincide with the angular velocities considered in rigid dynamics. 
A further reason for this definition will appear in Sect. III. 

Let us now investigate whether it is possible to determine x, 
y , z so that, considering only the relative velocities U, V, W, the 
line joining the points P, P shall have no angular motion. The 
conditions to be satisfied, in order that this may be the case, are 
evidently that the increments of the relative co-ordinates a? , y, z 
of the second point shall be ultimately proportional to those co 
ordinates. If e be the rate of extension of the line joining the two 
points considered, we shall therefore have 
Fx + liy 4- gz = ex , \ 

=ey A .............................. (3); 



s. 



82 ON THE FRICTION OF FLUIDS IN MOTION, 

where 

-^ du n dv jj. dio - dv dw 
F= dx G = dTy H = Tz V = dz + dj 

2g^ + ^,2h^ + f. 
dx dz dy ax 

If we eliminate from equations (3) the two ratios which exist 
between the three quantities x , y, z, we get the well known cubic 
equation 



which occurs in the investigation of the principal axes of a rigid 
body, and in various others. As in these investigations, it may be 
shewn that there are in general three directions, at right angles 
to each other, in which the point P may be situated so as to 
satisfy the required conditions. If two of the roots of (4) are 
equal, there is one such direction corresponding to the third root, 
and an infinite number of others situated in a plane perpendicular 
to the former; and if the three roots of (4) are equal, a line 
drawn in any direction will satisfy the required conditions. 

The three directions which have just been determined I shall 
call axes of extension. They will in general vary from one point 
to another, and from one instant of time to another. If we denote 
the three roots of (4) by e, e" } e ", and if we take new rectangular 
axes Ox t) Oy t , Oz f) parallel to the axes of extension, and denote 
by u tt U lt &c. the quantities referred to these axes corresponding 
to u, U, &c., equations (3) must be satisfied by ?//= 0, z, = 0, e = e, 
by <= 0, <= 0, e= e\ and by <= 0, y/= 0, e = c" ,, which requires 
that/ 7 = 0, g t 0, 7^ / = 0, and we have 

, & _du t _ _ dv, / _rr_^/ 

The values of U t , F, W /t which correspond to the residual 
motion after the elimination of the motion of rotation correspond 
ing to o> , &)" and &> ", are 

The angular velocity of which &> , &>", a/" are the components 
is independent of the arbitrary directions of the co-ordinate axes : 
the same is true of the directions of the axes of extension, and of 
the values of the roots of equation (4). This might be proved in 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 83 

various ways ; perhaps the following is the simplest. The condi 
tions by which co , ", ta>" are determined are those which express 
that the relative velocities U, V, W, which remain after eliminating 
a certain angular velocity, are such that Udx + Vdy + Wdz is 
ultimately an exact differential, that is to say when squares and 
products of x , ?/ and z are neglected. It appears moreover from 
the solution that there is only one way in which these conditions 
can be satisfied for a given system of co-ordinate axes. Let us 
take new rectangular axes, OK, Oy, Oz, and let U, V, W be the 
resolved parts along these axes of the velocities U, V, W, and 
x , y , z , the relative co-ordinates of P ; then 

U = Ucosicx + V cos xy -\- Wcos xz, 
dx = cosxxdx. + cosxydy + cosxzdz, &c. ; 

whence, taking account of the well known relations between the 
cosines involved in these equations, we easily find 

Udx + Vdy + Wdz = Urfx + Vdy +Wdz. 

It appears therefore that the relative velocities U, V, W, which 
remain after eliminating a certain angular velocity, are such that 
Ucx + Vdy + Wdz is ultimately an exact differential. Hence 
the values of U, V, W are the same as would have been obtained 
from equations (2) applied directly to the new axes, whence the 
truth of the proposition enunciated at the head of this paragraph 
is manifest. 

The motion corresponding to the velocities U tt V t , W t may be 
further decomposed into a motion of dilatation, positive or negative, 
which is alike in all directions, and two motions which I shall call 
motions of shifting, each of the latter being in two dimensions, and 
not affecting the density. For let S be the rate of linear extension 
corresponding to a uniform dilatation ; let <rx t cry/ be the velo 
cities parallel to^, y y , corresponding to a motion of shifting parallel 
to the plane x t y t , and let a f x lt a z t be the velocities parallel to 
x iy z tt corresponding to a similar motion of shifting parallel to the 
plane xz t . The velocities parallel to x lt y t , z t respectively corre 
sponding to the quantities 8, a and <r will be (8 + <r + a ) xj, (8 <r)y , 
(8 v)z , and equating these to 7, V , W t we shall get 

a = J(e + e" + e" / ) > ff = J(V + e " - 2e"), <r = J ( + e" - 2e"). 

Hence the most general instantaneous motion of an elementary 
portion of a fluid is compounded of a motion of translation, a 

02 



84- ON THE FRICTION OF FLUIDS IN MOTION, 

motion of rotation, a motion of uniform dilatation, and two motions 
of shifting of the kind just mentioned. 

3. Having determined the nature of the most general instan 
taneous motion of an element of a fluid, we are now prepared to 
consider the normal pressures and tangential forces called into 
play by the relative displacements of the particles. Let p be the 
pressure which would exist about the point P if the neighbouring 
molecules were in a state of relative equilibrium: let p+p, be 
the normal pressure, and t, the tangential action, both referred to 
a unit of surface, on a plane passing through P and having a given 
direction. By the hypotheses of Art. 1. the quantities p,, t t will 
be independent of the angular velocities &> , w" , " , depending 
only on the residual relative velocities U, V, W, or, which comes 
to the same, on e, e" and e", or on a, a and 8. Since this re 
sidual motion is symmetrical with respect to the axes of extension, 
it follows that if the plane considered is perpendicular to any one 
of these axes the tangential action on it is zero, since there is no 
reason why it should act in one direction rather than in the 
opposite ; for by the hypotheses of Art. 1 the change of density 
and temperature about the point P is to be neglected, the consti 
tution of the fluid being ultimately uniform about that point. 
Denoting then by p+p, p+p", p+p" the pressures on planes 
perpendicular to the axes of a? y , ?/ /5 #,, we must have 

p = <l>(e f ,e",e f "), p" = <j>(e", e", e \ p" = <b(e", e, e"), 
</>(> , e" y e" } denoting a function of e , e and e" which is sym 
metrical with respect to the two latter quantities. The question 
is now to determine, on whatever may seem the most probable 
hypothesis, the form of the function (p. 

Let t;s first take the simpler case in which there is no dilata 
tion, and only one motion of shifting, or in which e = e\ e" 0, 
and let us consider what would take place if the fluid consisted of 
smooth molecules acting on each other by actual contact. On 
this supposition, it is clear, considering the magnitude of the pres 
sures acting on the molecules compared with their masses, that 
they would be sensibly in a position of relative equilibrium, except 
when the equilibrium of any one of them became impossible from 
the displacement of the adjoining ones, in which case the molecule 
in question would start into a new position of equilibrium. This 
start would cause a corresponding displacement in the molecules 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 85 

immediately about the one which started, and this disturbance 
would be propagated immediately in all directions, the nature of 
the displacement however being different in different directions, 
and would soon become insensible. During the continuance of 
this disturbance, the pressure on a small plane drawn through the 
element considered would not be the same in all directions, nor 
normal to the plane: or, which comes to the same, we may sup 
pose a uniform normal pressure p to act, together with a normal 
pressure p ti , and a tangential force t ljt p n and t /t being forces of 
great intensity and short duration, that is being of the nature of 
impulsive forces. As the number of molecules comprised in the 
element considered has been supposed extremely great, we may 
take a time r so short that all summations with respect to such 
intervals of time may be replaced without sensible error by inte 
grations, and yet so long that a very great number of starts shall 
take place in it. Consequently we have only to consider the aver 
age effect of such starts, and moreover we may without sensible 
error replace the impulsive forces such as p n and t /f , which succeed 
one another with great rapidity, by continuous forces. For planes 
perpendicular to the axes of extension these continuous forces will 
be the normal pressures p , p", p". 

Let us now consider a motion of shifting differing from the 
former only in having e increased in the ratio of m to 1. Then, if 
we suppose each start completed before the starts which would be 
sensibly affected by it are begun, it is clear that the same series of 
starts will take place in the second case as in the first, but at 
intervals of time which are less in the ratio of 1 to m. Conse 
quently the continuous pressures by which the impulsive actions 
due to these starts may be replaced must be increased in the ratio 
of m to 1. Hence the pressures p t p" t p" must be proportional 
to e, or we must have 

p =Ce, p"=C e , p" =C"e . 

It is natural to suppose that these formulae hold good for nega 
tive as well as positive values of e. Assuming this to be true, let 
the sign of e be changed. This comes to interchanging x and y, 
and consequently p" must remain the same, and p and p" must 
be interchanged. We must therefore have G" 0, G C. Put 
ting then C= 2/j, we have 

p = - SIM , p" = fye, p" = 0. 



86 ON THE FRICTION OF FLUIDS IN MOTION, 

It has hitherto been supposed that the molecules of a fluid are 
in. actual contact. We have every reason to suppose that this is 
not the case. But precisely the same reasoning will apply if they 
are separated by intervals as great as we please compared with 
their magnitudes, provided only we suppose the force of restitution 
called into play by a small displacement of any one molecule to be 
very great. 

Let us now take the case of two motions of shifting which co 
exist, and let us suppose e = <r + a , e" = <r, e" a . Let the 
small time r be divided into 2n equal portions, and let us suppose 
that in the first interval a shifting motion corresponding to e = 2cr, 
e"= 2a takes place parallel to the plane x t y t , and that in the 
second interval a shifting motion corresponding to e = 2cr , e "= 2o- 
takes place parallel to the plane xz^ and so on alternately. On 
this supposition it is clear that if we suppose the time r/2?i to be 
extremely small, the continuous forces by which the effect of the 
starts may be replaced will be p = 2 //, (or + cr ), p"= 2/*o-, p "= 2fia. 
By supposing n indefinitely increased, we might make the motion 
considered approach as near as we please to that in which the two 
motions of shifting coexist ; but we are not at liberty to do so, for 
in order to apply the above reasoning we must suppose the time 
r/2n to be so large that the average effect of the starts which 
occur in it may be taken. Consequently it must be taken as an 
additional assumption, and not a matter of absolute demonstration, 
that the effects of the two motions of shifting are superimposed. 

Hence if 8 = 0, i.e. if e + e" + e" = 0, we shall have in general 
/ = -2^ , p"=-W, p" = -2f*e"! (5). 

It was by this hypothesis of starts that I first arrived at these 
equations, and the differential equations of motion which result 
from them. On reading Poisson s memoir however, to which I 
shall have occasion to refer in Section IV., I was led to reflect that 
however intense we may suppose the molecular forces to be, and 
however near we may suppose the molecules to be to their posi 
tions of relative equilibrium, we are not therefore at liberty to 
suppose them in those positions, and consequently not at liberty 
to suppose the pressure equal in all directions in the intervals of 
time between the starts. In fact, by supposing the molecular 
forces indefinitely increased, retaining the same ratios to each 
other, we may suppose the displacements of the molecules from 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 87 

their positions of relative equilibrium indefinitely diminished, but 
on the other hand the force of restitution called into action by a 
given displacement is indefinitely increased in the same proportion. 
But be these displacements what they may, we know that the 
forces of restitution make equilibrium with forces equal and oppo 
site to the effective forces ; and in calculating the effective forces 
we may neglect the above displacements, or suppose the molecules 
to move in the paths in which they would move if the shifting 
motion took place with indefinite slowness. Let us first consider 
a single motion of shifting, or one for which e" = e, e" 0, and 
let p t and t f denote the same quantities as before. If we now sup 
pose e increased in the ratio of m to 1, all the effective forces will 
be increased in that ratio, and consequently p t and t / will be in 
creased in the same ratio. We may deduce the values of p p", and 
p" just as before, and then pass by the same reasoning to the case 
of two motions of shifting which coexist, only that in this case the 
reasoning will be demonstrative, since we may suppose the time 
r/2n indefinitely diminished. If we suppose the state of things 
considered in this paragraph to exist along with the motions of 
starting already considered, it is easy to see that the expressions 
for p, p" and p " will still retain the same form. 

There remains yet to be considered the effect of the dilatation. 
Let us first suppose the dilatation to exist without any shifting : 
then it is easily seen that the relative motion of the fluid at the 
point considered is the same in all directions. Consequently the 
only effect which such a dilatation could have would be to intro 
duce a normal pressure p t , alike in all directions, in addition to 
that due to the action of the molecules supposed to be in a state 
of relative equilibrium. Now the pressure p t could only arise 
from the aggregate of the molecular actions called into play by 
the displacements of the molecules from their positions of relative 
equilibrium ; but since these displacements take place, on an 
average, indifferently in all directions, it follows that the actions 
of which p t is composed neutralize each other, so that p t = 0. The 
same conclusion might be drawn from the hypothesis of starts, 
supposing, as it is natural to do, that each start calls into action 
as much increase of pressure in some directions as diminution of 
pressure in others. 

If the motion of uniform dilatation coexists with two motions 



88 ON THE FKICTION OF FLUIDS IN MOTION, 

of shifting, I shall suppose, for the same reason as before, that the 
effects of these different motions are superimposed. Hence sub 
tracting S from each of the three quantities e, e" and e", and 
putting the remainders in the place of e, e" and e" in equations 
(5), we have 



p = fj<e + e~e, p = 

p" = $p(e +e"-2e ") ............ (G). 

If we had started with assuming </> (e, e" , e"} to be a linear func 
tion of e , e" and e" , avoiding all speculation as to the molecular 
constitution of a fluid, we should have had at onc&p =Ce+(J(e"+e"), 
since p is symmetrical with respect to e" and e" \ or, changing the 
constants, p f //, (e" + e" 2e) + K (e -f e" + e "}. The expressions 
for p" and p" would be obtained by interchanging the requisite 
quantities. Of course we may at once put K = if we assume 
that in the case of a uniform motion of dilatation the pressure at 
any instant depends only on the actual density and temperature at 
that instant, and not on the rate at which the former changes 
with the time. In most cases to which it would be interesting to 
apply the theory of the friction of fluids the density of the fluid is 
either constant, or may without sensible error be regarded as con 
stant, or else changes slowly with the time. In the first two cases 
the results would be the same, and in the third case nearly the 
same, whether K were equal to zero or not. Consequently, if 
theory and experiment should in such cases agree, the experiments 
must not be regarded as confirming that part of the theory which 
relates to supposing K to be equal to zero. 

4. It will be easy now to determine the oblique pressure, or 
resultant of the normal pressure and tangential action, on any 
plane. Let us first consider a plane drawn through the point P 
parallel to the plane yz. Let Ox, make with the axes of #, y, z 
angles whose cosines are I , m, n ; let I" , m", n" be the same for 
Oy,, and l " t m" , n" the same for Oz r Let P 1 be the pressure, 
and (xty), (xtz) the resolved parts, parallel to y, z respectively, of 
the tangential force on the plane considered, all referred to a unit 
of surface, (xty) being reckoned positive when the part of the 
fluid towards - x urges that towards + x in the positive direction 
of y, and similarly for (xtz). Consider the portion of the fluid 
comprised within a tetrahedron having its vertex in the point P, 
its base parallel to the plane yz, and its three sides parallel to the 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 89 

planes x t y lt yz ft zx t respectively. Let A be the area of the base, 
and therefore I A, I" A, I" A the areas of the faces perpendicular 
to the axes of x t , y t , z t . By D Alembert s principle, the pressures 
and tangential actions on the faces of this tetrahedron, the moving 
forces arising from the external attractions, not including the 
molecular forces, and forces equal and opposite to the effective 
moving forces will be in equilibrium, and therefore the sums of 
the resolved parts of these forces in the directions of x, y and z 
will each be zero. Suppose now the dimensions of the tetrahedron , 
indefinitely diminished, then the resolved parts of the external, 
and of the effective moving forces will vary ultimately as the 
cubes, and those of the pressures and tangential forces on the 
sides as the squares of homologous lines. Dividing therefore the 
three equations arising from equating to zero the resolved parts 
of the above forces by A, and taking the limit, we have 



the sign 2 denoting the sum obtained by taking the quantities 
corresponding to the three axes of extension in succession. Putting 
for p ,p" and p" their values given by (6), putting e +e"+e"=8S, 
and observing that 2 2 = 1, 2W = 0, 2ZV = 0, the above equa 
tions become 
1\ =p - ZfjL^re + 2/xS, (xty) = - 2^1 m e, (xtz) = - 2/*2/W. 

The method of determining the pressure on any plane from 
the pressures on three planes at right angles to each other, which 
has just been given, has already been employed by MM. Cauchy 
and Poisson. 

The most direct way of obtaining the values of 2?V &c. would 
be to express I , m and n in terms of e by any two of equations 
(3), in which x , y , z are proportional to I , m, n, together with 
the equation I 2 + m 2 + n" 2 = 1, and then to express the resulting 
symmetrical function of the roots of the cubic equation (4) in 
terms of the coefficients. But this method would be excessively 
laborious, and need not be resorted to. For after eliminating the 
angular motion of the element of fluid considered the remaining 
velocities are ex , e y, , e "z , parallel to the axes of x lt y t) z t . 
The sum of the resolved parts of these parallel to the axis of 
x is lex] + 1 e y + 1" e" z . Putting for x f , y , z t their values 
I x + my + n z &c., the above sum becomes 

x %1 V -f y i in e + z ^l n e ; 



90 



OX THE FRICTION OF FLUIDS IN MOTION, 



but this sum is the same thing as the velocity U in equation (2), 
and therefore we have 



du 



du 



du div\ 
dxj 



It may also be very easily proved directly that the value of 38, 
the rate of cubical dilatation, satisfies the equation 



dz 



(7). 



dx dy 

Let P 2 , (ytz), (ytr) be the quantities referring to the axis of y, 
and P 3 , (ztx), (zty) those referring to the axis of z, which corre 
spond to P l &c. referring to the axis of x. Then we see that 
( y tz) = (zty), (ztx) = (xtz), (xty) = (ytx). Denoting these three 
quantities by T 19 T 2 , T 3 , and making the requisite substitutions 
and interchanges, we have 



(8). 



It may also be useful to know the components, parallel to 
x, y, z t of the oblique pressure on a plane passing through the 
point P, and having a given direction. Let /, m, n be the cosines 
of the angles which a normal to the given plane makes with the 
axes of x, y, z ; let P, Q, R be the components, referred to a unit 
of surface, of the oblique pressure on this plane, P, Q, R being 
reckoned positive when the part of the fluid in which is situated 
the normal to which /, m and n refer is urged by the other part 
in the positive directions of x t y, z, when I, m and n are positive. 
Then considering as before a tetrahedron of which the base is 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 91 

parallel to the given plane, the vertex in the point P, and the 
sides parallel to the co-ordinate planes, we shall have 



(9). 



In the simple case of a sliding motion for which u 0, v =f(x), 
w = 0, the ojly forces, besides the pressure >, which act on planes 
parallel to the co-ordinate planes are the two tangential forces T 9 , 
the value of which in this case is ^ dvjdx. In this case it is 
easy to shew that the axes of extension are, one of them parallel 
to Oz, and the two others in a plane parallel to xy, and inclined 
at angles of 45 to Ox. We see also that it is necessary to suppose 
JJL to be positive, since otherwise the tendency of the forces would 
be to increase the relative motion of the parts of the fluid, and 
the equilibrium of the fluid would be unstable. 

5. Having found the pressures about the point F on planes 
parallel to the co-ordinate planes, it will be easy to form the 
equations of motion. Let X, Y, Z be the resolved parts, parallel 
to the axes, of the external force, not including the molecular 
force ; let p be the density, t the time. Consider an elementary 
parallelepiped of the fluid, formed by planes parallel to the co 
ordinate planes, and drawn through the point (x, y, z) and the 
point (x + Aa 1 , y + Ay, z + A#). The mass of this- element will be 
ultimately pA^AyAz, and the moving force parallel to x arising 
from the external forces will be ultimately pJTA^AyAz; the effec 
tive moving force parallel too; will be ultimately p Du/Dt. A^AyAz,, 
where D is used, as it will be in the rest of this paper, to denote 
differentiation in which the independent variables are t and thre& 
parameters of the particle considered, (such for instance as its- 
initial cordinates,) and not t, x, y, z. It is easy also to shew that 
the moving force acting on the element considered arising from* 
the oblique pressures on the faces is ultimately 

dP dl\ dT n 
c dy a 

acting in the negative direction. Hence we have by D Alembert v s* 
principle 

ID -X\ +* + * + %*=*, &c (10),. 



92 ON THE FRICTION OF FLUIDS IN MOTION, 

in which equations we must put for Du/Dt its value 
du du du, du 

-j- + U -j- + V -j- + W -j- , 

dt dx dy dz 

and similarly for Dojdt and Dw/dt. In considering the general 
equations of motion it will be needless to write down more than 
one, since the other two may be at once derived from it by inter 
changing the requisite quantities. The equations (10), the ordi 
nary equation of continuity, as it is called, 



i , dpu ^fjv ^^ " ,x /-. -. \ 

dt + ~da> + ~dy+~te = "I " 

which expresses the condition that there is no generation or 
destruction of mass in the interior of a fluid, the equation con 
necting p and p, or in the case of an incompressible fluid the 
equivalent equation Dp/Dt = 0, and the equation for the propa 
gation of heat, if we choose to take account of that propagation, 
are the only equations to be satisfied at every point of the interior 
of the fluid mass. 

As it is quite useless to consider cases of the utmost degree 
of generality, I shall suppose the fluid to be homogeneous, and of 
a uniform temperature throughout, except in so far as the 
temperature may be raised by sudden compression in the case of 
small vibrations. Hence in equations (10) //, may be supposed to 
be constant as far as regards the temperature ; for, in the case 
of small vibrations, the terms introduced by supposing it to vary 
with the temperature would involve the square of the velocity, 
which is supposed to be neglected. If we suppose /JL to be in 
dependent of the pressure also, and substitute in (10) the values 
of P l &c. given by (8), the former equations become 

tDu ^\ dp _ fd 2 u d z u 
P \I)t J dx \dx 2 dif 

__ dfdu dv dw\ 
3 dx \dx dy dz) 

Let us now consider in what cases it is allowable to suppose 
p to be independent of the pressure. It has been concluded by 
Dubuat, from his experiments on the motion of water in pipes 
and canals, that the total retardation of the velocity due to 
friction is not increased by increasing the pressure. The total 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 93 

retardation depends, partly on the friction of the water against 
the sides of the pipe or canal, and partly on the mutual friction, 
or tangential action, of the different portions of the water. Now 
if these two parts of the whole retardation were separately variable 
with p, it is very unlikely that they should when combined give 
a result independent of p. The amount of the internal friction 
of the water depends on the value of //.. I shall therefore suppose 
that for water, and by analogy for other incompressible fluids, 
fju is independent of the pressure. On this supposition, we have 
from equations (11) and (12) 

dp fd*u d 2 u d 2 u 



= 



du dv dw 

~7 1 7~ + > = 0. 

ax dy dz 



These equations are applicable to the determination of the motion 
of water in pipes and canals, to the calculation of the effect of 
friction on the motions of tides and waves, and such questions. 

If the motion is very small, so that we may neglect the square 
of the velocity, we may put Du/Dt = du/dt, &c. in equations (13). 
The equations thus simplified are applicable to the determination 
of the motion of a pendulum oscillating in water, or of that of 
a vessel filled with water and made to oscillate. They are also 
applicable to the determination of the motion of a pendulum 
oscillating in air, for in this case we may, with hardly any error, 
neglect the compressibility of the air. 

The case of the small vibrations by which sound is propagated 
in a fluid, whether a liquid or a gas, is another in which dp/dp 
may be neglected. For in the case of a liquid reasons have been 
shewn for supposing //, to be independent of p, and in the case 
of a gas we may neglect dp/dp, if we neglect the small change 
in the value of /*, arising from the small variation of pressure due 
to the forces X, Y, Z. 

6. Besides the equations which must hold good at any point 
in the interior of the mass, it will be necessary to form also the 
equations which must be satisfied at its boundaries. Let M be 
a point in the boundary of the fluid. Let a normal to the surface 
at M, drawn on the outside of the fluid, make with the axes 
angles whose cosines are I, m, n. Let P , Q , R be the components 



94 ON THE FRICTION OF FLUIDS IN MOTION, 

of the pressure of the fluid about M on the solid or fluid with 
which it is in contact, these quantities being reckoned positive 
when the fluid considered presses the solid or fluid outside it in 
the positive directions of #, y, 2, supposing 7, m and n positive. 
Let S be a very small element of the surface about M, which 
will be ultimately plane, S a plane parallel and equal to S, and 
directly opposite to it, taken within the fluid. Let the distance 
between S and S be supposed to vanish in the limit compared 
with the breadth of S, a supposition which may be made if we 
neglect the effect of the curvature of the surface at M; and let 
us consider the forces acting on the element of fluid comprised 
between S and S , and the motion of this element. If we suppose 
equations (8) to hold good to within an insensible distance from 
the surface of the fluid, we shall evidently have forces ultimately 
equal to PS, QS, US, (P } Q and It being given by equations (9),) 
acting on the inner side of the element in the positive directions 
of the axes, and forces ultimately equal to P S, Q S, H S acting 
on the outer side in the negative directions. The moving forces 
arising from the external forces acting on the element, and the 
effective moving forces will vanish in the limit compared with the 
forces PS, &c. ; the same will be true of the pressures acting 
about the edge of the element, if we neglect capillary attraction, 
and all forces of the same nature. Hence, taking the limit, we 
shall have 



The method of proceeding will be different according as the 
bounding surface considered is a free surface, tb,e surface of a 
solid, or the surface of separation of two fluids, and it will be 
necessary to consider these cases separately. Of course the surface 
of a liquid exposed to the air is really the surface of separation 
of two fluids, but it may in many cases be regarded as a free 
surface if we neglect the inertia of the air : it may always be 
so regarded if we neglect the friction of the air as well as its 
inertia, 

Let us first take the case of a free surface exposed to a pres- 
siare II, which is supposed to be the same at all points, but may 
vary with the time ; and let L = be the equation to the surface- 
In this case we shall have P = IH, Q = rail, jR = wII; and 
putting for P, Q, R their values given by (9), and for P l &c. their 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 95 

values given by (8), and observing that in this case 8 = 0, we 
shall have 



dx \ay dx) \dz 

in which equations /, m, n will have to be replaced by dL/dx, 
dL/dy, dL/dz, to which they are proportional. 

If we choose to take account of capillary attraction, we have 
only to diminish the pressure n by the quantity Hi I ) , where 



H is a positive constant depending on the nature of the fluid, and 
r lt r 2 , are the principal radii of curvature at the point considered, 
reckoned positive when the fluid is concave outwards. Equations 
(14) with the ordinary equation 

dl. dL dL^ dL 

~ r + U-r-+V-,-+W-j- = ..................... (15), 

dt dx dy dz 

are the conditions to be satisfied for points at the free surface. 
Equations (14) are for such points what the three equations of 
motion are for internal points, and (15) is for the former what (11) 
is for the latter, expressing in fact that there is no generation or 
destruction of fluid at the free surface. 

The equations (14) admit of being differently expressed, in a 
way which may sometimes be useful. If we suppose the origin to 
be in the point considered, and the axis of z to be the external 
normal to the surface, we have I m = 0, n 1, and the equations 
become 

dw du ~ dw dv dw 

-T- +y- = 0, + = (), n-p+2/A-T- = ......... (16). 

dx dz dy dz dz 

The relative velocity parallel to z of a point (# , y , 0) in the 
free surface, indefinitely near the origin, is dw/dx . x -f dwjdy . y : 
hence we see that dw/dx, dw/dy are the angular velocities, reckoned 
from x to z and from y to z respectively, of an element of the free 
surface. Subtracting the linear velocities due to these angular 
velocities from the relative velocities of the point (x, ?/, z), and 
calling the remaining relative velocities U, V, W, we shall have 



96 ON THE FRICTION OF FLUIDS IN MOTION, 

j-r_du , du , /du div\ , 
dx dy \dz dx) 

dv , dv , . ido . dw 



T , v , v , o 
V= -j-x + -j-y + , 
dx c?v \ dz 



W = ~z. 

dz 

Hence we see that the first two of equations (16) express the con 
ditions that dU/dz = and dV/dz = Q, which are evidently the 
conditions to be satisfied in order that there may be no sliding 
motion in a direction parallel to the free surface. It would be 
easy to prove that these are the conditions to be satisfied in order 
that the axis of z may be an axis of extension. 

The next case to consider is that of a fluid in contact with a 
solid. The condition which first occurred to me to assume for 
this case was, that the film of fluid immediately in contact with 
the solid did not move relatively to the surface of the solid. I 
was led to try this condition from the following considerations. 
According to the hypotheses adopted, if there was a very large 
relative motion of the fluid particles immediately about any imagi 
nary surface dividing the fluid, the tangential forces called into 
action would be very large, so that the amount of relative motion 
would be rapidly diminished. Passing to the limit, we might sup 
pose that if at any instant the velocities altered discontinuously 
in passing across any imaginary surface, the tangential force called 
into action would immediately destroy the finite relative motion 
of particles indefinitely close to each other, so as to render the 
motion continuous; and from analogy the same might be supposed 
to be true for the surface of junction of a fluid and solid. But 
having calculated, according to the conditions which I have men 
tioned, the discharge of long straight circular pipes and rectangular 
canals, and compared the resulting formulae with some of the 
experiments of Bossut and Dubuat, I found that the formulae did 
not at all agree with experiment. I then tried Poisson s conditions 
in the case of a circular pipe, but with no better success. In fact, 
it appears from experiment that the tangential force varies nearly 
as the square of the velocity with which the fluid flows past the 
surface of a solid, at least when the velocity is not very small. It 
appears however from experiments on pendulums that the total 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 97 

friction varies as the first power of the velocity, and consequently 
we may suppose that Poisson s conditions, which include as a 
particular case those which I first tried, hold good for very small 
velocities. I proceed therefore to deduce these conditions in a 
manner conformable with the views explained in this paper. 

First, suppose the solid at rest, and let L = be the equation 
to its surface. Let M f be a point within the fluid, at an insensible 
distance h from M. Let to- be the pressure which would exist 
about M if there were no motion of the particles in its neighbour 
hood, and let p f be the additional normal pressure, and t t the tan 
gential force, due to the relative velocities of the particles, both 
with respect to one another and with respect to the surface of the 
solid. If the motion is so slow that the starts take place independ 
ently of each other, on the hypothesis of starts, or that the mole 
cules are very nearly in their positions of relative equilibrium, 
and if we suppose as before that the effects of different relative 
velocities are superimposed, it is easy to shew that p t and t t are 
linear functions of u, v, w and their differential coefficients with 
respect to as, y and z\ u, v, &c. denoting here the velocities of the 
fluid about the point M , in the expressions for which however the 
co-ordinates of M may be used for those of M , since h is neglected. 
Now the relative velocities about the points M and M depending 
on du/dx, &c. are comparable with du/dx . h, while those depending 
on u, v and w are comparable with these quantities, and therefore 
in considering the action of the fluid on the solid it is only neces 
sary to consider the quantities u, v and w. Now since, neglecting 
h, the velocity at M is tangential to the surface at M, u, v, and w 
are the components of a certain velocity V tangential to the sur 
face. The pressure p t must be zero ; for changing the signs u, v, 
and w the circumstances concerned in its production remain the 
same, whereas its analytical expression changes sign. The tangen 
tial force at M will be in the direction of V, and proportional to it, 
and consequently its components along the axes of x, y, z will be 
proportional to u, v, w. Reckoning the tangential force positive 
when, I, m, and n being positive, the solid is urged in the positive 
directions of x, y y z, the resolved parts of the tangential force will 
therefore be vu, w, vw, where v must evidently be positive, since 
the effect of the forces must be to check the relative motion of the 
fluid and solid. The normal pressure of the fluid on the solid 
being equal to r, its components will be evidently far, me?, ntr. 
s. 7 



98 ON THE FRICTION OF FLUIDS IN MOTION, 

Suppose now the solid to be in motion, and let u, v, w be the 
resolved parts of the velocity of the point M of the solid, and w , 
to", w " the angular velocities of the solid. By hypothesis, the 
forces by which the pressure at any point differs from the normal 
pressure due to the action of the molecules supposed to be in a 
state of relative equilibrium about that point are independent of 
any velocity of translation or rotation. Supposing then linear and 
angular velocities equal and opposite to those of the solid impressed 
both on the solid and on the fluid, the former will be for an 
instant at rest, and we have only to treat the resulting velocities 
of the fluid as in the first case. Hence P =l-G>+v(u u), &c.; 
and in the equations (8) we may employ the actual velocities u, 
v, w, since the pressures P, Q, R are independent of any motion 
of translation and rotation common to the whole fluid. Hence 
the equations F = P, &c. gives us 

l(sr p) + v(u u) 

du ^\ fdu dv\ fdu dw 

--8 ) + m (T-+-T- } + n(-J- + 
dx J \dy ctxj \dz 

which three equations with (15) are those which must be satisfied 
at the surface of a solid, together with the equation L = 0. It 
will be observed that in the case of a free surface the pressures 
P , Q , R are given, whereas in the case of the surface of a solid 
they are known only by the solution of the problem. But on the 
other hand the form of the surface of the solid is given, whereas 
the form of the free surface is known only by the solution of the 
problem. 

Dubuat found by experiment that when the mean velocity of 
water flowing through a pipe is less than about one inch in a 
second, the water near the inner surface of the pipe is at rest. 
If these experiments may be trusted, the conditions to be satisfied 
in the case of small velocities are those which first occurred to me, 
and which are included in those just given by supposing j/= oo . 

I have said that when the velocity is not very small the tan 
gential force called into action by the sliding of water over the 
inner surface of a pipe varies nearly as the square of the velocity. 
This fact appears to admit of a natural explanation. When a cur 
rent of water flows past an obstacle, it produces a resistance varying 
nearly as the square of the velocity. Now even if the inner surface 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 99 

of a pipe is polished we may suppose that little irregularities 
exist, forming so many obstacles to the current. Each little pro 
tuberance will experience a resistance varying nearly as the square 
of the velocity, from whence there will result a tangential action 
of the fluid on the surface of the pipe, which will vary nearly as 
the square of the velocity ; and the same will be true of the equal 
and opposite reaction of the pipe on the fluid. The tangential 
force due to this cause will be combined with that by which the 
fluid close to the pipe is kept at rest when the velocity is suf 
ficiently small*. 

[* Except in the case of capillary tubes, or, in case the tube be somewhat wider, 
of excessively slow motions, the main part of the resistance depends upon the 
formation of eddies. This much appears clear; but the precise way in which the 
eddies act is less evident. The explanation in the text gives probably the correct 
account of what takes place in the case of a river flowing over a rough stony bed; 
but in the case of a pipe of fairly smooth interior surface the minute protuberances 
would be too small to produce much resistance of the same kind as that con 
templated in the paragraph beginning near the foot of p. 53. 

What actually happens appears to be this. The rolling motion of the fluid 
belonging to the eddies is continually bringing the more swiftly moving fluid which 
is found nearer to the centre of the pipe close to the surface. And in consequence 
the gliding or shifting motion of the fluid in the immediate neighbourhood of the 
surface in such places is very greatly increased, and with it the tangential pressure. 

Thus while in some respects these two classes of resistances are similar, in 
others they are materially different. As typical examples of the two classes we 
may take, for the first, that of a polished sphere of glass of some size descending 
by its weight in deep water ; for the second, that of a very long circular glass pipe 
down which water is flowing. In both cases alike eddies are produced, and the 
eddies once produced ultimately die away. In both cases alike the internal friction 
of the fluid, and the friction between the fluid and the solid, are intimately 
connected with the formation of eddies, and it is by friction that the eddies die 
away, and the kinetic energy of the mass is converted into molecular kinetic 
energy, that is, heat. But in the first case the resistance depends mainly on the 
clilerence of the pressure p in front and rear, the resultant of the other forces of which 
the expressions are given in equations (8) being comparatively insignificant, while 
in the second case it is these latter pressures that we are concerned with, the 
resultant of the pressure p in the direction of the axis of the tube being practically 
nil, even though the polish of the surface be not mathematically perfect. 

Hence if, the motion being what it actually is, the fluidity of the fluid were 
suddenly to become perfect, the immediate effect on the resistance in the first case 
would be insignificant, while in the second case the resistance would practically 
vanish. Of course if the fluidity were to remain perfect, the motion after some 
time would be very different from what it had been before ; but that is not a point 
under consideration. 

Some questions connected with the effect of friction in altering the motion of 
a nearly perfect fluid will be considered further on in discussing the case of motion 
given in Art. 55 of a paper On the Critical Values of the Sums of Periodic Series. ] 

72 



100 ON THE FRICTION OF FLUIDS IN MOTION, 

There remains to be considered the case of two fluids having a 
common surface. Let u y v , w , /A , 8 denote the quantities belong 
ing to the second fluid corresponding to u, &c. belonging to the 
first. Together with the two equations .L = and (15) we shall 
have in this case the equation derived from (15) by putting u, v t w 
for u, v, w ; or, which comes to the same, we shall have the two 
former equations with 

l(u-u )+m(v-v ) + n(w-w ) =0 (18). 

If we consider the principles on which equations (17) were formed 
to be applicable to the present case, we shall have six more equa 
tions to be satisfied, namely (17), and the three equations derived 
from (17) by interchanging the quantities referring to the two 
fluids, and changing the signs I, m, n. These equations give the 
value of CT, and leave five equations of condition. If we must 
suppose v oo , as appears most probable, the six equations above 
mentioned must be replaced by the six u u> v = v, w = w, and 

lp pf(u, v, w} = lp f pf(u t v , w ), &c., 

f(u, v,w) denoting the coefficient of //- in the first of equations (17). 
We have here six equations of condition instead of five, but then 
the equation (18) becomes identical. 

7. The most interesting questions connected with this subject 
require for their solution a knowledge of the conditions which 
must be satisfied at the surface of a solid in contact with the fluid, 
which, except perhaps in case of very small motions, are unknown. 
It may be well however to give some applications of the preceding 
equations which are independent of these conditions. Let us then 
in the first place consider in what manner the transmission of sound 
in a fluid is affected by the tangential action. To take the simplest 
case, suppose that no forces act on the fluid, so that the pressure 
and density are constant in the state of equilibrium, and conceive 
a series of plane waves to be propagated in the direction of the 
axis of x, so that u =/(a?, <), v = 0, w = 0. Let p / be the pressure, 
and p, the density of the fluid when it is in equilibrium, and put 
p=p / +p. Then we have from equations (11) and (12), omitting 
the square of the disturbance, 

I dp du du dp 4 d*u_ 

+ =0 -+ 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 101 

Let A&p be the increment of pressure due to a very small incre 
ment Ap of density, the temperature being unaltered, and let m 
be the ratio of the specific heat of the fluid when the pressure is 
constant to its specific heat when the volume is constant; then 
the relation between p and p will be 

p = mA(p- P/ ) .............................. (20). 

Eliminating > and p from (19) and (20) we get 



-~ -0 
df da* S P/ dtdx z " 

To obtain a particular solution of this equation, let 



. x ,,,.. 

u = <j) (t) cos - -- 1- ^r (f) sin 



. 
A A. 

Substituting in the above equation, we see that $ (t) and i/r (t) 
must satisfy the same equation, namely, 

<t>" ^ + y 

the integral of which is 



* fn ^fa nt 27T&A 

= e ~ ct (C cos - - + C sm - - 

\ A A / 



A y 

where 



9XV/ 

(7 and being arbitrary constants. Taking the same expression 
with different arbitrary constants for ty(t), replacing products of 
sines and cosines by sums and differences, and combining the 
resulting sines and cosines two and two, we see that the resulting 
value of u represents two series of waves propagated in opposite 
directions. Considering only those waves which are propagated 
in the positive direction of x, we have 



V" *~J i ^ 2 I \^*-) 

We see then that the effect of the tangential force is to make 
the intensity of the sound diminish as the time increases, and to 
render the velocity of propagation less than what it would other 
wise be. Both effects are greater for high, than for low notes; 
but the former depends on the first power of p, while the latter 
depends only on //, 2 . It appears from the experiments of M. Biot, 
made on empty water pipes in Paris, that the velocity of propaga- 



102 ON THE FRICTION OF FLUIDS IN MOTION, 

tion of sound is sensibly the same whatever be its pitch. Hence 
it is necessary to suppose that for air fjf/\*p* is insensible com 
pared with A or pjp r I am not aware of any similar experiments 
made on water, but the ratio of (/a/Xp,) 2 to A would probably be 
insensible for water also. The diminution of intensity as the time 
increases is, in the case of plane waves, due entirely to friction ; 
but as we do not possess any means of measuring the intensity of 
sound the theory cannot be tested, nor the numerical value of fju 
determined, in this way. 

The velocity of sound in air, deduced from the note given by 
a known tube, is sensibly less than that determined by direct 
observation. Poisson thought that this might be due to the 
retardation of the air by friction against the sides of the tube. 
But from the above investigation it seems unlikely that the effect 
produced by that cause would be sensible. 

The equation (21) may be considered as expressing in all 
cases the effect of friction; for we may represent an arbitrary 
disturbance of the medium as the aggregate of series of plane 
waves propagated in all directions. 

8. Let us now consider the motion of a mass of uniform 
inelastic fluid comprised between two cylinders having a common 
axis, the cylinders revolving uniformly about their axis, and the 
fluid being supposed to have attained its permanent state of 
motion. Let the axis of the cylinders be taken for that of z, and 
let q be the actual velocity of any particle, so that u q sin 0, 
v = q cos 0, w = 0, r and 6 being polar co-ordinates in a plane 
parallel to xy. 

Observing that 



^" J __ J I _ %/ I 

daf^dtf dr 2 * r dr r* d& 
where/ is any function of x and y, and that dp/d& = 0, we have 
from equations (13), supposing after differentiation that the axis 
of x coincides with the radius vector of the point considered, and 
omitting the forces, and the part of the pressure due to them, 

dr ^ r 

d*q 1 dq q _ / 22 >, 

-S-H ~7 2 "* \"*if 

dr r dr r 
and the equation of continuity is satisfied identically. 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 103 

The integral of (22) is 

q=- + C r. 
r 

If a is the radius of the inner, and b that of the outer cylinder, 
and if q l , q 2 are the velocities of points close to these cylinders 
respectively, we must have q = q l when r = a, and q = q 2 when 
r = b } whence 

a , - a?,) r} (23). 



J. A *.&t M 

If the fluid is infinitely extended, b oo , and 



These cases of motion were considered by Newton (Principia, 
Lib. II. Prop. 51). The hypothesis which I have made agrees in 
this case with his, but he arrives at the result that the velocity 
is constant, not, that it varies inversely as the distance. This 
arises from his having taken, as the condition of their being no 
acceleration or retardation of the motion of an annulus, that the 
force tending to turn it in one direction must be equal to that 
tending to turn it in the opposite, whereas the true condition is 
that the moment of the force tending to turn it one way must 
be equal to the moment of the force tending to turn it the other. 
Of course, making this alteration, it is easy to arrive at the above 
result by Newton s reasoning. The error just mentioned vitiates 
the result of Prop. 52. It may be shewn from the general equa 
tions that in this case a permanent motion in annuli is impossible, 
and that, whatever may be the law of friction between the solid 
sphere and the fluid. Hence it appears that it is necessary to 
suppose that the particles move in planes passing through the 
axis of rotation, while they at the same time move round it. In 
fact, it is easy to see that from the excess of centrifugal force in 
the neighbourhood of the equator of the revolving sphere the 
particles in that part will recede from the sphere, and approach 
it again in the neighbourhood of the poles, and this circulating 
motion will be combined with a motion about the axis. If how 
ever we leave the centrifugal force out of consideration, as Newton 
has done, the motion in annuli becomes possible, but the solution 
is different from Newton s, as might have been expected. 



104 ON THE FRICTION OF FLUIDS IN MOTION, 

The case of motion considered in this article may perhaps 
admit of being compared with experiment, without knowing the 
conditions which must be satisfied at the surface of a solid. A 
hollow, and a solid cylinder might be so mounted as to admit of 
being turned with different uniform angular velocities round their 
common axis, which is supposed to be vertical. If both cylinders 
are turned, they ought to be turned in opposite directions, if only 
one, it ought to be the outer one ; for if the inner were made to 
revolve too fast, the fluid near it would have a tendency to fly 
outwards in consequence of the centrifugal force, and eddies would 
be produced. As long as the angular velocities are not great, so 
that the surface of the liquid is very nearly plane, it is not of much 
importance that the fluid is there terminated ; for the conditions 
which must be satisfied at a free surface are satisfied for any sec 
tion of the fluid made by a horizontal plane, so long as the motion 
about that section is supposed to be the same as it would be if the 
cylinders were infinite. The principal difficulty would probably be 
to measure accurately the time of revolution, and distance from the 
axis, of the different annuli. This would probably be best done by 
observing motes in the fluid. It might be possible also to discover 
in this way the conditions to be satisfied at the surface of the 
cylinders ; or at least a law might be suggested, which could be 
afterwards compared more accurately with experiment by means 
of the discharge of pipes and canals. 

If the rotations of the cylinders are in opposite directions, 
there will be a certain distance from the axis at which the fluid 
will not revolve at all. Writing - q l for q t in equation (23), we 

have for this distance 



9. Although the discharge of a liquid through a long straight 
pipe or canal, under given circumstances, cannot be calculated 
without knowing the conditions to be satisfied at the surface of 
contact of the fluid and solid, it may be well to go a certain way 
towards the solution. 

Let the axis of z be parallel to the generating lines of the 
pipe or canal, and inclined at an angle a to the horizon ; let the 
plane yz be vertical, and let y and z be measured downwards. 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 105 

The motion being uniform, we shall have u = 0, v = 0, w =f(, y), 
and we have from equations (13) 

dp _ dp dp fd^w d*w\ 

7- = 0, -f- - gp cos a, -f- = #0 sin a + p,( -j-^ + . 
d dz \dx 2 d* J 



7- , - , , 

dx dy dz \dx 2 dy* J 

In the case of a canal dp/dz = Q; and the calculation of the 
motion in a pipe may always be reduced to that of the motion 
in the same pipe when dpjdz is supposed to be zero, as may be 
shewn by reasoning similar to Dubuat s. Moreover the motion 
in a canal is a particular case of the motion in a pipe. For 
consider a pipe for which dp/dz = Q, and which is divided sym 
metrically by the plane xz. From the symmetry of the motion, 
it is clear that we must have dw/dy = when z = ; but this is 
precisely the condition which would have to be satisfied if the 
fluid had a free surface coinciding with the plane xz ; hence we 
may suppose the upper half of the fluid removed, without affect 
ing the motion of the rest, and thus we pass to the case of a canal. 
Hence it is the same thing to determine the motion in a canal, 
as to determine that in the pipe formed by completing the canal 
symmetrically with respect to the surface of the fluid. 

We have then, to determine the motion, the equation 
d*w d z w gp sin a _ 

w 1 "^ 4 ~T~ 

In the case of a rectangular pipe, it would not be difficult to 
express the value of w at any point in terms of its values at the 
several points of the perimeter of a section of the pipe. In the 
case of a cylindrical pipe the solution is extremely easy : for if 
we take the axis of the pipe for that of z, and take polar co 
ordinates r, 6 in a plane parallel to xy, and observe that dw/d6 = 0, 
since the motion is supposed to be symmetrical with respect to 
the axis, the above equation becomes 

d z w 1 dw gp sin a 

7 u H ~~7 | -- = 0. 

ar r ar /JL 

Let a be the radius of the pipe, and U the velocity of the fluid 
close to the surface ; then, integrating the above equation, and 
determining the arbitrary constants by the conditions that w shall 
be finite when r = 0, and w = U when r = a, we have 

_ 
v 



106 ON THE FRICTION OF FLUIDS IN MOTION, 



SECTION II. 

Objections to Lagrange s proof of the theorem that if udx+vdy+wdz 
is an exact differential at any one instant it is always so, the 
pressure being supposed equal in all directions. Principles of 
M. Cauchys proof. A new proof of the theorem. A physical 
interpretation of the circumstance of the above expression 
being an exact differential. 

10. The proof of this theorem given by Lagrange depends 
on the legitimacy of supposing u, v and w capable of expansion 
according to positive integral powers of t, for a sufficiently small 
finite value of t. It is clear that the expansion cannot contain 
negative powers of t, since u, v and w are supposed to be finite 
when t = ; but it may be objected to Lagrange s proof that there 
are functions of t of which the expansion contains fractional 
powers of t } and that we do not know but that u, v and w may 
be such functions. This objection has been considered by Mr 
Power*, who has shewn that the theorem is true if we suppose 
u, v and w capable of expansion according to any powers of t. 
Still the proof remains unsatisfactory, in fact inconclusive, for 
these are functions of t, (for instance, e~ l / t2 , t log,) which do not 
admit of expansion according to powers of t, integral or fractional, 
and we do not know but that u, v and w may be functions of this 
nature. I do not here mention the proof which Poisson has 
given of the theorem in his Traite de Mecanique, because it 
appears to me liable to an objection to which I shall presently 
have occasion to refer : in fact, Poisson himself did not think the 
theorem generally true. 

It is remarkable that Mr Power s proof, if it were legitimate, 
would establish the theorem even when account is taken of the 
variation of pressure in different directions, according to the 
theory explained in Section I., if we suppose that d^/dp = 0. To 
shew this we have only got to treat equations (12) as Mr Power 
has treated the three equations of fluid motion formed on the 
ordinary hypothesis. Yet in this case the theorem is evidently 
untrue. Thus, conceive a mass of fluid which is bounded by 
a solid plane coinciding with the plane yz, and which extends 

* Cambridge Philosophical Transactions, Vol. vn. (Part 3) p. 455. 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 107 

infinitely in every direction on the positive side of the axis of x, 
and suppose the fluid at first to be at rest. Suppose now the 
solid plane to be moved in any manner parallel to the axis of y\ 
then, unless the solid plane exerts no tangential force on the fluid, 
(and we may suppose that it does exert some,) it is clear that at 
a given time we shall have u = Q, v=f(x) t t0 = 0, and therefore 
udx + vdy + wdz will not be an exact differential. It will be 
interesting then to examine in this case the nature of the function 
of t which expresses the value of v. 

Supposing X, Y, Z to be zero in equations (12), and observing 
that in the case considered we have dp/dy 0, we get 

dv _n d^v . ,. 

dt~p dx*" ( * ] 

Differentiating this equation n 1 times with respect to t, we 
easily get 

d*v 



dt n \pj dx in 

but when t 0, v = when x > 0, and therefore for a given value 
of x all the differential coefficients of v with respect to t are zero. 
Hence for indefinitely small values of t the value of v at a given 
point increases more slowly than if it varied ultimately as any 
power of t, however great ; hence v cannot be expanded in a series 
according to powers of t. This result is independent of the con 
dition to be satisfied at the surface of the solid plane. 

I think what has just been proved shews clearly that La- 
grange s proof of the theorem considered, even with Mr Power s 
improvement of it, is inadmissible. 

11. The theorem is however true, and a proof of it has been 
given by M. Cauchy*, which appears to me perfectly free from 
objection, and which is very simple in principle, although it 
depends on rather long equations. M. Cauchy first eliminates p 
from the three equations of motion by means of the conditions 
that d*p/dxdy = d*p/dydx, &c., he then changes the independent 
variables from x, y, z, t to a, b, c, t, where a, b, c are the initial 

* Memoire sur la Theorie des Ondcs, in the first volume of the Memoires des 
savans Etrangers. M. Cauchy has not had occasion to enunciate the theorem, but 
it is contained in his equations (16). This equation may be obtained in the same 
manner in the more general case in which p is supposed to be a function of p. 



108 ON THE FRICTION OF FLUIDS IN MOTION, 

co-ordinates of the particles. The three transformed equations 
admit each of being once integrated with respect to t ; and deter 
mining the arbitrary functions of a, 6, c by the initial values of 
u, v and w, the three integrals have the form 



G>; = Fto + Ga>" + Ha> ", &c., 

a) , w" and co" f denoting here the same as in Art. 2, and &> , &c. 
denoting the initial values of to , &c. for the same particle. Solving 
the above equations with respect to &> , w" and o> ", the resulting 
equations are 

, 1 dx , dx ,, dx 



CO = TV 



where S is a function of the differential coefficients of x, y and z 
with respect to a, b and c, which by the condition of continuity is 
shewn to be equal to pjp, p being the initial density about the 
particle whose density at the time considered is p. Since dx/da, &c. 
are finite, (for to suppose them infinite would be equivalent to 
supposing a discontinuity to exist in the fluid,) it follows at once 
from the preceding equations that if &> = 0, &> " = 0, &> " = 0, that 
is if u da + V db + w Q dc be an exact differential, either for the whole 
fluid or for any portion of it, then shall &> = 0, &>" = 0, &> " = 0, i.e. 
udx + vdy + wdz will be an exact differential, at any subsequent 
time, either for the whole mass or for the above portion of it. 

12. It is not from seeing the smallest flaw in M. Cauchy s 
proof that I propose a new one, but because it is well to view the 
subject in different lights, and because the proof which I am about 
to give does not require such long equations. It will be necessary 
in the first place to prove the following lemma. 

LEMMA. If o^, ft> 2 ,...a> n are n functions of t, which satisfy the 
n differential equations 



[ (25), 

^=P. t +/,... + F..,J 

where P,, Q l . . . V n may he functions of <,<,.. .co,, and if when <,= 0, 
o) 2 = 0...w H = 0, none of the quantities P I; ... F. is infinite for any 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 109 

value of t from to T, and if o^...^ are each zero when = 0, 
then shall each of these quantities remain zero for all values of t 
from to T. 

DEMONSTRATION. Let T be a finite value of t, then by hypo 
thesis T may be taken so small that the values of co l ...a) n are suf 
ficiently small to exclude all values which might render any one 
of the quantities ^...F^ infinite. Let L be a superior limit to 
the numerical values of the several quantities P l ...V n for all 
values of t from to T; then it is evident that a) l ...a) n cannot 
increase faster than if they satisfied the equations 

dw, T , . i 

(26), 



at J 

vanishing in this case also when t = 0. But if co l -f- o> 2 . . . + &> ? 
we have by adding together the above equations 

czn 



if now fl be not equal to zero, dividing this equation by H and 
integrating, we have 



but no value of C different from zero will allow 1 to vanish 
when t = 0, whereas by hypothesis it does vanish ; hence H = ; 
but fl is the sum of n quantities which evidently cannot be nega 
tive, and therefore each of these must be zero. Since then co l ...(o n 
would have to be equal to zero for all values of t from to T even 
if they satisfied equations (26), they must d fortiori be equal to 
zero in the actual case, since they satisfy equations (25). Hence 
there is no value of t from to T at which any one of the 
quantities co l ...co n can begin to differ from zero, and therefore 
these quantities must remain equal to zero for all values of t 
from to T. 

This lemma might be extended to the case in which n = oo , 
with certain restrictions as to the convergency of the series. We 
may also, instead of the integers 1, 2...W, have a continuous 
variable a which varies from to a, so that o> is a function of 



110 ON THE FRICTION OF FLUIDS IN MOTION, 

the independent variables a and t, satisfying the differential 
equation 

day 



where ^r(a, 0, t) does not become infinite for any value of a from 
to a combined with any value of t from to T. It may be shewn, 
just as before, that if <w = when = for all values of a from 
to a, then must a> = for all values of t from to T. The proposi 
tion might be further extended to the case in which a = oo , with 
a certain restriction as to the convergency of the integral, but 
equations (25) are already more general than I shall have occa 
sion to employ. 

It appears to me to be sometimes assumed as a principle that 
two variables, functions of another, t, are proved to be equal for 
all values of t when it is shewn that they are equal for a certain 
value of t, and that whenever they are equal for the same value 
of t their increments for the same increment of t are ultimately 
equal. But according to this principle, if two curves could be 
shewn always to touch when they meet they must always coincide, 
a conclusion manifestly false. I confess I cannot see that Newton 
in his Principia, Lib. I., Prop. 40, has proved more than that if 
the velocities of the two bodies are equal at equal distances, the in 
crements of those velocities for equal increments of the distances 
are ultimately equal : at least something additional seems re 
quired to put the pjoof quite out of the reach of objection. 
Again it is usual to speak of the condition, that the motion of 
a particle of fluid in contact with the surface of a solid at rest 
is tangential to the surface, as the same thing as the condition 
that the particle shall always remain in contact with the surface. 
That it is the same thing might be shewn by means of the lemma 
in this article, supposing the motion continuous; but independ 
ently of proof I do not see why a particle should not move in 
a curve not coinciding with the surface, but touching it where 
it meets it. The same remark will apply to the condition that 
a particle which at one instant lies in a free surface, or is in 
contact with a solid, shall ultimately lie in the free surface, or be 
in contact with the solid, at the consecutive instant. I refer here 
to the more general case in which the solid is at rest or in motion. 
For similar reasons Poisson s proof of the Hydrodynamical theorem 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. Ill 

which forms the principal subject of this section has always ap 
peared to me unsatisfactory, in fact far less satisfactory than 
Lagrange s. I may add that Poisson s proof, as well as Lagrange s, 
would apply to the case in which friction is taken into account, 
in which case the theorem is not true. 

13. Supposing p to be a function of p, I// (p), the ordinary 
equations of Hydrodynamics are 



df( p) _ Y tftPl V ~ *tf_(P) 7 _ P w /.o7\ 
"~dx~ 1H dy Dt dz ~Dt 

The forces X, Y, Z will here be supposed to be such that 
Xdx + Ydy + Zdz is an exact differential, this being the case 
for any forces emanating from centres, and varying as any func 
tions of the distances. Differentiating the first equations (27) 
with respect to y, and the second with respect to x, subtracting, 
putting for Du/Dt and Dv/Dt their values, adding and subtracting 
du/dz . dv/dz, and employing the notation of Art. 2, we obtain 

Dw" _du , dv (du dv\ ,,, , , 

~DT~~dz K dz ~(dx + dy) a: ^ " 

By treating the first and third, and then the second and third of 
equations (27) in the same manner, we should obtain two more 
equations, which may be got at once from that which has just 
been found by interchanging the requisite quantities. Now for 
points in the interior of the mass the differential coefficients 
du/dz, &c. will not be infinite, on account of the continuity of the 
motion, and therefore the three equations just obtained are a 
particular case of equations (25). If then udx + vdy + wdz is an 
exact differential for any portion of the fluid when = 0, that is, 
if w , a)" and CD " are each zero when t = 0, it follows from the 
lemma of the last article that ft/, co" and co " will be zero for any 
value of t, and therefore udx + vdy + wdz will always remain an 
exact differential. It will be observed that it is for the same 
portion of fluid, not for the fluid occupying the same portion of 
space, that this is true, since equations (28), &c. contain the 
differential coefficients Dco /Dt, &c., and not dw /dt, &c. 

14. The circumstance of udx + vdy + wdz being an exact 
differential admits of a physical interpretation which may be 



112 ON THE FRICTION OF FLUIDS IN MOTION, 

noticed, as it is well to view a subject of this nature in different 
lights. 

Conceive an indefinitely small element of a fluid in motion 
to become suddenly solidified, and the fluid about it to be suddenly 
destroyed ; let the form of the element be so taken that the re 
sulting solid shall be that which is the simplest with respect to 
rotatory motion, namely, that which has its three principal 
moments about axes passing through the centre of gravity equal 
to each other, and therefore every axis passing through that point 
a principal axis, and let us enquire what will be the linear and 
angular motion of this element just after solidification. 

By the instantaneous solidification, velocities will be suddenly 
generated or destroyed in the different portions of the element, 
and a set of mutual impulsive forces will be called into action. 
Let x, y, z be the co-ordinates of the centre of gravity G of the 
element at the instant of solidification, x + x, y + y , z + z those 
of any other point in it. Let u t v, w be the velocities of G along 
the three axes just before solidification, u, v , w the relative velo 
cities of the point whose relative co-ordinates are x, y, z . Let 
u, v, w be the velocities of G, u t , v,, w, the relative velocities of the 
point above mentioned, and o> , o>", w " the angular velocities just 
after solidification. Since all the impulsive forces are internal, we 
have 

u = u, v v, w = w. 

We have also, by the principle of conservation of areas, 

%m {y (w / w ) z (v t v)} = 0, &c., 

m denoting an element of the mass of the element considered. 
But u t = o/Y (o "y , u is ultimately equal to 

du , du , du , 
-r~ x + -7- v -f- -=- z , 
dx dy y dz 

and similar expressions hold good for the other quantities. Sub 
stituting in the above equations, and observing that 

= 0, and ^mx* = %m 2 = 2m/ 2 , 

dw dv 

i 
dy 

We see then that an indefinitely small element of the fluid, 
of which the three principal moments about the centre of gravity 



. , - fdw dv\ 

we nave o> = i --- r > 

2 \d dzj 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 113 

are equal, if suddenly solidified and detached from the rest of the 
fluid will begin to move with a motion simply of translation, 
which may however vanish, or a motion of translation combined 
with one of rotation, according as udx + vdy 4- wdz is, or is not an 
exact differential, and in the latter case the angular velocities 
will be the same as in Art. 2. 

The principle which forms the subject of this section might 
be proved, at least in the case of a homogeneous incompressible 
fluid, by considering the change in the motion of a spherical 
element of the fluid in the indefinitely small time dt. This 
method of proving the principle would shew distinctly its inti 
mate connexion with the hypothesis of normal pressure, or the 
equivalent hypothesis of the equality of pressure in all directions, 
since the proof depends on the impossibility of an angular velo 
city being generated in the element in the indefinitely small 
time dt by the pressure of the surrounding fluid, inasmuch as the 
direction of the pressure at any point of the surface ultimately 
passes through the centre of the sphere. The proof I speak of 
is however less simple than the one already given, and would 
lead me too far from my subject. 

SECTION III. 

Application of a method analogous to that of Sect. I. to the 
determination of the equations of equilibrium and motion of 
elastic solids. 

15. All solid bodies are more or less elastic, as is shewn by 
the capability they possess of transmitting sound, and vibratory 
motions in general. The solids considered in this section are 
supposed to be homogeneous and uncrystallized, so that when in 
their natural state the average arrangement of their particles is 
the same at one point as at another, and the same in one direction 
as in another. The natural state will be taken to be that in which 
no forces act on them, from which it may be shewn that the pres 
sure in the interior is zero at all points and in all directions, 
neglecting the small pressure depending on attractions of the 
nature of capillary attraction. 

Let x, y, 2 be the co-ordinates of any point P in the solid con 
sidered when in its natural state, a, /3, 7 the increments of those 
S. 8 



114 ON THE FRICTION OF FLUIDS IN MOTION, 

co-ordinates at the time considered, whether the body be in a state 
of constrained equilibrium or of motion. It will be supposed that 
a, @ and 7 are so small that their squares and products may be 
neglected. All the theorems proved in Art. 2 with reference to 
linear and angular velocities will be true here with reference to 
linear and angular displacements, since these two sets of quantities 
are resolved according to the same laws, as long as the angular 
displacements are supposed to be very small. Thus, the most 
general displacement of a very small element of the solid consists 
of a displacement of translation, an angular displacement, and three 
displacements of extension in the direction of three rectangular 
axes, which may be called in this case, with more propriety than in 
the former, axes of extension. The three displacements of extension 
may be resolved into two displacements of shifting, each in two 
dimensions, and a displacement of uniform dilatation, positive or 
negative. The pressures about the element considered will depend 
on the displacements of extension only; there may also, in the 
case of motion, be a small part depending on the relative velocities, 
but this part may be neglected, unless we have occasion to consider 
the effect of the internal friction in causing the vibrations of solid 
bodies to subside. It has been shewn (Art. 7) that the effect of 
this cause is insensible in the case of sound propagated through 
air; and there is no reason to suppose it greater in the case of 
solids than in the case of fluids, but rather the contrary. The 
capability which solids possess of being put into a state of isochro 
nous vibration shews that the pressures called into action by small 
displacements depend on homogeneous functions of those displace 
ments of one dimension. I shall suppose moreover, according to 
the general principle of the superposition of small quantities, that 
the pressures due to different displacements are superimposed, and 
consequently that the pressures are linear functions of the dis 
placements. Since squares of or, (3 and 7 are neglected, these 
pressures may be referred to a Unit of surface in the natural state 
or after displacement indifferently, and a pressure which is normal 
to any surface after displacement may be regarded as normal to 
the original position of that surface. Let -AS be the pressure 
corresponding to a uniform linear dilatation & when the solid is in 
equilibrium, and suppose that it becomes - mAS, in consequence 
of the heat developed, when the solid is in a state of rapid vibra 
tion. Suppose also that a displacement of shifting parallel to 



AND THE EQUILIBKIUM AND MOTION OF ELASTIC SOLIDS. 115 

the plane xy, for which a. = kx, /3 = ky, 7 = 0, calls into action a 
pressure Bk on a plane perpendicular to the axis of x, and a 
pressure Bk on a plane perpendicular to that of y\ the pressures 
on these planes being equal and of opposite signs, that on a plane 
perpendicular to the axis of z being zero, and the tangential forces 
on those planes being zero, for the same reasons as in Sect. I. It 
may also be shewn as before that it is necessary to suppose B 
positive, in order that the equilibrium of the solid medium may 
be stable, and it is easy to see that the same must be the case 
with A for the same reason. 

It is clear that we shall obtain the expressions for the pressures 
from those already found for the case of a fluid by merely putting 
a, /3, 7, B for u, v, w, //-, and AS or mAB for p, according as we 
are considering the case of equilibrium or of vibratory motion, the 
body being in the latter case supposed to be constrained only in 
so far as depends on the motion. 

For the case of equilibrium then we have from equations (8) 



* -U i i /^ a djB dy\ , , ,. .... . 

o being here = -k ( -=- + -, \--r-\l and the equations ot equilibrium 
3 \dx dy dzj 

will be obtained from (12) by putting Du/Dt = 0, p = AS, making 
the same substitution as before for u, v, w and p. We have there 
fore, for the equations of equilibrium, 

Y if A m <L l drj - ^ d 1\ 

P iJ \ * ) ~l~ I 7 ~7 i T~ I 

ax \ax dy dzj 



In the case of a vibratory motion, when the body is in its 
natural state except so far as depends on the motion, we have 
from equations (8) 



and the equations of motion will be derived from (12) as before, 
only Du/Dt &c. must be replaced by d*x/d(? &c., and X, Y, Z put 
equal to zero. The equations of motion, then, are 

82 



116 ON THE FRICTION OF FLUIDS IN MOTION, 



d f d * , d & i < 

- - + + 



16. The conditions to be satisfied at the surface of the solid 
may be easily deduced from the analogous conditions in the case 
of a fluid with a free surface, only it will be necessary to replace 
the normal pressure II by an oblique pressure, of which the com 
ponents will be denoted by X lt Y I} Z^ We have then, making 
the necessary changes in the quantities involved in (14), 



X, + IAS + B + + + n + = 0, &c., 

{ dx \dy dx) \dz dxj) 

for the case of equilibrium, and for the case of motion such as that 
just considered it will only be necessary to replace A by mA in 
these equations. If we measure the angles of which I, m, n are 
the cosines from the external normal, the forces Jf 1? Y lt Z 1 must be 
reckoned positive when, Z, m and n being positive, the surface of 
the solid is urged in the negative directions of x, y, z, and in other 
cases the signs must be taken conformably. 

If the solid considered is in a state of constraint when at rest, 
and is moreover put into a state of vibration, the pressures and 
displacements due to these two causes must be calculated separately 
and added together. If m were equal to 1, they could be calcu 

lated together from the same equations. 

r f 

SECTION IV. 

Principles of Poissons theory of elastic solids, and of the oblique 
pressures existing in fluids in motion. Objections to one of his 
hypotheses. Reflections on the constitution, and equations of 
motion of the luminiferous ether in vacuum. 

17. In the twentieth Colder of the Journal de I Ecole Polytech- 
nique may be found a memoir by Poisson, entitled Memoire sur les 
Equations generates de VEquilibre et du Mouvement des Corps 
solides tlastiques et des Fluides, which contains the substance of 
two memoirs presented by him to the Academy, brought together 
with some additions. In this memoir the author treats principally 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 117 

of the equations of equilibrium and motion of elastic solids, of the 
equations of equilibrium of fluids, with reference especially to 
capillary attraction, and of the equations of motion of fluids, sup 
posing the pressure not to be equal in all directions, 

It is supposed by Poisson that all bodies, whether solid or 
fluid, are composed of ultimate molecules, separated from each 
other by vacant spaces. In the cases of an uncrystallized solid 
in its natural state, and of a fluid in equilibrium, he supposes 
that the molecules are arranged irregularly, and that the average 
arrangement is the same in all directions. These molecules he 
supposes to act on each other with forces, of which the main 
part is a force in the direction of the line joining the centres of 
gravity, and varying as some function of the distance of these 
points, and the remainder a secondary force, or it may be two 
secondary forces, depending on the molecules not being mathe 
matical points. He supposes that it is on these secondary forces 
that the solidity of solid bodies depends. He supposes however 
that in calculating the pressures these secondary forces may be 
neglected, partly because they become insensible at much smaller 
distances than the main part of the forces, and partly because they 
act, on the average, alike in all directions. He supposes that the 
molecular force decreases very rapidly as the distance increases, 
yet not so rapidly but that the sphere in which the molecular 
action is sensible contains an immense number of molecules. He 
supposes consequently that in estimating the resultant force of a 
hemisphere of the medium on a molecule in the centre of its base 
the action of the neighbouring molecules, which are situated 
irregularly, may be neglected compared with the action of those 
more remote, of which the average may be taken. The consequence 
of this supposition of course is that the total action is normal to 
the base of the hemisphere, and sensibly the same for one molecule 
as for an adjacent one. 

The rest of the reasoning by which Poisson establishes the 
equations of motion and equilibrium of elastic solids is purely 
mathematical, sufficient data having been already assumed. It 
might appear that the reasoning in Art. 16 of his memoir, by 
which the expression for N is simplified, required the fresh hypo 
thesis of a symmetrical arrangement of the molecules ; but it really 
does not, being admissible according to the principle of averages. 



118 ON THE FRICTION OF FLUIDS IN MOTION, 

Taking for the natural state of the body that in which the pressure 
is zero, the equations at which Poisson arrives contain only one 
unknown constant k, whereas the equations of Sect. in. of this 
paper contain two, A or mA and B. This difference depends on 
the assumption made by Poisson that the irregular part of the 
force exerted by a hemisphere of the medium on a molecule in the 
centre of its base may be neglected in comparison with the whole 
force. As a result of this hypothesis, Poisson finds that the change 
in direction, and the proportionate change in length, of a line 
joining two molecules are continuous functions of the co-ordinates 
of one of the molecules and the angles which determine the direc 
tion of the line ; whereas in Sect, ill., if we adopt the hypothesis 
of ultimate molecules at all, it is allowable to suppose that these 
quantities vary irregularly in passing from one pair of molecules 
to an adjacent pair. Of course the equations of Sect. ill. ought to 
reduce themselves to Poisson s equations for a particular relation 
between A and B. Neglecting the heat developed by compression, 
as Poisson has done, and therefore putting m = 1, this relation is 
4 = 55. 

18. Poisson s theory of fluid motion is as follows. The time 
t is supposed to be divided into a number n of equal parts, each 
equal to r. In the first of these the fluid is supposed to be dis 
placed as an elastic solid would be, according to Poisson s previous 
theory, and therefore the pressures are given by the same equa 
tions. If the causes producing the displacement were now to 
cease, the fluid would re-arrange itself, so that the average arrange 
ment about each point should be the same in all directions after 
a very short time. During this time, the pressures would have 
altered, in an unknown manner, from those corresponding to a 
displaced solid to a normal pressure equal to p + Dp/Dt . r, the 
pressures during the alteration involving an unknown function of 
the time elapsed since the end of the interval r. Another dis 
placement and another re-arrangement may now be supposed to 
to take place, and so on. But since these very small relative mo 
tions will take place independently of each other, we may suppose 
each displacement to begin at the expiration of the time during 
which the preceding one is supposed to remain, and we may sup 
pose each re-arrangement to be going on during the succeeding 
displacements. Supposing now n to become infinite, we pass to 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 119 

the case in which the fluid is supposed to be continually beginning 
to be displaced as a solid would, and continually re-arranging itself 
so as to make the average arrangement about each point the same 
in all directions. 

Poisson s equations (9), page 152, which are applicable to the 
motion of a liquid, or of an elastic fluid in which the change of 
density is small, agree with equations (12) of this paper. For the 
quantity -fyt is the pressure p which would exist at any instant if 

the motion were then to cease, and the increment, j r or -~ r, 

Cut JJt 

of this quantity in the very small time r will depend only on 
the increment, -?? T or - T, of the density yt or p. Consequently 

the value of J r will be the same as if the density of the par- 
ctt 



tide considered passed from %t to %t + - T in the time T by a 

uniform motion of dilatation. I suppose that according to Pois 
son s views such a motion would not require a re- arrangement of 
the molecules, since the pressure remains equal in all directions. 

On this supposition we shall get the value of -J- from that of 

ut 

R t Kin the equations of page 140 by putting 



du dv dw 1 dyt 

. . A/ 

doc dy dz 3%t dt 



We have therefore 



_ _ 
a ~dt ~3 (K blG - 



Putting now for ft + ft its value 2a&, and for J*- its value given 

% 
by equation (2), the expression for r, page 152, becomes 



j. a /JFJ. f^ J.-2E j. dw \ 

= ^ + o(^- + ^)-r~ + ^ r T~ / 
" 3 v \dx dy dz) 



Observing that a (K + Jc) = /9, this value of CT reduces Poisson s 
equations (9) to the equations (12) of this paper. 

Poisson himself has not made this reduction of his equations, 
nor any equivalent one, so that his equations, as he has left them, 



120 ON THE FRICTION OF FLUIDS IN MOTION, 

involve two arbitrary constants. The reduction of these two to 
one depends on the assumption that a uniform expansion of any 
particle does not require a re-arrangement of the molecules, as it 
leaves the pressure still equal in all directions. If we do not 
make this assumption, but retain the two arbitrary constants, the 
equations will be the same as those which would be obtained by 
the method of this paper, supposing the quantity K of Art. 3 not 
to be zero. 

19. There is one hypothesis made in the common theory of 
elastic solids, the truth of which appears to me very questionable. 
That hypothesis is the one to which I have already alluded in 
Art. 17, respecting the legitimacy of neglecting the irregular part 
of the action of the molecules in the immediate neighbourhood of 
the one considered, in comparison with the total action of those 
more remote, which is regular. It is from this hypothesis that it 
follows as a result that the molecules are not displaced among one 
another in an irregular manner, in consequence of the directive 
action of neighbouring molecules. Now it is obvious that the 
molecules of a fluid admit of being displaced among one another 
with great readiness. The molecules of solids, or of most solids 
at any rate, must admit of new arrangements, for most solids 
admit of being bent, permanently, without being broken. Are we 
then to suppose that when a solid is constrained it has no tendency 
to relieve itself from the state of constraint, in consequence of its 
molecules tending towards new relative positions, provided the 
amount of constraint be very small ? It appears to me to be much 
more natural to suppose a priori that there should be some such 
tendency. 

In the case of a uniform dilatation or contraction of a particle, 
a re-arrangement of its molecules would be of little or no avail 
towards relieving it from constraint, and therefore it is natural to 
suppose that in this case there is little or no tendency towards such 
a re-arrangement. It is quite otherwise, however, in the case of 
what I have called a displacement of shifting. Consequently B 
will be less than if there were no tendency to a re-arrangement. 
On the hypothesis mentioned in this article, of which the absence 
of such tendency is a consequence, I have, said that a relation has 
been found between A and B, namely A = oB. It is natural 
then to expect to find the ratio of A to B greater than 5, ap- 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 121 

preaching more nearly to 5 as the solid considered is more hard 
and brittle, but differing materially from 5 for the softer solids, 
especially such as India rubber, or, to take an extreme case, jelly. 
According to this view the relation A 5B belongs only to an 
ideal elastic solid, of which the solidity, or whatever we please to 
call the property considered, is absolutely perfect. 

To shew how implicitly the common theory of elasticity seems 
to be received by some, I may mention that MM. Lame* and 
Clapeyron mention Indian rubber among the substances to which 
it would seem they consider their theory applicable*. I do not 
know whether the coefficient of elasticity, according to that theory, 
has been determined experimentally for India rubber, but one 
would fancy that the cubical compressibility thence deduced, by a 
method which will be seen in the next article, would turn out com 
parable with that of a gas. 

20. I am not going to enter into the solution of equations (30), 
but I wish to make a few remarks on the results in some simple 
cases. 

If k be the cubical contraction due to a uniform pressure P, 
then will 



If a wire or rod, of which the boundary is any cylindrical sur 
face, be pulled in the direction of its length by a force of which 
the value, referred to a unit of surface of a section of the rod, in P, 
the rod will extend itself uniformly in the direction of its length, 
and contract uniformly in the perpendicular direction ; and if e 
be the extension in the direction of the length, and c the contraction 
in any perpendicular direction, both referred to a unit of length, 
we shall have 



~ SAB GAB 

p 

also, the cubical dilatation = e 2c = -r . 

If a cylindrical wire of radius r be twisted by a couple of which 

* Memoires dcs savam Etranyers, Tom. iv. p. 4G9. 



122 ON THE FRICTION OF FLUIDS IN MOTION, 

the moment is M y and if be the angle of torsion for a length z of 
the wire, we shall have 

ZM* 
- 



The expressions for k, c, e and 0, and of course all expressions 
of the same nature, depend on the reciprocals of A and B. Sup 
pose now the value of e, or 0, or any similar quantity not depending 
on A alone, be given as the result of observation. It will easily 
be conceived that we might find very nearly the same value for B 
whether we supposed A = 5B or A = nB, where n may be consider 
ably greater than 5, or even infinite. Consequently the observation 
of two such quantities, giving very nearly the same value of B, 
might be regarded as confirming the common equations. 

If we denote by E the coefficient of elasticity when A is 
supposed to be equal to 5B we have, neglecting the atmospheric 

pressure*, 

_2P fl _2M* 
~ ~ 



If now we denote by E l the value of E deduced from observation 
of the value of e, and by E 9 the value of E obtained by observing 
the value of 6, or else, which comes to the same, by observing the 
time of oscillation of a known body oscillating by torsion, we shall 
have 



If A be greater than oB, E l ought to be a little greater than E z . 
This appears to agree with observation. Thus the following num 
bers are given by M. Lame f E^ = 8000, E t = 7500 for iron; E^= 2510, 
E z 2250 for brass J. The difference between the values of E t and 
E 2 is attributed by M. Lame to the errors to which the obser 
vation of the small quantity e is liable. If the above numbers 

may be trusted, we shall have 

A 
A = 60000, B = 7500, -= = 8 for iron; 

A = 29724, B = 2250, ^ = 13 21 for brass. 

* Lam6, Cours de Physique, Tom. i. 
t Lamd, Cotirs de Physique, Tom. i. 
$ These numbers refer to the French units of length and weight. 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 123 

The cubical contraction k is almost too small to be made the 
subject of direct observation*, it is therefore usually deduced from 
the value of e y or from the coefficient of elasticity E found in some 
other way. On the supposition of a single coefficient E, we have 
Jc/e = f, but retaining the two, A and B, we have 

* U* *(-* 



e A + B \ A 

which will differ greatly from f if A/B be much greater than 5. 
The whole subject therefore requires, I think, a careful examina 
tion, before we can set down the values of the coefficients of cubical 
contraction of different substances in the list of well ascertained 
physical data. The result, which is generally admitted, that the 
ratio of the velocity of propagation of normal, to that of tangential 
vibrations in a solid is equal to \/3, is another which depends en 
tirely on the supposition that A = oB. The value of m, again, as 
deduced from observation, will depend upon the ratio of A to B ; 
and it would be highly desirable to have an accurate list of the 
values of m for different substances, in hopes of thereby discover 
ing in what manner the action of heat on those substances is 
related to the physical constants belonging to them, such as their 
densities, atomic weights, &c. 

The observations usually made on elastic solids are made on 
slender pieces, such as wires, rods, and thin plates. In such pieces, 
all the particles being at no great distance from the surface, it is easy 
to see that when any small portion is squeezed in one direction it 
has considerable liberty of expanding itself in a direction perpen 
dicular to this, and consequently the results must depend mainly 
on the value of B, being not very different from what they would 
be if A were infinite. This is not so much the case with thick, 
stout pieces. If therefore such pieces could be put into a state of 
isochronous vibration, so that the musical notes and nodal lines 
could be observed, they would probably be better adapted than 
slender pieces for determining the value of mA. The value of 

* I find however that direct experiments have been made by Prof. Oersted. 
According to these experiments the cubical compressibility of solids which would 
be obtained from Poisson s theory is in some cases as much as 20 or 30 times too 
great. See the Report of the British Association for 1833, p. 353, or Archives des 
decouvertes, &c. for 1834, p. 94. [It is to be noted that Oersted s method gives only 
differences of compressibility.] 



124 ON THE FRICTION OF FLUIDS IN MOTION, 

m might be determined by comparing the value of mA, deduced 
from the observation of vibrations, with the value of A, deduced 
from observations made in cases of equilibrium, or, perhaps, of very 
slow motion. 

21. The equations (32) are the same as those which have 
been obtained by different authors as the equations of motion of 
the luminiferous ether in vacuum. Assuming for the present 
that the equations of motion of this medium ought to be deter 
mined on the same principles as the equations of motion of an 
elastic solid, it will be necessary to consider whether the equations 
(32) are altered by introducing the consideration of a uniform 
pressure II existing in the medium when in equilibrium; for we 
have evidently no right to assume, either that no such pressure 
exists, or, supposing it to exist, that the medium would expand 
itself but very slightly if it were removed. It will now no longer 
be allowable to confound the pressure referred to a unit of surface 
as it was, in the position of equilibrium of the medium, with 
the pressure referred to a unit of surface as it actually is. The 
latter mode of referring the pressure is more natural, and will 
be more convenient. Let the pressure, referred to a unit of 
surface at it is, be resolved into a normal pressure H+p l and a 
tangential pressure ^. All the reasoning of Sect. in. will apply 
to the small forces p^ and t l ; only it must be remembered that 
in estimating the whole oblique pressure a normal pressure II 
must be compounded with the pressures given by equations (31). 
In forming the equations of motion, the pressure II will not 
appear, because the resultant force due to it acting on the element 
of the medium which is considered is zero. The equations (32) 
will therefore be the equations of motion required. 

If we had chosen to refer the pressure to a unit of surface in 
the original state of the surface, and had resolved the whole 
pressure into a pressure II + p l normal to the original position 
of the surface, and a pressure ^ tangential to that position, the 
reasoning of Sec. III. would still have applied, and we should 
have obtained the same expressions as in (31) for the pressures 
P lt T l) &c., but the numerical value of A would have been 
different. According to this method, the pressure II would have 
appeared in the equations of motion. It is when the pressures 
are measured according to the method which I have adopted that 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 125 

it is true that the equilibrium of the medium would be unstable 
if either A or B were negative. I must here mention that from 
some oversight the right-hand sides of Poisson s equations, at 
page 68 of the memoir to which I have referred, are wrong. The 

*. A .. . . Tl(d*u d*u 

first ot these equations ought to contain I -=- 2 -f -j- ^ 



p \dx z dy* dz*. 

instead of =-5 , and similar changes must be made in the other 

p dx 

two equations. 

It is sometimes brought as an objection to the equations of 
motion of the luminiferous ether, that they are the same as those 
employed for the motion of solid bodies, and that it seems un 
natural to employ the same equations for substances which must 
be so differently constituted. It was, perhaps, in consequence 
of this objection that Poisson proposes, at page 147 of the memoir 
which I have cited, to apply to the calculation of the motion of 
the lurniferous ether the same principles, with a certain modifica 
tion, as those which he employed in arriving at his equations (9) 
page 152, i.e. the equations (12) of this paper. That modification 
consists in supposing that a certain function of the time < (t) does 
not vary very rapidly compared with the variation of the pressure. 
Now the law of the transmission of a motion transversal to the 
direction of propagation depending on equations (12) of this paper 
is expressed, in the simplest case, by the equation (24) ; and we 
see that this law is the same as that of the transmission of heat, 
a law extremely different from that of the transmission of vi 
bratory motions. It seems therefore unlikely that these principles 
are applicable to the calculation of the motion of light, unless 
the modification which I have mentioned be so great as wholly 
to alter the character of the motion, that is, unless we suppose the 
pressure to vary extremely fast compared with the function cj> (t), 
whereas in ordinary cases of the motion of fluids the function <f> (t) 
is supposed to vary extremely fast compared with the pressure. 

Another view of the subject may be taken which I think 
deserves notice. Before explaining this view however it will be 
necessary to define what I mean in this paragraph by the word 
elasticity. There are two distinct kinds of elasticity ; one, that 
by which a body which is uniformly compressed tends to 
regain its original volume, the other, that by which a body 
which is constrained in a manner independent of compres- 



126 ON THE FRICTION OF FLUIDS IN MOTION, 

sion tends to assume its original form. The constants A and 
B of Sec. Hi. may be taken as measures of these two kinds 
of elasticity. In the present paragraph, the word will be used 
to denote the second kind. Now many highly elastic substances, 
as iron, copper, &c., are yet to a very sensible degree plastic. The 
plasticity of lead is greater than that of iron or copper, and, as 
appears from experiment, its elasticity less. On the whole it 
is probable that the greater the plasticity of a substance the less 
its elasticity, and vice versa, although this rule is probably far 
from being without exception. When the plasticity of the sub 
stance is still further increased, and its elasticity diminished, 
it passes into a viscous fluid. There seems no line of demarcation 
between a solid and a viscous fluid. In fact, the practical dis 
tinction between these two classes of bodies seems to depend on 
the intensity of the extraneous force of gravity, compared with 
the intensity of the forces by which the parts of the substance 
are held together. Thus, what on the Earth is a soft solid might, 
if carried to the Sun, and retained at the same temperature, be 
a viscous fluid, the force of gravity at the surface of the Sun 
being sufficient to make the substance spread out and become 
level at the top : while what on the Earth is a viscous fluid might 
on the surface of Pallas be a soft solid. The gradation of viscous, 
into what are called perfect fluids seems to present as little ab 
ruptness as that of solids into viscous fluids; and some experiments 
which have been made on the sudden conversion of water and 
ether into vapour, when enclosed in strong vessels and exposed 
to high temperatures, go towards breaking down the distinction 
between liquids and gases. 

According to the law of continuity, then, we should expect 
the property of elasticity to run through the whole series, only, 
it may become insensible, or else may be masked by some other 
more conspicuous property. It must be remembered that the 
elasticity here spoken of is that which consists in the tangential 
force called into action by a displacement of continuous sliding : 
the displacements also which will be spoken of in this paragraph 
must be understood of such displacements as are independent 
of condensation or rarefaction. Now the distinguishing property 
of fluids is the extreme mobility of their parts. According to 
the views explained in this article, this mobility is merely an 
extremely great plasticity, so that a fluid admits of a finite, but 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 127 

exceedingly small amount of constraint before it will be relieved 
from its state of tension by its molecules assuming new positions 
of equilibrium. Consequently the same oblique pressures can 
be called into action in a fluid as in a solid, provided the amount 
of relative displacement of the parts be exceedingly small. All 
we know for certain is that the effect of elasticity in fluids, 
(elasticity of the second kind be it remembered,) is quite insensible 
in cases of equilibrium, and it is probably insensible in all ordinary 
cases of fluid motion. Should it be otherwise, equations (8) and 
(12) will not be true, or only approximately true. But a little 
consideration will shew that the property of elasticity may be 
quite insensible in ordinary cases of fluid motion, and may yet 
be that on which the phenomena of light entirely depend. When 
we find a vibrating string, the small extent of vibration of which 
can be actually seen, filling a whole room with sound, and re 
member how rapidly the intensity of the vibrations of the air 
must diminish as the distance from the string increases*, we may 
easily conceive how small in general must be the amount of the 
relative motion of adjacent particles of air in the case of sound. 
Now the extent of the vibration of the ether, in the case of light, 
may be as small compared with the length of a wave of light 
as that of the air is compared with the length of a wave of sound : 
we have no reason to suppose it otherwise. When we remember 
then that the length of a wave of sound in air varies from some 
inches to several feet, while the greatest length of a wave of 
light is about 00003 of an inch, it is easy to imagine that the 
relative displacement of the particles of ether may be so small 
as not to reach, nor even come near to the greatest relative dis 
placement which could exist without the molecules of the medium 
assuming new positions of equilibrium, or, to keep clear of the 
idea of molecules, without the medium assuming a new arrange 
ment which might be permanent. 

It has been supposed by some that air, like the luminiferous 
ether, ought to admit of transversal vibrations. According to 
the views of this article such would, mathematically speaking, 
be the case ; but the extent of such vibrations would be necessarily 
so very small as to render them utterly insensible, unless we had 

* [In all ordinary cases it is to the vibrations of the sounding-board, or of 
the supporting body acting as a sounding-board, and not to those of the string 
directly, that the sound is almost wholly due.] 



128 ON THE FRICTION OF FLUIDS IN MOTION, 

organs with a delicacy equal to that of the retina adapted to 
receive them. 

It has been shewn to be highly probable that the ratio of A 
to B increases rapidly according as the medium considered is 
softer and more plastic. For fluids therefore, and among them 
for the luminiferous ether, we should expect the ratio of A to B 
to be extremely great. Now if N be the velocity of propagation 
of normal vibrations in the medium considered in Sect, in., and 
T that of transversal vibrations, it may be shewn from equations 
(32) that 

AT2 _ m A + 4>B B 

\ , 
3p p 

This is very easily shewn in the simplest case of plane waves : for 
if /3 = 7 = 0, a=f(x), the equations (32) give 



whence a = <j> (Nt - a) + ty (Nt + a) ; and if a = 7 = 0, /3 =/(#), 

A , .. d*/3 .,<?* , 

the same equations give p - = o - , whence 



Consequently we should expect to find the ratio of N to T ex 
tremely great. This agrees with a conclusion of the late Mr 
Green s*. Since the equilibrium of any medium would be 
unstable if either A or B were negative, the leas,t possible value 
of the ratio of N 2 to T 2 is f, a result at which Mr Green also 
arrived. As however it has been shewn to be highly probable 
that A>5B even for the hardest solids, while for the softer ones A/B 
is much greater than 5, it is probable that N/T is greater than ^3 
for the hardest solids, and much greater for the softer ones. 

If we suppose that in the luminiferous ether A/B may be con 
sidered infinite, the equations of motion admit of a simplification. 

For if we put mA ( d ,~ + ^ + ^] =-p in equations (32), and 
\dx dy dzJ 

suppose m^4 to become infinite while p remains finite, the equa 
tions become 

* Cambridge Philosophical Transactions, Vol. vii. Part I. p. 2. 



AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 129 

d a dp , fi (d*a <Za d a 
= - B 



, . 

* d0 dy 

and -j- + -j- + -j- = 0. 

a,c ay as; 

When a vibratory motion is propagated in a medium of which 
(33) are the equations of motion, it may be shewn that p ^(t) 
if the medium be indefinitely extended, or else if there be no 
motion at its boundaries. In considering therefore the trans 
mission of light in an uninterrupted vacuum the terms involving 
p will disappear from equations (33) ; but these terms are, I 
believe, important in explaining Diffraction, which is the principal 
phenomenon the laws of which depend only on the equations of 
motion of the luminiferous ether in vacuum. It will be observed 
that putting A = GO comes to the same thing as regarding the 
ether as incompressible with respect to those motions which 
constitute Light. 



ON THE PROOF OF THE PROPOSITION THAT (Mx + Ny)~ l is AN 
INTEGRATING FACTOR OF THE HOMOGENEOUS DIFFERENTIAL 
EQUATION M + N dyjdx = 0. 

[From the Cambridge Mathematical Journal, Vol. iv. p. 241. (May, 1845.)] 

A FALLACIOUS proof is sometimes given of this proposition, 
which ought to be examined. The substance of the proof is as 
follows. 

Let us see whether it is possible to find a multiplier V, a 
homogeneous function of x and y, which shall render Mdx + Ndy 
an exact differential. Let M and N be of n, and V of p dimen 
sions; let 

&U=V(Mdx + Ndy) ..................... (1); 



then, on properly choosing the arbitrary constant in U,} ... 
7 will be a homogeneous function of n + p + 1 dimensions,] ^ 
whence, by a known theorem, 



....... (2); 

therefore, dividing (1) by (2), 

dU = Mdxj^Ndy . 

(n +p + 1) U " MX + Ny~ 

and the first side of this equation being an exact differential, it 
follows that the second side is so also, and consequently that 
(Mas + Ny}~ 1 is an integrating factor. 

Now the factor so found is of n l dimensions ; so that 
the first side of (2) is zero. In fact, we shall see that the state 
ment (A) is not true as applied to the case in question, unless 

MX + Ny = 0. 



FACTOR OF HOMOGENEOUS DIFFERENTIAL EQUATION. 131 

The general form of a function of x of n dimensions is Ax n . 

The general form of a homogeneous function of x and y of n di- 

- 
. The integral of the first is in general 

Ax n ^j(n + 1),. omitting the arbitrary constant; and consequently 
the dimensions of the function are increased by unity by inte 
gration. But in the particular case in which n = 1, the integral 
is A logx, which is not a quantity of dimensions, at least accord 
ing to the definition just given, according to which definition only 
is the proposition with reference to homogeneous functions as 
sumed in (2) true. Let us now examine in what cases U will be 
of n -\-p +1 dimensions. 

Putting M=M Q x n , N=N^x n , y = xz, M Q and N will be func 
tions of z alone, and we shall have 

Mdx + Ndy = x n {(M Q + JV dx + N x dz}. 

If M Q + N Q z = 0, i.e. if MX + Ny = 0, we see that af*" 1 will be 
an integrating factor. The integral, being a function of z, will 
be of dimensions, and both sides of (2) will be zero. 

If MX + Ny is not equal to 0, we may multiply and divide by 
(M -f N Q z) x, and we have 



Hence we see that {x n+1 (M + N Q z)}~* or (Mx + Ny} 
integrating factor. For this factor we have 



(f> denoting the function arising from the integration with respect 
to*. 

dU dU 

In this case we have x -j \- y = = 1, not = 0. 
dx y dy 

It may be of some interest to enquire in what cases an exact 
differential of any number of independent variables, in which 
the differential coefficients are homogeneous functions of n dimen 
sions, has an integral which is a homogeneous function of n + 1 
dimensions. 

92 



132 INTEGRATING FACTOR OF 

Let d U=Mdx -\-Ndy + Pdz -f ... be the exact differential. Let 
y = yx, z = z x . . . , M=M x*, N =N x n . . . , so that M 0) N Q . . . are 
functions of y , z . . . only ; then 

= x n {(M Q + N,y + P/. . .) dx + (N Q dy + Pdz . . .) x}. 



First, suppose the coefficient of dx in this equation to be zero, 
or Mx + Ny + Pz ... =0; then the expression for dU cannot be 
an exact differential unless n 1. In this case U will be a 
function of y, z ..., and will therefore be a homogeneous function 
of n + 1 or dimensions. 

Secondly, suppose the coefficient of dx not to be zero ; then 



Now I say that jf^- > p / is the exact differential of 

a function of the independent variables y , z ..., or, taking y, z ... 
for the independent variables instead of y, z ..., x being supposed 
constant, and putting for- , N ,... their values, that 



Mx + Ny + Pz ... 
is an exact differential. 

For, putting MX + Ny + Pz ... = D, in order that the quantity 
considered should be an exact differential, it is necessary and 
sufficient that the system of equations of which the type is 

j J P 

d D d D 

- = - should be satisfied. This equation gives 
dz dy 

D (^p\ +P f- N f-^. 

\dz dy) dy dz 

Now, since dN/dz = dP/dy, by the conditions of M dx 4- Ndy 
\-Pdz .... being an exact differential, the above equation becomes 



HOMOGENEOUS DIFFERENTIAL EQUATION. 133 

Keplacing dM/dy, dP/dy ... by dN/dx, dN/dz... and dM/dz, 
dN/dz ... by dP/dx, dP/dy . . ., this equation becomes 

dN dN dN \ ^fdP dP dP 
l x+^r- y + -T-z...}-N(-j~x 
dx dy 9 d* J \dx 

dN dN 

Now up* +-&* + 

dP dP 



and therefore the above equation is satisfied. Hence 

z... orits 
..> 

is an exact differential dty(y,z ...}. Consequently equation (3) 
becomes 



which equation being by hypothesis integrable, it follows that 



and Mx+Ny... being moreover a homogeneous function of 
dimensions, it is clear that we must have </> (a) = Ae^ +l)a . Hence 

we have 

dU = Ax n+l e<+* d (log x + ^r). 

If now 7i + 1 is not equal to 0, we have 






omitting the constant ; but if n = 1, we have 



We see then that if Mx + Ny + Pz... =0, which can only 
happen when w = 1, U will be a homogeneous function of n + 1 
or dimensions. If Mx + Ny + Pz ...... is not equal to 0, then, 

if n + 1 is not equal to 0, and the constant in V is properly chosen, 
U will be a homogeneous function of n + 1 dimensions, but if 
n + l =0, 7 will not be a homogeneous function of dimensions, 
but will contain log x. Of course it might equally have contained 
the logarithm of y or z t &c.; in fact, 



z ... 



[From the Philosophical Magazine, Vol. xxvii. p. 9. (July, 1845.)] 

ON THE ABEKRATION OF LIGHT. 

THE general explanation of the phenomenon of aberration is 
so simple, and the coincidence of the value of the velocity of 
light thence deduced with that derived from the observations of 
the eclipses of Jupiter s satellites so remarkable, as to leave no 
doubt on the mind as to the truth of the explanation. But when 
we examine the cause of the phenomenon more closely, it is far 
from being so simple as it appears at first sight. On the theory 
of emissions, indeed, there is little difficulty ; and it would seem 
that the more particular explanation of the cause of aberration 
usually given, which depends on the consideration of the motion 
of a telescope as light passes from its object-glass to its cross 
wires, has reference especially to this theory ; for it does not apply 
to the theory of undulations, unless we make the rather startling 
hypothesis that the luminiferous ether passes freely through the 
sides of the telescope and through the earth itsetf. The undu- 
latory theory of light, however, explains so simply and so beauti 
fully the most complicated phenomena, that we are naturally led 
to regard aberration as a phenomenon unexplained by it, but not 
incompatible with it. 

The object of the present communication is to attempt an 
explanation of the cause of aberration which shall be in accordance 
with the theory of undulations. I shall suppose that the earth 
and the planets carry a portion of the ether along with them so 
that the ether close to their surfaces is at rest relatively to those 
surfaces, while, its velocity alters as we recede from the surface, 
till, at no great distance, it is at rest in space. According to the 
undulatory theory, the direction in which a heavenly body is seen 



ON THE ABERRATION OF LIGHT. 135 

is normal to the fronts of the waves which have emanated from 
it, and have reached the neighbourhood of the observer, the ether 
near him being supposed to be at rest relatively to him. If 
the ether in space were at rest, the front of a wave of light at any 
instant being given, its front at any future time could be found 
by the method explained in Airy s tracts. If the ether were in 
motion, and the velocity of propagation of light were infinitely 
small, the wave s front would be displaced as a surface of parti 
cles of the ether. Neither of these suppositions is however true, 
for the ether moves while light is propagated through it. In the 
following investigation I suppose that the displacements of a 
wave s front in an elementary portion of time due to the two 
causes just considered take place independently. 

Let u, v, w be the resolved parts along the rectangular axes of 
x, y, z, of the velocity of the particle of ether whose co-ordinates 
are x, y, z, and let V be the velocity of light supposing the ether 
at rest. In consequence of the distance of the heavenly bodies, it 
will be quite unnecessary to consider any waves except those which 
are plane, except in so far as they are distorted by the motion of 
the ether. Let the axis of z be taken in, or nearly in the direction 
of propagation of the wave considered, so that the equation of 
a wave s front at any time will be 

z = c+Vt + t; ........................... (i), 

C being a constant, t the time, and f . a small quantity, a function 
of x, y and t. Since u, v, w and f are of the order of the aberra 
tion, their squares and products may be neglected. 

Denoting by a, 0, 7 the angles which the normal to the wave s 
front at the point (x, y, z) makes with the axes, we have, to the 
first order of approximation, 



~ 
cosa = ~ , co$p = ~, 0087 = 1 ............... (2); 

and if we take a length Vdt along this normal, the co-ordinates 
of its extremity will be 



i,_KJt yvdt, z+Vdt. 
dx dy 

If the ether were at rest, the locus of these extremities would be 
the wave s front at the time t + dt, but since it is in motion, the 



136 ON THE ABERRATION OF LIGHT. 

co-ordinates of those extremities must be further increased by udt, 
vdt, wdt. Denoting then by x, y, z the co-ordinates of the point 
of the wave s front at the time t + dt which corresponds to the 
point (x, y, z] at the time t, we have 



z = z -f (w 4- V) dt ; 

and eliminating x t y and z from these equations and (1), and de 
noting % by f(x, y, t), we have for the equation to the wave s front 
at the time t + dt, 

C + Vt 



or, expanding, neglecting dt 2 and the square of the aberration, and 
suppressing the accents of x, y and z > 

z=C+Vt + Z+(w+V)dt .................. (3). 

But from the definition of it follows that the equation to the 
wave s front at the time t + dt will be got from (1) by putting 
t + dt for t, and we have therefore for this equation 



Comparing the identical equations (3) and (4), we have 

d 
_ ? /i/i 

dt" 

This equation gives f = [?$; but in the small term f we may 
replace I wdt by \wdz+ V\ this comes to taking the approximate 

value of z given by the equation z = 0+ Ftf instead of * for the 
parameter of the system of surfaces formed by the wave s front in 
its successive positions. Hence equation (1) becomes 



Combining the value of f just found with equations (2), we 
get, to a first approximation, 



ON THE ABERRATION OF LIGHT. 137 

equations which might very easily be proved directly in a more 
geometrical manner. 

If random values are assigned to u, v and w, the law of aber 
ration resulting from these equations will be a complicated one; 
but if u, v and w are such that udx + vdy + wdz is an exact dif 
ferential, we have, 

dw _ du dw _dv t 
dx dz dy dz 

whence, denoting by the suffixes 1, 2 the values of the variables 
belonging to the first and second limits respectively, we obtain 



If the motion of the ether be such that udx + vdy + wdz is an 
exact differential for one system of rectangular axes, it is easy to 
prove, by the transformation of co-ordinates, that it is an exact 
differential for any other system. Hence the formulae (6) will 
hold good, not merely for light propagated in the direction first 
considered, but for light propagated in any direction, the direc 
tion of propagation being taken in each case for the axis of jg. If 
we assume that udx + vdy + wdz is an exact differential for that 
part of the motion of the ether which is due to the motion of 
translation of the earth and planets, it does not therefore follow 
that the same is true for that part which depends on their motions 
of rotation. Moreover, the diurnal aberration is too small to be 
detected by observation, or at least to be measured with any ac 
curacy, and I shall therefore neglect it. 

It is not difficult to shew that the formulas (6) lead to the 
known law of aberration. In applying them to the case of a star, 
if we begin the integrations in equations (5) at a point situated 
at such a distance from the earth that the motion of the ether, 
and consequently the resulting change in the direction of the 
light, is insensible, we shall have u t = 0, v t = ; and if, moreover, 
we take the plane xz to pass through the direction of the earth s 
motion, we shall- have 



, u 

and ~ a - 



138 ON THE ABERRATION OF LIGHT. 

that is, the star will appear displaced towards the direction in 
which the earth is moving, through an angle equal to the ratio of 
the velocity of the earth to that of light, multiplied by the sine of 
the angle between the direction of the earth s motion and the line 
joining the earth and the star. 



ADDITIONAL NOTE. 

[In what precedes waves of light are alone considered, and the 
course of a ray is not investigated, the investigation not being 
required. There follows in the original paper an investigation 
having for object to shew that in the case of a body like the 
moon or a planet which is itself in motion, the effect of the dis 
tortion of the waves in the neighbourhood of the body in altering 
the apparent place of the body as determined by observation is 
insensible. For this, the orthogonal trajectory of the wave in its 
successive positions from the body to the observer is considered, 
a trajectory which in its main part will be a straight line, from 
which it will not differ except in the immediate neighbourhood of 
the body and of the earth, where the ether is distorted by their 
respective motions. The perpendicular distance of the further 
extremity of the trajectory from the prolongation of the straight 
line which it forms in the intervening quiescent ether is shewn to 
subtend at the earth an angle which, though not actually 0, is so 
small that it may be disregarded. 

The orthogonal trajectory of a wave in its successive positions 
does not however represent the course of a ray, as it would do if 
the ether were at rest. Some remarks made by Professor Challis 
in the course of discussion suggested to me the examination of 
the path of a ray, which in the case in which udx + vdy + wdz 
is an exact differential proved to be a straight line, a result which 
I had not foreseen when I wrote the above paper, which I may 
mention was read before the Cambridge Philosophical Society on 
the 18th of May, 1845 (see Philosophical Magazine, vol. XXIX., 
p. 62). The rectilinearity of the path of a ray in this case, though 
not expressly mentioned by Professor Challis, is virtually con 
tained in what he wrote. The problem is rather simplified by 
introducing the consideration of rays, and may be treated from 
the beginning in the following manner. 



ON THE ABERRATION OF LIGHT. 139 

The notation in other respects being as before, let a , /3 be the 
small angles by which the direction of the wave-normal at the 
point (xy y, z) deviates from that of Oz towards Ox, Oy, respec 
tively, so that a , ft are the complements of a, /3, and let a,, ft / be 
the inclinations to Oz of the course of a ray at the same point. 
By compounding the velocity of propagation through the ether 
with the velocity of the ether we easily see that 



Let us now trace the changes of a /t @, during the time dt. 
These depend first on the changes of a , /3 , and secondly on those 
of u, v. 

As regards the change in the direction of the wave-normal, we 
notice that the seat of a small element of the wave in its suc 
cessive positions is in a succession of planes of particles nearly 
parallel to the plane of x, y. Consequently the direction of the 
element of the wave will be altered during the time dt by the 
motion of the ether as much as a plane of particles of the ether 
parallel to the plane of the wave, or, which is the same to the 
order of small quantities retained, parallel to the plane xy. Now 
if we consider a particle of ether at the time t having for co 
ordinates x, y, z, another at a distance dx parallel to the axis 
of x y and a third at a distance dy parallel to the axis of y, we see 
that the displacements of these three particles parallel to the axis 
of z during the time dt will be 

7 / dw .. / dw 

wdt, 



and dividing the relative displacements by the relation distances, 
we have dw/dx. dt, dw/dy . dt for the small angles by which the 
normal is displaced, in the planes of xz t yz t from the axes x t y, so 

that 

dw -., 7/v dw 7 , 
dy. =----- dt, dp=--j-dt. 
dx dy 

We have seen already that the changes of u, v are dujdz . Vdt, 
dv/dz . Vdt, so that 

dv 



140 ON THE ABERRATION OF LIGHT. 

Hence, provided the motion of the ether be such that 

udx + vdy + wdz 

is an exact differential, the change of direction of a ray as it 
travels along is nil, and therefore the course of a ray is a straight 
line notwithstanding the motion of the ether. The rectilinearity 
of propagation of a ray of light, which a priori would seem very 
likely to be interfered with by the motion of the ether produced by 
the earth or heavenly body moving through it, is the tacit as 
sumption made in the explanation of aberration given in treatises 
of Astronomy, and provided that be accounted for the rest follows 
as usual*. It follows further that the angle subtended at the 
earth by the perpendicular distance of the point where a ray leaves 
a heavenly body from the straight line prolonged which represents 
its course through the intervening quiescent ether, is not merely 
too small to be observed, but actually nil.] 

* To make this explanation quite complete, we should properly, as Professor 
Challis remarks, consider the light coming from the wires of the observing telescope, 
in company with the light from the heavenly body. 



[From the Philosophical Magazine, Vol. xxvm. p. 76. (Feb. 1846.)] 

ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 

THE theory of the aberration of light, and of the absence of 
any influence of the motion of the earth on the laws of refraction, 
&c., given by Fresnel in the ninth volume of the Annales de 
Chimie, p. 57, is really very remarkable. If we suppose the 
diminished velocity of propagation of light within refracting media 
to arise solely from the greater density of the ether within them, 
the elastic force being the same as without, the density which it 
is necessary to suppose the ether within a medium of refractive 
index //, to have is yu, 2 , the density in vacuum being taken for unity. 
Fresnel supposes that the earth passes through the ether without 
disturbing it, the ether penetrating the earth quite freely. He 
supposes that a refracting medium moving with the earth carries 
with it a quantity of ether, of density yu, 2 1, which constitutes the 
excess of density of the ether within it over the density of the 
ether in vacuum. He supposes that light is propagated through 
this ether, of which part is moving with the earth, and part is 
at rest in space, as it would be if the whole were moving with the 
velocity of the centre of gravity of any portion of it, that is, with 
a velocity (1 /-T 2 ) v, v being the velocity of the earth. It may 
be observed however that the result would be the same if we 
supposed the whole of the ether within the earth to move to 
gether, the ether entering the earth in front, and being im 
mediately condensed, and issuing from it behind, where it is 
immediately rarefied, undergoing likewise sudden condensation or 
rarefaction in passing from one refracting medium to another. 
On this supposition, the evident condition that a mass v of the 
ether must pass in a unit of time across a plane of area unity, 



142 ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 

drawn anywhere within the earth in a direction perpendicular 
to that of the earth s motion, gives (1 /^~ 2 ) v for the velocity 
of the ether within a refracting medium. As this idea is rather 
simpler than Fresnel s, I shall adopt it in considering his theory. 
Also, instead of considering the earth as in motion and the ether 
outside it as at rest, it will be simpler to conceive a velocity equal 
and opposite to that of the earth impressed both on the earth and 
on the ether. On this supposition the earth will be at rest ; the 
ether outside it will be moving with a velocity v, and the ether 
in a refracting medium with a velocity v//j?, in a direction contrary 
to that of the earth s real motion. On account of the smallness of 
the coefficient of aberration, we may also neglect the square of 
the ratio of the earth s velocity to that of light ; and if we resolve 
the earth s velocity in different directions, we may consider the 
effect of each resolved part separately. 

In the ninth volume of the Comptes Rendus of the Academy 
of Sciences, p. 774, there is a short notice of a memoir by M. 
Babinet, giving an account of an experiment which seemed to 
present a difficulty in its explanation. M. Babinet found that 
when two pieces of glass of equal thickness were placed across 
two streams of light which interfered and exhibited fringes, in 
such a manner that one piece was traversed by the light in the 
direction of the earth s motion, and the other in the contrary 
direction, the fringes were not in the least displaced. This result, 
as M.. Babinet asserts, is contrary to the theory of aberration 
contained in a memoir read by him before the Academy in 1829, 
as well as to the other received theories on the subject. I have 
not been able to meet with this memoir, but it is easy to shew 
that the result of M. Babinet s experiment is in perfect accordance 
with Fresnel s theory. 

Let T be the thickness of one of the glass plates, V the ve 
locity of propagation of light in vacuum, supposing the ether 
at rest. Then V/p would be the velocity with which light would 
traverse the glass if the ether were at rest; but the ether 
moving with a velocity v/fjf, the light traverses the glass with a 

velocity - + -o , and therefore in a time 
J ~ 



ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 143 

But if the glass were away, the light, travelling with a velocity 
V v, would pass over the space T in the time 



T 

Hence the retardation, expressed in time, =(/u, 1) ^, the same 

as if the earth were at rest. But in this case no effect would be 
produced on the fringes, and therefore none will he produced in 
the actual case. 

I shall now shew that, according to Fresnel s theory, the laws 
of reflexion and refraction in singly refracting media are un 
influenced by the motion of the earth. The method which I 
employ will, I hope, be found simpler than Fresnel s ; besides 
it applies easily to the most general case. Fresnel has not given 
the calculation for reflexion, but has merely stated the result; 
and with respect to refraction, he has only considered the case 
in which the course of the light within the refracting medium 
is in the direction of the earth s motion. This might still leave 
some doubt on the mind, as to whether the result would be the 
same in the most general case. 

If the ether were at rest, the direction of light would be that 
of a normal to the surfaces of the waves. When the motion 
of the ether is considered, it is most convenient to define the 
direction of light to be that of the line along which the same 
portion of a wave moves relatively to the earth. For this is in 
all cases the direction which is ultimately observed with a tele 
scope furnished with cross wires. Hence, if A is any point in 
a wave of light, and if we draw AB normal to the wave, and 
proportional to V or V/JJL, according as the light is passing through 
vacuum or through a refracting medium, and if we draw EG in 
the direction of the motion of the ether, and proportional to 
v or v/fjf, and join AC, this line will give the direction of the ray. 
Of course, we might equally have drawn AD equal and parallel to 
BC and in the opposite direction, when DB would have given the 
direction of the ray. 

Let a plane P be drawn perpendicular to the reflecting or 
refracting surface and to the waves of incident light, which in this 
investigation may be supposed plane. Let the velocity v of the 
ether in vacuum be resolved into p perpendicular to the plane P, 



14-t ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 



and q in that plane ; then the resolved parts of the velocity v/fj? 
of the ether within a refracting medium will be pip?, y/f^- Let 
us first consider the effect of the velocity p. 

It is easy to see that, as far as regards this resolved part of 
the velocity of the ether, the directions of the refracted and 
reflected waves will be the same as if the ether were at rest. 
Let BAG (fig. 1) be the intersection of the refracting surface 
and the plane P\ DAE a normal to the refracting surface; AF, 
A Gr, AH normals to the incident, reflected and refracted waves. 
Hence AF, AG, AH will be in the plane P, and 

^ GAD = FAD, p sin HAE = sin FAD. 
Take 

AH=-AF. 




Draw Gg> Hh perpendicular to the plane P, and iri the direction 
of the resolved part p of the velocity of the ether, and Ff in the 
opposite direction ; and take 

Ff : Hh : FA :: p : 4 : V, and Gg = Ff, 

and join A with / g and h. Then fA t Ag, Ah will be the di 
rections of the incident, reflected and refracted rays. Draw FD, 
HE perpendicular to DE, and join/D, hE. ThenfDF, hEH will 
be the inclinations of the planes fAD t hAE to the plane P. 
Now 



tan 



and wa.FAD-ii.wa.HAE; therefore tavFDf=ianHEh, and 



ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 145 



therefore the refracted ray A h lies in the plane of incidence 
fAD. It is easy to see that the same is true of the reflected ray 
Ag. Also t gAD =/AD; and the angles fAD, hAE are sensibly 
equal to FAD, HAE respectively, and we therefore have without 
sensible error, sin fAD = /j,smhAE. Hence the laws of reflexion 
and refraction are not sensibly affected by the velocity p. 

Let us now consider the effect of the velocity q. As far as 
depends on this velocity, the incident, reflected and refracted 
rays will all be in the plane P. Let AH, AK, AL be the in 
tersections of the plane P with the incident, reflected and refracted 
waves. Let ty, ^, ->|r be the inclinations of these waves to the 
refracting surface ; let NA. be the direction of the resolved part 
q of the velocity of the ether, and let the angle NA C = a. 

The resolved part of q in a direction perpendicular to AH 
is (7 sin (-\|r -}- a). Hence the wave AH travels with the velocity 




F-t- q sin (^r + a) ; and consequently the line of its intersection 
with the refracting surface travels along AB with the velocity 
coseCA/r [V+ q sin (^ + a)}. Observing that q/fj? is the velocitv 
of the ether within the refracting medium, and V/fju the velocity 
of propagation of light, we shall find in a similar manner that 
the lines of intersection of the refracting surface with the reflected 
and refracted waves travel along AB with velocities 

coseci/rj V+ q sin (^ a.)}, cosec 



sn 



But since the incident, reflected and refracted waves intersect 
the refracting surface in the same line, we must have 

sin^ { V+q sin (^ + a)} = sin i/r {F + q sin (^ a)} 



sin ^ { F+ q sin (\fr + a) } 



sn 



F+ - sin 



a) 



...(A) 



10 



146 ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 

Draw HS perpendicular to AH, ST parallel to NA, take 
ST : HS : : q : V, and join HT. Then HT is the direction of 
the incident ray; and denoting the angles of incidence, reflexion 
and refraction by <, <,, </> , we have 

(f)-ty = SIIT = J*? = y x resolved part of q along AH 



Similarly, 



whence 



sin -^ = sin ^> -^-cos <^> cos (</> + a), 
sin 1^= sin (j> cos ^ cos (<, a), 
sin ^r = sin < cos $ cos (</> + a). 



On substituting these values in equations (A), and observing 
that in the terms multiplied by q we may put </=<> p sin?<jb = sin0, 
the small terms destroy each other, and we have sin $, = sin <, 
yu,sin (j) =siia.(f). Hence the laws of reflexion and refraction at 
the surface of a refracting medium will not be affected by the 
motion of the ether. 

In the preceding investigation it has been supposed that the 
refraction is out of vacuum into a refracting medium. But the 
result is the same in the general case of refraction out of one 
medium into another, and reflexion at the common surface. For 
all the preceding reasoning applies to this case if we merely 
substitute p/p z , q/p* for p t q, V/p for V, and p/p for M, fi being 
the refractive index of the first medium. Of course refraction 
out of a medium into vacuum is included as a particular case. 

It follows from the theory just explained, that the light coming 
from any star will behave in all cases of reflexion and ordinary 
refraction precisely as it would if the star were situated in the 
place which it appears to occupy in consequence of aberration, 
and the earth were at rest. It is, of course, immaterial whether 
the star is observed with an ordinary telescope, or with a telescope 
having its tube filled with fluid. It follows also that terrestrial 



ON FRESNEL S THEORY OF THE ABERRATION OF LIGHT. 147 

objects are referred to their true places. All these results would 
follow immediately from the theory of aberration which I pro 
posed in the July number of this Magazine ; nor have I been able 
to obtain any result, admitting of being compared with experi 
ment, which would be different according to which theory we 
adopted. This affords a curious instance of two totally different 
theories running parallel to each other in the explanation of phe 
nomena. I do not suppose that many would be disposed to main 
tain Fresnel s theory, when it is shewn that it may be dispensed 
with, inasmuch as we would not be disposed to believe, without 
good evidence, that the ether moved quite freely through the solid 
mass of the earth. Still it would have been satisfactory, if it had 
been possible, to have put the two theories to the test of some 
decisive experiment. 



102 



[From the Cambridge and Dublin Mathematical Journal, 
Vol. I. p. 183 (May, 1846).] 

ON A FOKMULA FOR DETERMINING THE OPTICAL CONSTANTS 
OF DOUBLY REFRACTING CRYSTALS. 

IN order to explain the object of this formula, it will be neces 
sary to allude to the common method of determining the optical 
constants. Two plane faces of the crystal are selected, which 
are parallel to one of the axes of elasticity; or if such do not 
present themselves, they are obtained artificially by grinding. 
A pencil of light is transmitted across these faces in a plane per 
pendicular to them both, as in the case of an ordinary prism. 
This pencil is by refraction separated into two, of which one is 
polarized in the plane of incidence, and follows the ordinary law 
of refraction, while the other is polarized in a plane perpendicular 
to the plane of incidence, and follows a different law. It will 
be convenient to call these pencils respectively the ordinary and 
the extraordinary, in the case of biaxal, as well as uniaxal crystals. 
The minimum deviation of the ordinary pencil is tnen observed, 
and one of the optical constants, namely that which relates to 
the axis of elasticity parallel to the refracting edge, is thus de 
termined by the same formula which applies to ordinary media. 
This formula will also give one of the other constants, by means 
of the observation of the minimum deviation of the extraordinary 
pencil, in the particular case in which one of the principal planes 
of the crystal bisects the angle between the refracting planes : 
but if this condition be not fulfilled it will be necessary to employ 
either two or three prisms, according as the crystal is uniaxal 
or biaxal, to determine all the constants. The extraordinary 
pencil, however, need not in any case be rejected, provided only a 
formula be obtained connecting the minimum deviation observed 



ON A FORMULA FOR DETERMINING THE OPTICAL, ETC. 149 

with the optical constants. It will thus be possible to determine 
all the constants with a smaller number of prisms ; the necessity 
of using artificial faces may often be obviated ; or if two faces 
are cut as nearly as may be equally inclined to one of the axes of 
elasticity lying in the plane of incidence, or one cut face is used 
with a natural face, the errors of cutting may be allowed for. 

Let AEB be a section of the prism by the plane of refraction, 
(the reader will have no difficulty in drawing a figure,) E being 
the refracting edge; let i be the refracting angle; OA, OB, OG 
the directions of the axes of elasticity, being any point within 
the prism, the two former of these lines being in, and the latter 
perpendicular to, the plane of refraction ; a, b, c the optical con 
stants referring to them, that is, according to Fresnel s theory, 
the velocities of propagation of waves in which the vibrations 
are parallel to the three axes respectively. Everything being 
symmetrical with respect to the plane of incidence, we need only 
consider what takes place in that plane. This plane will cut 
the wave surface in a circle of radius c, and an ellipse whose 
semiaxes are a along OB and b along OA. We have only got to 
consider the ellipse, since it is it that determines the direction 
of the extraordinary ray. The form of the crystal will very often 
make known the directions of the axes of elasticity. Supposing 
these directions known, let a, ft denote the inclinations of OA, OB 
to the produced parts of EA, EB respectively ; a, /? and i being 
of course connected by the equation a. + /3 = JTT + i. 

Let 0, T|T be the angles of incidence and emergence, the light 
being supposed incident on the face EA ; $ the inclination of the 
refracted wave to EA, ty its inclination to EB, D the deviation, 
v the velocity of the wave within the crystal, u its velocity in 
the outer medium, which may be supposed to be either air, or a 
liquid of known refractive power. Then we have 

D = <j> + ^r-i* (1), 

* + * = (2), 

v sin $ = wsin </> (3), 

vsin ty u sin-^r (4), 

v* = a? cos 2 (a -</> )+ 6 2 sin 2 (a - < ) (5). 

* I am indebted to the Rev. P. Frost for the suggestion of employing equations 
(1)...(4), rather than making use of the ellipse in which the wave surface is cut by 
the plane of incidence. 



150 ON A FORMULA FOR DETERMINING THE 

From (2), (3), (4), 
u sin ijr = v sin \Jr = u sin (i ( ) = u sin i cos $ v cos i* sin <> ; 

. . cos d> f = = . (sin ilr 4 cos i sin d>) : 
u sin ^ x 

, . ,, v . . . , 

and sin cf> = : . sin i sin 6 : 

w sin i 

substituting in (5), 

w 2 sin 2 i = a 2 {cos a (sin ty 4 cos t sin </>) 4 sin a sin i sin </>) 2 
.4 & 2 (sin a (sin i|r 4 cos i sin 0) cos a sin 4 sin </>) 2 , 
or w* sin 2 * = a 2 (cos a sin i|r 4 sin /? sin (/>) 2 

4 & 2 (sin a sin ty 4 cos ^ sin (/)) 2 ...................... (6), 

the relation between < and ^. Putting ty $ 0, and taking 

account of (1), (6) becomes 

2w* sin 8 i = [a 2 cos 2 a 4 6 2 sin 2 a} (1 - cos (D + {+0)} 

4 {a 2 sin 2 /3 4 6 2 cos 2 ] {1 - cos (D 4 * - 6>)} 
4 2 (a 2 cos a sin y8 4 6 2 sin a cos /3) {cos ^ cos (D + i)}, 
or .Fcos04sin04# = .................. (7), 

Avhere 

F= a? {(cos* a 4 sin 2 /3) cos (D 4 i) - 2 cos a sin /5} 

4 6 2 {(sin 2 a 4 cos 2 j3) cos (D 4 ) - 2 sin a cos /5}, 
(7 = ( a _ &) ( S i n 2 ^ _ C os 2 a ) sin (D 4 i), 

JI= 2w* sin 2 i a 2 {cos 2 a 4 sin 2 ft 2 cos a sin $ cos (Z> 4 i)} 
b 2 {sin 2 a 4 cos 2 /3 2 sin a cos /3 cos (D 4 )}. 
Now when D, regarded as a function of 0, is a maximum or mini 

mum -^ = 0, whence from (7) 
do 



and eliminating 6 from this equation and (7), we have 



Putting for F t G and H their values, and reducing, this equation 
becomes 

bin 2 (D 4 a*V - {cos 2 a 4 sin 2 /3 - 2 cos (D 4 cos a sin /3} wV 
- {sin 2 a 4 cos 2 - 2 cos (D 4 1) sin a cos /3J iftf 4 sin 2 . %* = . . . (8). 

This equation will be rendered more convenient for numerical 
calculation by replacing products and powers of sines and cosines 



OPTICAL CONSTANTS OF DOUBLY REFRACTING CRYSTALS. 151 

by sums and differences. Treated in this manner, the equation 
becomes 

versin 2 (D + i) cfb 2 -(A+B) u z c? - (A - B] iftf 

+ versin 2i.w 4 = 0... (9), 
where A = versin D + versin (D + 2i), 

B = cos 2a - cos 2/3 - cos (D + 2a) + cos (D + 2/3). 

If the principal plane A OC of the crystal bisects the angle 
between the refracting faces, we have 

i Q 7T i 

a = 2 P^Z+Z* 
whence from (8), putting D + i = A, 

(9 . 9 A 9 oA/79 9 A o 9 \ 

a 2 sm 2 -g tr BUT ~ J f D cos 2 ^ - u 2 cos 2 ^ ) = 0. 

The former of these factors is evidently that which corresponds to 
the problem ; the latter corresponds to refraction through a prism 
having its faces parallel to those of the actual prism, and having 
its refracting angle supplemental to /. We have therefore 

. i 

Sm 2 



so that the constant a is given by the same formula that applies to 
ordinary media, as it should. 

If the refracting faces are perpendicular to the axes of elas 
ticity which lie in the plane of incidence, the formula (8) or (9) 
takes a very simple form. In this case we have a=/3 = i=%7r, 
and therefore 

cos 2 D . ct 2 6 2 - wV- tftf + w 4 = 0. 

Mathematically speaking, one prism would be sufficient for 
determining the three constants a, 6, c. For c would be deter 
mined by means of the ordinary pencil; and by observing the 
extraordinary pencil with the crystal in air, and again with the 
crystal in some liquid, we should have two equations of the form 
(8), by combining which we should obtain a 2 and 6 2 by the 
solution of a quadratic equation. But since a is usually nearly 
equal to 6, it is evident that the course of the extraordinary ray 
within the crystal would be nearly the same in the two observa- 



152 ON A FORMULA FOR DETERMINING THE OPTICAL, ETC. 

tions, being in each case inclined at nearly equal angles to the 
refracting faces, and consequently the errors of observation would 
be greatly multiplied in the result. Even if a differed greatly 
from b, only one of these constants could be accurately determined 
in this manner if the refracting angle were nearly bisected by 
a principal plane. But two prisms properly chosen appear amply 
sufficient for determining accurately the three constants by the 
method of minimum deviations, even should neither prism have 
its angle exactly bisected by a principal plane of the crystal. 

It is not necessary to observe the deviation when it is a 
minimum, as Professor Miller has remarked to me, since the angle 
of incidence may be measured very accurately by moving the 
telescope employed till the luminous slit, seen directly, appears 
on the cross wires, and then turning it till the slit, seen by re 
flection at the first face of the prism, again appears on the cross 
wires, the prism meanwhile remaining fixed*. The angle through 
which the telescope has been turned is evidently the supplement 
of twice the angle of incidence. If this method of observation be 
adopted, <, D, and i will be known by observation, whence ^ 
will be got immediately from (1). Thus all the coefficients in 
(6) will be known quantities, and this equation furnishes a very 
simple relation between a and b. The coefficients may easily be 
calculated numerically by treating them like those in equation 
(8), or else by employing subsidiary angles. 

[* A method of measuring the refractive indices of isotropic media depending on 
the measurement of the deviation and angle of incidence is described by Professor 
Swan in the Edinburgh New Philos( phical Journal , Vol. xxxvi. (1844) p. 102.] 



[From the Philosophical Magazine^ Vol. xxix. p. 6 (July, 1846)]. 

ON THE CONSTITUTION OF LUMINIFEROUS ETHER, VIEWED WITH 

REFERENCE TO THE PHENOMENON OF THE ABERRATION OF 

LIGHT. 

IN a former communication to this Magazine (July, 1845),* 
I shewed that the phenomenon of aberration might be explained 
on the undulatory theory of light, without making the startling 
supposition that the earth in its motion round the sun offers 
no resistance to the ether. It appeared that the phenomenon 
was fully accounted for, provided we supposed the motion of the 
ether such as to make 

udx + vdy + wdz (a) 

an exact differential, Avhere u, v, w are the resolved parts, along 
three rectangular axes, of the velocity of the particle of ether 
whose co-ordinates are x, y, z. It appeared moreover that it 
was necessary to make this supposition in order to account in 
this way for the phenomenon of aberration. I did not in that 
paper enter into any speculations as to the physical causes in 
consequence of which (a) might be an exact differential. The 
object of the present communication is to consider this question. 

The enquiry naturally divides itself into two parts : First, 
In what manner does one portion of ether act on another be 
yond the limits of the earth s atmosphere ? Secondly, What 
takes place in consequence of the mutual action of the air and 
the ether ? 

In order to separate these two questions, let us first conceive 
the earth to be destitute of an atmosphere. Before considering 
the motion of the earth and the ether, let us take the case of 

* Ante, p. 134. 



154 ON THE CONSTITUTION OF LUMINIFEROUS ETHER. 

a solid moving in an ordinary incompressible fluid, which may 
be supposed to be infinitely extended in all directions about the 
solid. If we suppose the solid and fluid to be at first at rest, 
and the solid to be then moved in any manner, it follows from 
the three first integrals of the ordinary equations of fluid motion, 
obtained by M. Cauchy, that the motion of the fluid at any 
time will be such that (a) is an exact differential. From this 
it may be easily proved, that if at any instant the solid be re 
duced to rest, the whole of the fluid will be reduced to rest 
likewise ; and that the motion of the fluid is the same as it would 
have been if the solid had received by direct impact the motion 
which it has at that instant. Practically however the motion 
of the fluid after some time would differ widely from what would 
be thus obtained, at least if the motion of the solid be progressive 
and not oscillatory. This appears to be due to two causes : first, 
the motion considered would probably be unstable in the part 
of the fluid behind the solid; and secondly, a tangential force 
is called into play by the sliding of one portion of fluid along 
another ; and this force is altogether neglected in the common 
equations of hydrodynamics, from which equations the motion 
considered is deduced. If, instead of supposing the solid to 
move continuously, we supposed it first to be in motion for a 
very small interval of time, then to be at rest for another equal 
interval, then to be in motion for a third interval equal to the 
former, and so on alternately, theoretically the fluid ought to 
be at rest at the expiration of the first, third, &c. intervals, but 
practically a very slight motion would remain at the end of the 
first interval, would last through the second and third, and would 
be combined with a slight motion of the same kind, which would 
have been left at the end of the third interval, even if the fluid 
immediately before the commencement of it had been at rest ; 
and the accumulation of these small motions would soon become 
sensible. 

Let us now return to the ether. We know that the trans 
versal vibrations constituting light are propagated with a velocity 
about 10,000 times as great as the velocity of the earth; and 
Mr Green has shewn that the velocity of propagation of normal 
vibrations is in all probability incomparably greater than that 
of transversal vibrations (Cambridge Philosophical Transactions, 
vol. VII. p. 2). Consequently, in considering the motion of the 



ON THE CONSTITUTION OF LUMINIFEROUS ETHER. 155 

ether due to the motion of the earth, we may regard the ether 
as perfectly incompressible. To explain dynamically the pheno 
mena of light, it seems necessary to suppose the motion of the 
ether subject to the same laws as the motion of an elastic solid. 
If the views which I have explained at the end of a paper On 
the Friction of Fluids, &c. (Cambridge Philosophical Transactions, 
vol. viii. part 8)* be correct, it is only for extremely small vi 
bratory motions that this is the case, while if the motion be 
progressive, or not very small, the ether will behave like an 
ordinary fluid. According to these views, therefore, the earth 
will set the ether in motion in the same way as a solid would 
set an ordinary incompressible fluid in motion. 

Instead of supposing the earth to move continuously, let us 
first suppose it to move discontinuously, in the same manner 
as the solid considered above, the ether being at rest just before 
the commencement of the first small interval of time. By what 
precedes, the ether will move during the first interval in the 
same, or nearly the same, manner as an incompressible fluid 
would ; and when, at the end of this interval, the earth is reduced 
to rest, the whole of the ether will be reduced to rest, except 
as regards an extremely small motion, of the same nature as 
that already considered in the case of an ordinary fluid. But 
in the present case this small motion will be propagated into 
space with the velocity of light; so that just before the com 
mencement of the third interval the ether may be considered 
as at rest, and everything will be the same as before. Supposing 
now the number of intervals of time to be indefinitely increased, 
and their magnitude indefinitely diminished, we pass to the case 
in which the earth is supposed to move continuously. 

It appears then, from these views of the constitution of the 
ether, that (a) must be an exact differential, if it be not pre 
vented from being so by the action of the air on the ether. We 
know too little about the mutual action of the ether and material 
particles to enable us to draw any very probable conclusion 
respecting this matter; I would merely hazard the following 
conjecture. Conceive a portion of the ether to be filled with a 
great number of solid bodies, placed at intervals, and suppose 
these bodies to move with a velocity which is very small compared 

* Ante, p. 125. 



156 ON THE CONSTITUTION OF LUMINIFEROUS ETHER. 

with the velocity of light, then the motion of the ether between 
the bodies will still be such that (a) is an exact differential. But 
if these bodies are sufficiently close and numerous, they must 
impress either the whole, or a considerable portion of their own 
velocity on the ether between them. Now the molecules of air 
may act the part of these solid bodies. It may thus come to pass 
that (a) is an exact differential, and yet the ether close to the 
surface of the earth is at rest relatively to the earth. The latter 
of these conditions is however not necessary for the explanation of 
aberration*. 

[* A short demonstration that the path of a ray in the moving ether is a 
straight line, which here followed, is omitted, as the proposition has already been 
proved in the additional note printed at p. 138.] 



[From the Report of the British Association for 1846, Part I. p. 1.] 

REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

AT the meeting of the British Association held at Cambridge last 
year, the Committee of the Mathematical Section expressed a wish 
that a Report on Hydrodynamics should be prepared, in continua 
tion of the reports which Prof. Challis had already presented to 
the Association on that subject. Prof. Challis having declined the 
task of preparing this report, in consequence of the pressure of 
other engagements, the Committee of the Association did me the 
honour to entrust it to me. In accordance with the wishes of the 
Committee, the object of the present report will be to notice re 
searches in this subject subsequent to the date of the reports of 
Prof. Challis. It will sometimes however be convenient, for the 
sake of giving a connected view of certain branches of the subject, 
to refer briefly to earlier investigations. 

The fundamental hypothesis on which the science of hydro 
statics is based may be considered to be, that the mutual action 
of two adjacent portions of a fluid at rest is normal to the surface 
which separates them. The equality of pressure in all directions 
is not an independent hypothesis, but a necessary consequence of 
the former. This may be easily proved by the method given in 
the Exercises of M. Cauchy*, a method which depends on the con 
sideration of the forces acting on a tetrahedron of the fluid, the 
dimensions of which are in the end supposed to vanish. This 
proof applies equally to fluids at rest and fluids in motion ; and 
thus the hypothesis above-mentioned may be considered as the 
fundamental hypothesis of the ordinary theory of hydrodynamics, 
as well as hydrostatics. This hypothesis is fully confirmed by 

* Tom. ii. p. 42. 



158 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

experiment in the case of the equilibrium of fluids ; but the com 
parison of theory and experiment is by no means so easy in the 
case of their motion, on account of the mathematical difficulty of 
treating the equations of motion. Still enough has been done to 
shew that the ordinary equations will suffice for the explanation 
of a great variety of phenomena; while there are others the 
laws of which depend on a tangential force, which is neglected in 
the common theory, and in consequence of which the pressure is 
different in different directions about the same point. The linear 
motion of fluids in uniform pipes and canals is a simple instance*. 
In the following report I shall first consider the common theory 
of hydrodynamics, and then notice some theories which take ac 
count of the inequality of pressure in different directions. It 
will be convenient to consider the subject under the following 
heads : 

I. General theorems connected with the ordinary equations of 
fluid motion. 

II. Theory of waves, including tides. 

III. The discharge of gases through small orifices. 

IV. Theory of sound. 

V. Simultaneous oscillations of fluids and solids. 

VI. Formation of the equations of motion when the pressure 
is not supposed equal in all directions. 

I. Although the common equations of hydrodynamics have 
been so long known, their complexity is so great f that little has 
been done with them except in the case in which the expression 
usually denoted by 

udx + vdy + wdz (A) 

is the exact differential of a function of the independent variables 
x, y, ( I* becomes then of the utmost importance to inquire in 
what cases this supposition may be made. Now Lagrange enun 
ciated two theorems, by virtue of which, supposing them true, the 
supposition may be made in a great number of important cases, 
in fact, in nearly all those cases which it is most interesting to 

[* See the footnote at p. 99.] 

t In nearly all the investigations of Mr Airy it will be found that (A) is an 
exact differential, although he does not start with assuming it to be so. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 159 

investigate. It must be premised that in these theorems the 
accelerating forces X, Y, Z are supposed to be such that 

Xdx + Ydy + Zd* 

is an exact differential, supposing the time constant, and the 
density of the fluid is supposed to be either constant, or a function 
of the pressure. The theorems are 

First, that (A) is approximately an exact differential when the 
motion is so small that squares and products of u, v, w and their 
differential coefficients may be neglected. By calling (A) approxi 
mately an exact differential, it is meant that there exists an ex 
pression u t dx + v t dy + w t dz, which is accurately an exact differential, 
and which is such that u^ v^ w t differ from u, v, w respectively by 
quantities of the second order only. 

Secondly, that (A) is accurately an exact differential at all 
times when it is so at one instant, and in particular when the 
motion begins from rest. 

It has been pointed out by Poisson that the first of these 
theorems is not true*. In fact, the initial motion, being arbitrary, 
need not be such as to render (A) an exact differential. Thus 
those cases coming under the first theorem in which the assertion 
is true are merged in those which come under the second, at least 
if we except the case of small motions kept up by disturbing 
causes, a case in which we have no occasion to consider initial 
motion at all. This case it is true is very important. 

The validity of Lagrange s proof of the second theorem depends 
on the legitimacy of supposing u, v and w capable of expansion 
according to positive, integral powers of the time t, for a sufficiently 
small value of that variable. This proof lies open to objection ; 
for there are functions of t the expansions of which contain frac 
tional powers, and there are others which cannot be expanded 
according to ascending powers of t, integral or fractional, even 
though they may vanish when t = 0. It has been shewn by Mr 
Power that Lagrange s proof is still applicable if u, v and w admit 
of expansion according to ascending powers of t of any kindf. The 
second objection however still remains : nor does the proof which 
Poisson has substituted for Lagrange s in his { Traite de Me cani- 
que appear at all more satisfactory. Besides, it does not appear 

* Memoires de VAcadCmie des Sciences, torn. x. p. 554. 

t Transactions of the Cambridge Philosophical Society, vol. vii. p. 455. 



160 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

from these proofs what becomes of the theorem if it is only for a 
certain portion of the fluid that (A) is at one instant an exact 
differential. 

M. Cauchy has however given a proof of the theorem *, which 
is totally different from either of the former, and perfectly satis 
factory. M. Cauchy first eliminates the pressure by differentiation 
from the three partial differential equations of motion. He then 
changes the independent variables in the three resulting equations 
from x, y, z, t to a, b, c, t, where a, b, c are the initial co-ordinates 
of the particle whose co-ordinates at the time t are x, y, z. The 
three transformed equations admit each of being once integrated 
with respect to t, and the arbitrary functions of a, b, c introduced 
by integration are determined by the initial motion, which is sup 
posed to be given. The theorem in question is deduced without 
difficulty from the integrals thus obtained. It is easily proved 
that if the velocity is suddenly altered by means of impulsive 
forces applied at the surface of the fluid, the alteration is such as 
to leave (A) an exact differential if it were such before impact. 
M. Cauchy s proof shews moreover that if (A) be an exact diffe 
rential for one portion of the fluid, although riot for the whole, it 
will always remain so for that portion. It should be observed, 
that although M. Cauchy has proved the theorem for an incom 
pressible fluid only, the same method of proof applies to the more 
general case in which the density is a function of the pressure. 

In a paper read last year before the Cambridge Philosophical 
Society, I have given a new proof of the same theorem f. This 
proof is rather simpler than M. Cauchy s, inasmuch as it does not 
require any integration. 

In a paper published in the Philosophical Magazine J, Prof. 
Challis has raised an objection to the application of the theorem 
to the case in which the motion of the fluid begins from rest. 
According to the views contained in this paper, we are not in 
general at liberty to suppose (A) to be an exact differential when 
u, v and w vanish : this supposition can only be made when the 
limiting value of t~ a (udx + vdy + wdz) is an exact differential, where 
a is so taken as that one at least of the terms in this expression 
does not vanish when t vanishes. 

* M6moires des Savans Etrangers, torn. i. p. 40. 

t Transactions of the Cambridge Philosophical Society, vol. viii. p. 307. 

J Vol. xxiv. New Series, p. 94. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 161 

It is maintained by Prof. Challis that the received equations 
of hydrodynamics are not complete, as regards the analytical prin 
ciples of the science, and he has given a new fundamental equation, 
in addition to those received, which he calls the equation of con 
tinuity of the motion*. On this equation Prof. Challis rests a result 
at which he has arrived, and which all must allow to be most 
important, supposing- it correct, namely that whenever (A) is 
an exact differential the motion of the fluid is necessarily recti 
linear, one peculiar case of circular motion being excepted. As I 
have the misfortune to differ from Professor Challis on the points 
mentioned in this and the preceding paragraph, for reasons which 
cannot be stated here, it may be well to apprise the reader that 
many of the results which will be mentioned further on as satis 
factory lie open to Professor Challis s objections. 

By virtue of the equation of continuity of a homogeneous 
incompressible fluid, the expression udy vdx will always be the 
exact differential of a function of x and y. In the Cambridge 
Philosophical Transactions^ there will be found some applications 
of this function, and of an analogous function for the case of 
motion which is symmetrical about an axis, and takes place in 
planes passing through the axis. The former of these functions 
had been previously employed by Mr Earnshaw. 

II. In the investigations which come under this head, it is to 
be understood that the motion is supposed to be very small, so 
that first powers only of small quantities are retained, unless the 
contrary is stated. 

The researches of MM. Poisson and Cauchy were directed to 
the investigation of the waves produced by disturbing causes 
acting arbitrarily on a small portion of the fluid, which is then left 
to itself. The mathematical treatment of such cases is extremely 
difficult ; and after all, motions of this kind are not those which 
it is most interesting to investigate. Consequently it is the 
simpler cases of wave motion, and those which are more nearly con 
nected with the phenomena which it is most desirable to explain, 
that have formed the principal subject of more recent investiga 
tions. It is true that there is one memoir by M. Ostrogradsky, 



* Transactions of the Cambridge Philosophical Society, vol. viii. p. 31; and 
Philosophical Magazine,, vol. xxvi. New Series, p. 425. 
t Vol. vii. p. 439. (Ante, p. 1.) 

S. 11 



162 REPORT OX RECENT RESEARCHES IN HYDRODYNAMICS. 

read before the French Academy in 1826*, to which this character 
does not apply. In this memoir the author has determined the 
motion of the fluid contained in a cylindrical basin, supposing the 
fluid at first at rest, but its surface not horizontal. The interest 
of the memoir however depends almost exclusively on the mathe 
matical processes employed ; for the result is very complicated, 
and has not been discussed by the author. There is one circum 
stance mentioned by M. Plana*)- which increases the importance of 
the memoir in a mathematical point of view, which is that Poisson 
met with an apparent impossibility in endeavouring to solve the 
same problem. I do not know whether Poisson s attempt w is 
ever published. 

Theory of Long Waves. When the length of the waves whose 
motion is considered is very great compared with the depth of the 
fluid, we may without sensible error neglect the difference between 
the horizontal motions of different particles in the same vertical 
line, or in other words suppose the particles once in a vertical 
line to remain in a vertical line : we may also neglect the vertical, 
compared with the horizontal effective force. These considerations 
extremely simplify the problem ; and the theory of long waves is 
very important from its bearing on the theory of the tides. La- 
grange s solution of the problem in the case of a fluid of uniform 
depth is well known. It is true that Lagrange fell into error in 
extending his solution to cases to which it does not apply ; but 
there is no question as to the correctness of his result when 
properly restricted, that is when applied to the case of long waves 
only. There are however many questions of interest connected 
with this theory which have not been considered by Lagrange. 
For instance, what will be the velocity of propagation in a uniform 
canal whose section is not rectangular ? How will the form of the 
wave be altered if the depth of the fluid, or the dimensions of the 
canal, gradually alter ? 

In a paper read before the Cambridge Philosophical Society in 
May 1837 + , the late Mr Green has considered the motion of long 
waves in a rectangular canal whose depth and breadth alter very 
slowly, but in other respects quite arbitrarily. Mr Green arrived 
at the following results : If & be the breadth, and 7 the depth of 

* Mtmoires des Savans Etrangers, torn. iii. p. 23. 

t Turin Memoirs for 1835, p. 253. 

J Transactions of the Cambridge Philosophical Society, vol. vi. p. 457. 



REPORT OX RECEXT RESEARCHES IX HYDRODYXAMICS. 163 

the canal, then the height of the wave ccyS ^ i, the horizontal 
velocity of the particles in a given phase of their motion oc -4 7"^, 
the length of the wave oc 7?, and the velocity of propagation = Jg^> 
With respect to the height of the wave, Mr Russell was led by his 
experiments to the same law of its variation as regards the breadth 
of the canal, and with respect to the effect of the depth he observes 
that the height of the wave increases as the depth of the fluid 
decreases, but that the variation of the height of the wave is very 
slow compared with the variation of the depth of the canal. 

In another paper read before the Cambridge Philosophical 
Society in February 1839*, Mr Green has given the theory of the 
motion of long waves in a triangular canal with one side vertical. 
Mr Green found the velocity of propagation to be the same as that 
in a rectangular canal of half the depth. 

In a memoir read before the Royal Society of Edinburgh in 
April 1839 f, Prof. Kelland has considered the case of a uni 
form canal whose section is of any form. He finds that the velo 



city of propagation is given by the very simple formula AjT > 

where A is the area of a section of the canal, and I the breadth 
of the fluid at the surface. This formula agrees with the experi 
ments of Mr Russell, and includes as a particular case the formula 
of Mr Green for a triangular canal. 

Mr Airy, the Astronomer Royal, in his excellent treatise on 
Tides and Waves, has considered the case of a variable canal with 
more generality than Mr Green, inasmuch as he has supposed the 
section to be of any formj. If A, b denote the same things as in 
the last paragraph, only that now they are supposed to vary slowly 
in passing along the canal, the coefficient of horizontal displace 
ment oc A~%$, and that of the vertical displacement oc A~^b~^ 9 
while the velocity of propagation at any point of the canal is that 
given by the formula of the preceding paragraph. Mr Airy has 
proved the latter formula in a more simple manner than Prof. 
Kelland, and has pointed out the restrictions under which it is 



* Transactions of the Cambridge Philosophical Society, vol. vii. p. 87. 
t Transactions of the Royal Society of Edinburgh, vol. xiv. pp. 524, 530. 
J Encyclopedia Metropolitan^ article Tides and Waves. Art. 260 of the 
treatise. 

Art. 218, &c. 

112 



164 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

true. Other results of Mr Airy s will be more conveniently con 
sidered in connection with the tides. 

Theory of Oscillatory Waves. When the surface of water is 
covered with an irregular series of waves of different sizes, the 
longer waves will be continually overtaking the shorter, and the 
motion will be very complicated, and will offer no regular laws. 
In order to obtain such laws we must take a simpler case: we 
may for instance propose to ourselves to investigate the motion of 
a series of waves which are propagated with a constant velocity, 
and without change of form, in a fluid of uniform depth, the 
motion being in two dimensions and periodical. A series of waves 
of this sort may be taken as the type of oscillatory waves in 
general, or at least of those for which the motion is in two dimen 
sions: to whatever extent a series of waves propagated in fluid 
of a uniform depth deviates from this standard form, to the same 
extent they fail in the characters of uniform propagation and in 
variable form. 

The theory of these waves has long been known. In fact each 
element of the integrals by which MM. Poisson and Cauchy ex 
pressed the disturbance of the fluid denotes what is called by Mr 
Airy a standing oscillation, and a progressive oscillation of the 
kind under consideration will result from the superposition of two 
of these standing oscillations properly combined. Or, if we merely 
replace products of sines and cosines under the integral signs by 
sums and differences, each element of the new integrals will denote 
a progressive oscillation of the standard kind. The theory of these 
waves however well deserves a more detailed investigation. The 
most important formula connected with them is that which gives 
the relation between the velocity of propagation, the length of the 
waves, and the depth of the fluid. If c be the velocity of propa 
gation, X the length of the waves, measured from crest to crest, h 

9 

the depth of the fluid, and ra = , then 



m 



If the surface of the fluid be cut by a vertical plane perpen 
dicular to the ridges of the waves, the section of the surface will 
be the curve of sines. Each particle of the fluid moves round and 
round in an ellipse, whose major axis is horizontal. The particle 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 165 

is in its highest position when the crest of the wave is passing 
over it, and is then moving in the direction of propagation of the 
wave ; it is in its lowest position when the hollow of the wave is 
passing over it, and is then moving in a direction contrary to the 
direction of propagation. At the bottom of the fluid the ellipse is 
reduced to a right line, along which the particle oscillates. When 
the length of waves is very small compared with the depth of the 
fluid, the motion at the bottom is insensible, and all the expres 
sions will be sensibly the same as if the depth were infinite. On 

this supposition the expression for c reduces itself to A / >~~ The 

ellipses in which the particles move are replaced by circles, and 
the motion in each circle is uniform. The motion decreases with 
extreme rapidity as the point considered is further removed from 
the surface ; in fact, the coefficients of the horizontal and vertical 
velocity contain as a factor the exponential e~ y , where y is the 
depth of the particle considered below the surface. When the depth 
of the fluid is finite, the law (as to time) of the horizontal and 
vertical displacements of the particles is the same as when the depth 
is infinite. When the length of the waves is very great compared 
with the depth of the fluid, the horizontal motion of different 
particles in the same vertical line is sensibly the same. The ex 
pression for c reduces itself to Jgh, the same as would have been 
obtained directly from the theory of long waves. The whole 
theory is given very fully in the treatise of Mr Airy*. The nature 
of the motion of the individual particles, as deduced from a rigor 
ous theory, was taken notice of, I believe for the first time, by 
Mr Green f, who has considered the case in which the depth is in 
finite. 

The oscillatory waves just considered are those which are pro 
pagated uniformly in fluid of which the depth is everywhere the 
same. When this condition is not satisfied, as for instance when 
the waves are propagated in a canal whose section is not rectangu 
lar, it is desirable to know how the velocity of propagation and 
the form of the waves are modified by this circumstance. There 
is one such case in which a solution has been obtained. In a 
paper read before the Eoyal Society of Edinburgh in January 1841, 

* Tides and Waves, art. 160, &c. 

t Transactions of the Cambridge Philosophical Society, vol. vii. p. 95. 



ICG REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

Prof. Kelland has arrived at a solution of the problem in the case 
of a triangular canal whose sides are inclined at an angle of 45 
to the vertical, or of a canal with one side vertical and one side 
inclined at an angle of 45, in which the motion will of course be the 
same as in one half of the complete canal*. The velocity of propa 
gation is given by the formula (B), which applies to a rectangular 
canal, or to waves propagated without lateral limitation, provided 
we take for h half the greatest depth in the triangular canal, and 
for X, or 27T/m, a quantity less than the length of the waves in the 
triangular canal in the ratio of 1 to *J2. As to the form of the 
waves, a section of the surface made by a vertical plane parallel 
to the edges of the canal is the curve of sines ; a section made by 
a vertical plane perpendicular to the former is the common cate 
nary, with its vertex in the plane of the middle of the canal 
(supposed complete), and its concavity turned upwards or down 
wards according as the section is taken where the fluid is elevated 
or where it is depressed. Thus the ridges of the waves do not 
bend forwards, but are situated in a vertical plane, and they rise 
higher towards the slanting sides of the canal than in the middle. 
I shall write down the value of <, the integral of (A), and then any 
one who is familiar with the subject can easily verify the preceding 
results. In the following expression x is measured along the 
bottom line of the canal, y is measured horizontally, and z verti 
cally upwards : 

(f> = A(ey-}-e- a y)(e aZ + - aZ *)smj2z(x-ct) (C). 

I have mentioned these results under the head, of oscillatory 
waves, because it is to that class only that the investigation strictly 
applies. The length of the waves is however perfectly arbitrary, 
and when it bears a large ratio to the depth of the fluid, it seems 
evident that the circumstances of the motion of any one wave can 
not be materially affected by the waves which precede and follow 
it, especially as regards the form of the middle portion, or ridge, 
of the wave. Now the solitary waves of Mr Russell are long com 
pared with the depth of the fluid ; thus in the case of a rect 
angular canal he states that the length of the wave is about six 
times the depth. Accordingly Mr Russell finds that the form of 
the ridge agrees well with the results of Prof. Kelland. 

* Transactions of the Royal Society of Edinburgh, vol. xv. p. 121. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 167 

It appears from Mr Russell s experiments that there is a certain 
limit to the slope of the sides of a triangular canal, beyond which 
it is impossible to propagate a wave in the manner just considered. 
Prof. Kelland has arrived at the same result -from theory, but his 
mathematical calculation does not appear to be quite satisfactory. 
Nevertheless there can be little doubt that such a limit does 
exist, and that if it be passed, the wave will be either continually 
breaking at the sides of the canal, or its ridge will become bow- 
shaped, in consequence of the portion of the wave in the middle 
of the canal being propagated more rapidly than the portions 
which lie towards the sides. When once a wave has become suf 
ficiently curved it may be propagated without further change, as 
Mr Airy has shewn*. Thus the case of motion above considered 
is in nowise opposed to the circumstance that the tide wave as 
sumes a curved form when it is propagated in a broad channel in 
which the water is deepest towards the centre. 

It is worthy of remark, that if in equation (C), we transfer the 
origin to either of the upper edges of the canal (supposed com 
plete), and then suppose h to become infinite, having previously 
written Ae~ iah for A, the result will express a series of oscillatory 
waves propagated in deep water along the edge of a bank having 
a slope of 45, the ridges of the waves being perpendicular to the 
edge of the fluid. It is remarkable that the disturbance of the 
fluid decreases with extreme rapidity as the perpendicular distance 
from the edge increases, and not merely as the distance from the 
surface increases. Thus the disturbance is sensible only in the 
immediate neighbourhood of the edge, that is at a distance from 
it which is a small multiple of X. The formula may be accommo 
dated to the case of a bank having any inclination by merely 
altering the coefficients of y and z, without altering the sum of the 
squares of the coefficients. If i be the inclination of the bank to 
the vertical, it will be easily found that the velocity of propagation 

is equal to ( | cos i ) . When i vanishes these waves pass into those- 

already mentioned as the standard case of oscillatory waves ; and 
when i becomes negative, or the bank overhangs the fluid, a motion 
of this sort becomes impossible. 

I have had occasion to refer to what Mr Airy calls a standing 

* Tides and Waves, art. 359. 



168 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

oscillation or standing wave. To prevent the possibility of con 
fusion, it may be well to observe that Mr Airy uses the term in 
a totally different sense from Mr Russell. The standing wave of 
Mr Airy is the oscillation which would result from the co-existence 
of two series of progressive waves, which are equal in every respect, 
but are propagated in opposite directions. With respect to the 
standing wave of Mr Russell, it cannot be supposed that the ele 
vations observed in mountain streams can well be made the sub 
ject of mathematical calculation. Nevertheless in so far as the 
motion can be calculated, by taking a simple case, the theory does 
not differ from that of waves of other classes. For if we only sup 
pose a velocity equal and opposite to that of the stream impressed 
both on the fluid and on the stone at the bottom which produces 
the disturbance, we pass to the case of a forced wave produced in 
still water by a solid dragged through it. There is indeed one 
respect in which the theory of these standing waves offers a pecu 
liarity, which is, that the velocity of a current is different at 
different depths. But the theory of such motions is one of great 
complexity and very little interest. 

Theory of Solitary Waves. It has been already remarked that 
the length of the solitary wave of Mr Russell is considerable com 
pared with the depth of the fluid. Consequently we might expect 
that the theory of long waves would explain the main phenomena 
of solitary waves. Accordingly it is found by experiment that the 
velocity of propagation of a solitary wave in a rectangular canal 
is that given by the formula of Lagrange, the height of the wave 
being very small, or that given by Prof. Kelland s, formula when 
the canal is not rectangular. Moreover, the laws of the motion of 
a solitary wave, deduced by Mr Green from the theory of long 
waves, agree with the observations of Mr Russell. Thus Mr Green 
found, supposing the canal rectangular, that the particles in a 
vertical plane perpendicular to the length of the canal remain in 
a vertical plane ; that the particles begin to move when the wave 
reaches them, remain in motion while the wave is passing over 
them, and are finally deposited in new positions ; that they move 
in the direction of propagation of the wave, or in the contrary 
direction, according as the wave consists of an elevation or a de 
pression*. But when we attempt to introduce into our calculations 

* Transactions of the Cambridge Philosophical Society, vol. vii. p. 87. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 169 

the finite length of the wave, the problem becomes of great 
difficulty. Attempts have indeed been made to solve it by the 
introduction of discontinuous functions. But whenever such func 
tions are introduced, there are certain conditions of continuity 
to be satisfied at the common surface of two portions of fluid to 
which different analytical expressions apply; and should these 
conditions be violated, the solution will be as much in fault as it 
would be if the fluid were made to penetrate the bottom of the 
canal. No doubt, the theory is contained, to a first approximation, 
in the formulas of MM. Poisson and Cauchy ; but as it happens 
the obtaining of these formula? is comparatively easy, their discus 
sion forms the principal difficulty. When the height of the wave 
is not very small, so that it is necessary to proceed to a second 
approximation, the theory of long waves no longer gives a velocity 
of propagation agreeing with experiment. It follows, in fact, from 
the investigations of Mr Airy, that the velocity of propagation of a 



long wave is, to a second approximation, *g(h + 3k), where h is 
the depth of the fluid when it is in equilibrium, and h + k the 
height of the crest of the wave above the bottom of the canal *. 

The theory of the two great solitary waves of Mr Russell forms 
the subject of a paper read by Mr Earnshaw before the Cambridge 
Philosophical Society in December last-f. Mr Russell found by 
experiment that the horizontal motion of the fluid particles was 
sensibly the same throughout the whole of a vertical plane per 
pendicular to the length of the canal. He attributed the observed 
degradation of the wave, and consequent diminution of the velocity 
of propagation, entirely to the imperfect fluidity of the fluid, and 
its adhesion to the sides and bottom of the canal. Mr Earnshaw 
accordingly investigates the motion of the fluid on the hypotheses, 
first, that the particles once in a vertical plane, perpendicular to 
the length of the canal, remain in a vertical plane ; secondly, that 
the wave is propagated with a constant velocity and without 

* Tides and Waves, art. 208. In applying this formula to a solitary wave, it is 
necessary to take for h the depth of the undisturbed portion of the fluid. In the 
treatise of Mr Airy the formula is obtained for a particular law of disturbance, but 
the same formula would have been arrived at, by the same reasoning, had the law 
not been restricted. This formula is given as expressing the velocity of propagation 
of the phase of high water, which it is true is not quite the same as the velocity of 
propagation of the crest of the wave ; but the two velocities are the same to the 
second order of approximation. 

t Transactions of the Cambridge Philosophical Society, vol. viii. p. 326. 



170 REPORT OX RECENT RESEARCHES IN HYDRODYNAMICS. 

change of form. It is important to observe that these hypotheses 
are used not as a foundation for calculation, but as a means of 
selecting a particular kind of motion for consideration. The equa 
tions of fluid motion admit of integration in this case in finite 
terms, without any approximation, and it turns out that the motion 
is possible, so far as the wave itself is concerned, and everything is 
determined in the result except two constants, which remain arbi 
trary. However, in order that the motion in question should 
actually take place, it is necessary that there should be an instan 
taneous generation or destruction of a finite velocity, and likewise 
an abrupt change of pressure, at the junction of the portion of 
fluid which constitutes the wave with the portions before and 
behind which are at rest, both which are evidently impossible. It 
follows of course that one at least of the two hypotheses must be 
in fault. Experiment shewing that the first hypothesis is very 
nearly true, while the second (from whatever cause) is sensibly 
erroneous, the conclusion is that in all probability the degradation 
of the wave is not to be attributed wholly to friction, but that it 
is an essential characteristic of the motion. Nevertheless the 
formula for the velocity of propagation of the positive wave, at 
which Mr Earnshaw has arrived, agrees very well with the experi 
ments of Mr Russell; the formula for the negative wave also agrees, 
but not closely. These two formula can be derived from each 
other only by introducing imaginary quantities. 

It is the opinion of Mr Russell that the solitary wave is a 
phenomenon swi generis, in nowise deriving its character from the 
circumstances of the generation of the wave. His experiments 
seem to render this conclusion probable. Should it be correct, 
the analytical character of the solitary wave remains to be dis 
covered. A complete theory of this wave should give, not only 
its velocity of propagation, but also the law of its degradation, 
at least of that part of the degradation which is independent of 
friction, which is probably by far the greater part. With respect 
to the importance of this peculiar wave however, it must be re 
marked that the term solitary wave, as so defined, must not be 
extended to the tide wave, which is nothing more (as far as 
regards the laws of its propagation) than a very long wave, of 
which the form may be arbitrary. It is hardly necessary to re 
mark that the mechanical theories of the solitary wave and of the 
aerial sound wave are altogether different. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 171 

Theory of River and Ocean Tides. The treatise of Mr Airy 
already referred to is so extensive, and so full of original matter, 
that it will be impossible within, the limits of a report like the 
present to do more than endeavour to give an idea of the nature 
of the calculations and methods of explanation employed, and to 
mention some of the principal results. 

On account of the great length of the tide wave, the horizontal 
motion of the water will be sensibly the same from top to bottom. 
This circumstance most materially simplifies the calculation. The 
partial differential equation for the motion of long waves, when 
the motion is very small, is in the simplest case the same as that 
which occurs in the theory of the rectilinear propagation of sound ; 
and in Mr Airy s investigations the arbitrary functions which occur 
in its integral are determined by the conditions to be satisfied at 
the ends of the canal in which the waves are propagated, in a 
manner similar to that in which the arbitrary functions are deter 
mined in the case of a tube in which sound is propagated. When 
the motion is not very small, the partial differential equation of 
wave motion may be integrated by successive approximations, the 
arbitrary functions being determined at each order of approxima 
tion as before. 

To proceed to some of the results. The simplest conceivable 
case of a tidal river is that in which the river is regarded as a 
uniform, indefinite canal, without any current. The height of the 
water at the mouth of the canal will be expressed, as in the open 
sea, by a periodic function of the time, of the form a sin (nt + a). 
The result of a first approximation of course is that the disturb 
ance at the mouth of the canal will be propagated uniformly up 
it, with the velocity due to half the depth of the water. But on 
proceeding to a second approximation*, Mr Airy finds that the 
form of the wave will alter as it proceeds up the river. Its front 
will become shorter and steeper, and its rear longer and more 
gently sloping. When the wave has advanced sufficiently far up 
the river, its surface will become horizontal at one point in the 
rear, and further on the wave will divide into two. At the mouth 
of the river the greatest velocities of the ebb and flow of the tide 
are equal, and occur at low and high water respectively; the time 
during which the water is rising is also equal to the time during 

* Art. 198, &c. 



172 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

which it is falling. But at a station up the river the velocity of 
the ebb-stream is greater than that of the flow-stream, and the 
rise of the water occupies less time than its fall. If the station 
considered is sufficiently distant from the mouth of the river, and 
the tide sufficiently large, the water after it has fallen some way 
will begin to rise again : there will in fact be a double rise and 
fall of the water at each tide. This explains the double tides 
observed in some tidal rivers. The velocity with which the phase of 
high water travels up the river is found to be Jgk(I + ob), k being 
the depth of the water when in equilibrium, and bk the greatest 
elevation of the water at the mouth of the river above its mean 
level. The same formula will apply to the case of low water if we 
change the sign of b. This result is very important, since it shews 
that the interval between the time of the moon s passage over the 
meridian of the river station and the time of high water will be 
affected by the height of the tide. Mr Airy also investigates the 
effect of the current in a tidal river. He finds that the difference 
between the times of the water s rising and falling is increased by 
the current. 

When the canal is stopped by a barrier the circumstances are 
altered. When the motion is supposed small, and the disturbing 
force of the sun and moon is neglected, it is found in this case 
that the tide-wave is a stationary wave*, so that there is high or 
low water at the same instant at every point of the canal; but 
if the length of the canal exceeds a certain quantity, it is high 
water in certain parts of the canal at the instant when it is low 
water in the remainder, and vice versa. The height of high water 
is different in different parts of the canal : it increases from the 
mouth of the canal to its extremity, provided the canal s length 
does not exceed a certain quantity. If four times the length of 
the canal be any odd multiple of the length of a free wave whose 
period is equal to that of the tide, the denominator of the expres 
sion for the tidal elevation vanishes. Of course friction would 
prevent the elevation from increasing beyond a certain amount, 
but still the tidal oscillation would in such cases be very large. 

When the channel up which the tide is propagated decreases 
in breadth or depth, or in both, the height of the tide increases in 
ascending the channel. This accounts for the great height of the 

* Art. 307. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 173 

tides observed at the head of the Bristol Channel, and in such 
places. In some of these cases however the great height may 
be partly due to the cause mentioned at the end of the last 
paragraph. 

When the tide-wave is propagated up a broad channel, which 
becomes shallow towards the sides, the motion of the water in the 
centre will be of the same nature as the motion in a free canal, so 
that the water will be flowing up the channel with its greatest 
velocity at the time of high water. Towards the coasts however 
there will be a considerable flow of water to and from the shore ; 
and as far as regards this motion, the shore will have nearly the 
same effect as a barrier in a canal, and the oscillation will be of 
the nature of a stationary wave, so that the water will be at rest 
when it is at its greatest height. If, now, we consider a point at 
some distance from the shore, but still not near the middle of the 
channel, the velocity of the water up and down the channel will 
be connected with its height in the same way as in the case of a 
progressive wave, while the velocity to and from the shore will be 
connected with the height of the water in the same way as in 
a stationary wave. Combining these considerations, Mr Airy is 
enabled to explain the apparent rotation of the water in such 
localities, which arises from an actual rotation in the direction of 
its motion*. 

When the motion of the water is in two dimensions the mathe 
matical calculation of the tidal oscillations is tolerably simple, at 
least when the depth of the water is uniform. But in the case of 
nature the motion is in .three dimensions, for the water is distri 
buted over the surface of the earth in broad sheets, the boundaries 
of which are altogether irregular. On this account a complete 
theory of the tides appears hopeless, even in the case in which the 
depth is supposed uniform. Laplace s theory, in which the whole 
earth is supposed to be covered with water, the depth of which 
follows a very peculiar law, gives us no idea of the effect of the 
limitation of the ocean by continents. Mr Airy consequently in 
vestigates the motion of the water on the supposition of its being 
confined to narrow canals of uniform depth, which in the calcula 
tion are supposed circular. The case in which the canal forms a 
great circle is especially considered. This method enables us in 

* Art. 360, &c. 



174 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

some degree to estimate the effect of the boundaries of the sea ; 
aad it has the great advantage of leading to calculations which 
can be worked out. There can be 110 doubt, too, that the con 
clusions arrived at will apply, as to their general nature, to the 
actual case of the earth. 

With a view to this application of the theory, Mr Airy calcu 
lates the motion of the water in a canal when it is under the 
action of a disturbing force, which is a periodic function of the 
time. The disturbing force at a point whose abscissa, measured 
along the canal from a fixed point, is x, is supposed to be expressed 
by a function of the form A sin (nt mx -f- a). This supposition is 
sufficiently general for the case of the tides, provided the canal on 
the earth be supposed circular. In all cases the disturbing force 
will give rise to an oscillation in the water having the same period 
as the force itself. This oscillation is called by Mr Airy a forced 
wave. It will be sufficient here to mention some of the results of 
this theory as applied to the case of the earth. 

In all cases the expression for the tidal elevation contains as a 
denominator the difference of the squares of two velocities, one 
the velocity of propagation of a free wave along the canal, the 
other the velocity with which a particular phase of the disturbing 
force travels along the canal, or, which is the same, the velocity of 
propagation of the forced wave. Hence the height of the tides 
will not depend simply on the magnitude of the disturbing force, 
but also on its period. Thus the mass of the moon cannot be in 
ferred directly from the comparison of spring and neap tides, since 
the heights of the solar and lunar tides are affected by the different 
motions of the sun arid moon in right ascension, and consequently 
in hour-angle. When the canal under consideration is equatorial 
the diurnal tide vanishes. The height of high water is the same 
at all points of the canal, and there is either high or low water at 
the point of the canal nearest to the attracting body, according as 
the depth of the water is greater or less than that for which a 
free wave would be propagated with the same velocity as the 
forced wave. In the general case there is both a diurnal and a 
semidiurnal tide, and the height of high water, as well as the 
interval between the transit of the attracting body over the meri 
dian of the place considered and the time of high water, is different 
at different points of the canal. When the canal is a great circle 
passing through the poles, the tide-wave is a stationary wave. 



REPORT ON RECENT RESEARCHES IX HYDRODYNAMICS. 175 

When the coefficient of the disturbing force is supposed to vary 
slowly, in consequence of the change in declination, &c. of the 
disturbing body, it is found that the greatest tide occurs on the 
day on which the disturbing force is the greatest. 

The preceding results have been obtained on the supposition 
of the absence of all friction ; but Mr Airy also takes friction into 
consideration. He supposes it to be represented by a horizontal 
force, acting uniformly from top to bottom of the water, and vary 
ing as the first power of the horizontal velocity. Of course this 
supposition is not exact : still there can be no doubt that it 
represents generally the effect of friction. When friction is taken 
into account, the denominator of the expressions for the tidal 
elevation is essentially positive, so that the motion can never 
become infinite. In the case of a uniform tidal river stopped by 
a barrier, the high wa,ter is no longer simultaneous at all points, 
but the phase of high water always travels up the river. But of 
all the results obtained by considering friction, the most important 
appears to be, that when the slow variation of the disturbing 
force is taken into account, the greatest tide, instead of happening 
on the day when the disturbing force is greatest, will happen later 
by a certain time p v Moreover, in calculating the tides, we must 
use, not the relative positions of the sun and moon for the instant 
for which the tide is calculated, but their relative positions for a 
time earlier by the same interval p l as in the preceding case. The 
expression for p l depends both on the depth of the canal and on 
the period of the tide, and therefore its value for the diurnal tide 
cannot be inferred from its value for the semidiurnal. It appears 
also that the phase of the tide is accelerated by friction. 

The mechanical theory of the tides of course belongs to hydro 
dynamics; but I do not conceive that the consideration of the 
reduction and discussion of tidal observations falls within the 
province of this report. 

Before leaving the investigations of Mr Airy, I would call at 
tention to a method which he sometimes employs very happily in 
giving a general explanation of phenomena depending on motions 
which are too complicated to admit of accurate calculation. It is 
evident that any arbitrary motion may be assigned to a fluid, 
(with certain restrictions as to the absence of abruptness,) provided 
we suppose certain forces to act so as to produce them. The 
values of these forces are given by the equations of motion. In 



176 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

some cases the forces thus obtained will closely resemble some 
known forces ; while in others it will be possible to form a clear 
conception of the kind of motion which must take place in the 
absence of such forces. For example, supposing that there is pro 
pagated a series of oscillatory waves of the standard kind, except 
that the height of the waves increases proportionably to their 
distance from a fixed line, remaining constant at the same point 
as the time varies, Mr Airy finds for the force requisite to maintain 
such a motion an expression which may be assimilated to the force 
which wind exerts on water. This affords a general explanation 
of the increase in the height of the waves in passing from a wind 
ward to a lee shore*. Again, by supposing a series of waves, as 
near the standard kind as circumstances will admit, to be pro 
pagated along a canal whose depth decreases slowly, and examin 
ing the force requisite to maintain this motion, he finds that a 
force must be applied to hold back the heads of the waves. In 
the absence, then, of such a force the heads of the waves will have 
a tendency to shoot forwards. This explains the tendency of waves 
to break over a sunken shoal or along a sloping beach~f*. The 
word tendency is here used, because when a wave comes at all 
near breaking, but little reliance can be placed in any investigation 
which depends upon the supposition of the motion being small. 
To take one more example of the application of this method, by 
supposing a wave to travel, unchanged in form, along a canal, with 
a velocity different from that of a free wave, and examining the 
force requisite to maintain such a motion, Mr Airy is enabled to 
give a general explanation of some very curious circumstances 
connected with the motion of canal boats J, which have been ob 
served by Mr Russell. 

III. In the 16th volume of the Journal de 1 Ecole Polytech- 
nique , will be found a memoir by MM. Barre* de Saint- Venant 
and Wantzel, containing the results of some experiments on the 
discharge of air through small orifices, produced by considerable 
differences of pressure. The formula for the velocity of efflux 
derived from the theory of steady motion, and the supposition 
that the mean pressure at the orifice is equal to the pressure at a 
distance from the orifice in the space into which the discharge 

* Art. 265, &e. + Art. 238, &c. 

+ Art. 405, &c. Cahier xxvii. p. 85. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 177 

takes place, leads to some strange results of such a nature as to 
make us doubt its correctness. If we call the space from which 
the discharge takes place i\iQ first space, and that into which it 
takes place the second space, and understand by the term reduced 
velocity the velocity of efflux diminished in the ratio of the density 
in the second space to the density in the first, so that the reduced 
velocity measures the rate of discharge, provided the density in 
the first space remain constant, it follows from the common for 
mula that the reduced velocity vanishes when the density in the 
second space vanishes, so that a gas cannot be discharged into a 
vacuum. Moreover, if the density of the first space is given, the 
reduced velocity is a maximum when the density in the second 
space is rather more than half that in the first. The results 
remain the same if we take account of the contraction of the 
vein, and they are not materially altered if we take into account 
the cooling of the air by its rapid dilatation. The experiments 
above alluded to were made by allowing the air to enter an ex 
hausted receiver through a small orifice, and observing simul 
taneously the pressure and temperature of the air in the receiver, 
and the time elapsed since the opening of the orifice. It was 
found that when the exhaustion was complete the reduced velocity 
had a certain value, depending on the orifice employed, and that 
the velocity did not sensibly change till the pressure of the air in 
the receiver became equal to about Jths of the atmospheric pres 
sure. The reduced velocity then began to decrease, and finally 
vanished when the pressure of the air in the receiver became 
equal to the atmospheric pressure. 

These experiments shew that when the difference of pressure 
in the first and second spaces is considerable, we can by no means 
suppose that the mean pressure at the orifice is equal to the 
pressure at a distance in the second space, nor even that there 
exists a contracted vein, at which we may suppose the pressure to 
be the same as at a distance. The authors have given an empiri 
cal formula, which represents very nearly the reduced velocity, 
whatever be the pressure of the air in the space into which the 
discharge takes place. 

The orifices used in these experiments were generally about 

one millimetre in diameter. It was found that widening the 

mouth of the orifice, so as to make it funnel-shaped, produced a 

much greater proportionate increase of velocity when the velocity 

S. 12 



178 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

of efflux was small than when it was large. The authors have 
since repeated their experiments with air coming from a vessel in 
which the pressure was four atmospheres: they have also tried 
the effect of using larger orifices of four or five millimetres 
diameter. The general results were found to be the same as 
before*. 

IV. In the 6th volume of the Transactions of the Cambridge 
Philosophical Society, p. 403, will be found a memoir by Mr Green 
on the reflection and refraction of sound, which is well worthy of 
attention. This problem had been previously considered by Pois- 
son in an elaborate memoir -f\ Poisson treats the subject with 
extreme generality, and his analysis is consequently very compli 
cated. Mr Green, on the contrary, restricts himself to the case of 
plane waves, a case evidently comprising nearly all the phenomena 
connected with this subject which are of interest in a physical 
point of view, and thus is enabled to obtain his results by a very 
simple analysis. Indeed Mr Green s memoirs are very remarkable, 
both for the elegance and rigour of the analysis, and for the ease 
with which he arrives at most important results. This arises in a 
great measure from his divesting the problems he considers of all 
unnecessary generality: where generality is really of importance 
he does not shrink from it. In the present instance there is one 
important respect in which Mr Green s investigation is more general 
than Poisson s, which is, that Mr Green has taken the case of any 
two fluids, whereas Poisson considered the case of two elastic fluids, 
in which equal condensations produce equal increments of pressure. 
It is curious, that Poisson, forgetting this restriction, applied his 
formulae to the case of air and water. Of course his numerical 
result is altogether erroneous. Mr Green easily arrives at the 
ordinary laws of reflection and refraction. He obtains also a very 
simple expression for the intensity of the reflected sound. If A is 
the ratio of the density of the second medium to that of the first, 
and B the ratio of the cotangent of the angle of refraction to the 
cotangent of the angle of incidence, then the intensity of the 
reflected sound is to the intensity of the incident as A B to 
A + B. In this statement the intensity is supposed to be mea 
sured by the first power of the maximum displacement. When 

* Comptes Rendus, torn. xvii. p. 1140. 

t Memoires de VAcadtmie des Sciences, torn. x. p. 317. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 179 

the velocity of propagation in the first medium is less than in the 
second, and the angle of incidence exceeds what may be called the 
critical angle, Mr Green restricts himself to the case of vibrations 
following the cycloidal law. He finds that the sound suffers total 
internal reflection. The expression for the disturbance in the 
second medium involves an exponential with a negative index, 
and consequently the disturbance becomes quite insensible at a 
distance from the surface equal to a small multiple of the length 
of a wave. The phase of vibration of the reflected sound is also 
accelerated by a quantity depending on the angle of incidence. 
It is remarkable, that when the fluids considered are ordinary 
elastic fluids, or rather when they are such that equal condensa 
tions produce equal increments of pressure, the expressions for 
the intensity of the reflected sound, and for the acceleration of 
phase when the angle of incidence exceeds the critical angle, are 
the same as those given by Fresnel for light polarized in a plane 
perpendicular to the plane of incidence. 

V. Not long after the publication of Poisson s memoir on the 
simultaneous motions of a pendulum and of the surrounding air*, 
a paper by Mr Green was read before the Royal Society of Edin 
burgh, which is entitled Researches on the Vibration of Pendulums 
in Fluid Media [. Mr Green does not appear to have been at that 
time acquainted with Poisson s memoir. The problem which he 
has considered is one of the same class as that treated by Poisson. 
Mr Green has supposed the fluid to be incompressible, a suppo 
sition, however, which will apply without sensible error to air, in 
considering motions of this sort. Poisson regarded the fluid as 
elastic, but in the end, in adapting his formula to use, he has 
neglected as insensible the terms by which the effect of an elastic 
differs from that of an inelastic fluid. The problem considered by 
Mr Green is, however, in one respect much more general than 
that solved by Poisson, since Mr Green has supposed the oscil 
lating body to be an ellipsoid, whereas Poisson considered only a 
sphere. Mr Green has obtained a complete solution of the pro 
blem in the case in which the ellipsoid has a motion of translation 
only, or in which the small motion of the fluid due to its motion 

* M&moires de V Academic des Sciences, torn. xi. p. 521. 

t This paper was read in December, 1833, and is printed in the 13th volume of 
the Society s Transactions, p. 54, &c. 

122 



180 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

of rotation is neglected. The result is that the resistance of the 
fluid will be allowed for if we suppose the mass of the ellipsoid 
increased by a mass bearing a certain ratio to that of the fluid 
displaced. In the general case this ratio depends on three trans 
cendental quantities, given by definite integrals. If, however, 
the ellipsoid oscillates in the direction of one of its principal axes, 
the ratio depends on one only of these transcendents. When the 
ellipsoid passes into a spheroid, the transcendents above mentioned 
can be expressed by means of circular or logarithmic functions. 
When the spheroid becomes a sphere, Mr Green s result agrees 
with Poisson s. It is worthy of remark, that Mr Green s formula 
will enable us to calculate the motion of an ellipse or circle oscil 
lating in a fluid, in a direction perpendicular to its plane, since a 
material ellipse or circle may be considered as a limiting form of 
an ellipsoid. In this case, however, the motion would probably 
have to be extremely small, in order that the formula should apply 
with accuracy. 

In a paper On the Motion of a small Sphere acted on by the 
Vibrations of an Elastic Medium, read before the Cambridge 
Philosophical Society in April 1841*, Prof. Challis has considered 
the motion of a ball pendulum, retaining in his solution small 
quantities to the second order. The principles adopted by Prof. 
Challis in the solution of this problem are at variance with those 
of Poisson, and have given rise to a controversy between him and 
Mr Airy, which will be found in the 17th, 18th, and 19 volumes 
of the Philosophical Magazine (New Series). In the paper just 
referred to, Prof. Challis finds that when the fluid is incompressible 
there is no decrement in the arc of oscillation, except what arises 
from friction and capillary attraction. In the case of air there is 
a slight theoretical decrement ; but it is so small that Prof. Challis 
considers the observed decrement to be mainly owing to friction. 
This result follows also from Poisson s solution. Prof. Challis also 
finds that a small sphere moving with a uniform velocity experi 
ences no resistance, and that when the velocity is partly uniform 
and partly variable, the resistance depends on the variable part 
only. The problem, however, referred to in the title of this paper, 
is that of calculating the motion of a small sphere situated in an 
elastic fluid, and acted on by no forces except the pressure of the 

* Transactions of the Cambridge Philosophical Society, vol. vii. p. 333. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 181 

fluid, in which an indefinite series of plane condensing and rarefy 
ing waves is supposed to be propagated. This problem is solved 
by the author on principles similar to those which he has adopted 
in the problem of an oscillating sphere. The views of Prof. Challis 
with respect to this problem, which he considers a very important 
one, are briefly stated at the end of a paper published in the 
Philosophical Magazine*. 

In a paper On some Cases of Fluid Motion, published in the 
Transactions of the Cambridge Philosophical Society^, I have 
considered some modifications of the problem of the ball pendu 
lum, adopting in the main the principles of Poisson, of the 
correctness of which I feel fully satisfied, but supposing the fluid 
incompressible from the first. In this paper the effect of a distant 
rigid plane interrupting the fluid in which the sphere is oscillating 
is given to the lowest order of approximation with which the 
effect is sensible. It is shewn also that when the ball oscillates 
in a concentric spherical envelope, the effect of the resistance of 
the fluid is to add to the mass of the sphere a mass equal to 

b* + 2a* m 
~F^ 2 

where a is the radius of the ball, b that of the envelope, and m 
the mass of the fluid displaced. Poisson, having reasoned on the 
very complicated case of an elastic fluid, had come to the con 
clusion that the envelope would have no effect. 

One other instance of fluid motion contained in this paper will 
here be mentioned, because it seems to afford an accurate means 
of comparing theory and experiment in a class of motions in 
which they have not hitherto been compared, so far as I am 
aware. When a box of the form of a rectangular parallelepiped, 
filled with fluid and closed on all sides, is made to perform small 
oscillations, it appears that the motion of the box will be the. 
same as if the fluid were replaced by a solid having the same 
mass, centre of gravity, and principal axes as the solidified fluid 
but different principal moments of inertia. These moments are 
given by infinite series, which converge with extreme rapidity, so 
that the numerical calculation is very easy. The oscillations most 
convenient to employ would probably be either oscillations by 
torsion, or bifilar oscillations. 

* Vol. xviii., New Series, p. 481. t Vol. viii. p. 105. 



182 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

VI. M. Navier was, I believe, the first to give equations for 
the motion of fluids without supposing the pressure equal in all 
directions. His theory is contained in a memoir read before 
the French Academy in 1822*. He considers the case of a 
homogeneous incompressible fluid. He supposes such a fluid 
to be made up of ultimate molecules, acting on each other by 
forces which, when the molecules are at rest, are functions simply 
of the distance, but which, when the molecules recede from, or 
approach to each other, are modified by this circumstance, so 
that two molecules repel each other less strongly when they are 
receding, and more strongly when they are approaching, than 
they do when they are at rest")*. The alteration of attraction or 
repulsion is supposed to be, for a given distance, proportional to 
the velocity with which the molecules recede from, or approach 
to each other; so that the mutual repulsion of two molecules 
will be represented by f(r) VF(r), where r is the distance of 
the molecules, V the velocity with which they recede from each 
other, and f(r), F(r) two unknown functions of r depending on 
the molecular force, and as such becoming insensible when r 
has become sensible. This expression does not suppose the 
molecules to be necessarily receding from each other, nor their 
mutual action to be necessarily repulsive, since V and F (r) may 
be positive or negative. It is not absolutely necessary that f(r) 
and F (r) should always have the same sign. In forming the 
equations of motion M. Navier adopts the hypothesis of a sym 
metrical arrangement of the particles, or at least, which leads 
to the same result, neglects the irregular part <of the mutual 
action of neighbouring molecules. The equations at which he 
arrives are those which would be obtained from the common 

dp 4 (d"u d*u d*u\ . , f dp . 

equations by wntmg ~A ^ + ^ + -^ m place of ^ m 

the first, and making similar changes in the second and third. 
A is here an unknown constant depending on the nature of the 
fluid. 

The same subject has been treated on by PoissonJ, who has 
adopted hypotheses which are very different from those of M. 

* Memoires de V Academic des Sciences, torn. vi. p. 389. 

t This idea appears to have been borrowed from Dubuat. See his Principes 
d Hydraulique, torn. ii. p. 60. 

J Journal de VEcole Poly technique, torn. xiii. cah. 20, p. 139. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 183 

Navier. Poisson s theory is of this nature. He supposes the 
time t to be divided into n equal parts, each equal to r. In 
the first of these he supposes the fluid to be displaced in the same 
manner as an elastic solid, so that the pressures in different 
directions are given by the equations which he had previously 
obtained for elastic solids. If the causes producing the dis 
placement were now to cease to act, the molecules would very 
rapidly assume a new arrangement, which would render the 
pressure equal in all directions, and while this re-arrangement 
was going on, the pressure would alter in an unknown manner 
from that belonging to a displaced elastic solid to the pressure 
belonging to the fluid in its new state. The causes of dis 
placement are however going on during the second interval r; 
but since these different small motions will take place inde 
pendently, the new displacement which will take place in the 
second interval r will be the same as if the molecules were not 
undergoing a re-arrangement. Supposing now n to become in 
finite, we pass to the case in which the fluid is continually be 
ginning to be displaced like an elastic solid, and continually 
re-arranging itself so as to make the pressure equal in all direc 
tions. The equations at which Poisson arrived are, in the cases 
of a homogeneous incompressible fluid, and of an elastic fluid 
in which the change of density is small, those which would be 
derived from the common equations by replacing dp/dx in the 
first by 

dp , fd\i d*u d*u\ p d fdu dv dw\ 
dx \dx* dy* dz 2 J dx \dx dy dzj 

and making similar changes in the second and third. In these 
equations A and B are two unknown constants. It will be 
observed that Poisson s equations reduce themselves to Navier s 
in the case of an incompressible fluid. 

The same subject has been considered in a quite different 
point of view by M. Barre de Saint- Venant, in a communication 
to the French Academy in 1843, an abstract of which is contained 
in the Comptes Rendus*. The principal difficulty is to connect 
the oblique pressures in different directions about the same point 
with the differential coefficients dujdx, du/dy, &c., which express 
the relative motion of the fluid particles in the immediate neigh- 

* Tom. xvii. p. 1240. 



184 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

bourhood of that point. This the author accomplishes by as 
suming that the tangential force on any plane passing through 
the point in question is in the direction of the principal sliding 
(glissement] along that plane. The sliding along the plane xy 

, , dw du . , ,. , dw dv . 

is measured by -j h j- in the direction of x, and -j- + -y- in the 
7 ax dz dy dz 

direction of y. These two slidings may be compounded into one, 
which will form the principal sliding along the plane xy. It 
is then shewn, by means of M. Cauchy s theorems connecting 
the pressures in different directions in any medium, that the 
tangential force on any plane passing through the point considered, 
resolved in any direction in that plane, is proportional to the 
sliding along that plane resolved in the same direction, so that 
if T represents the tangential force, referred to a unit of surface, 
and 8 the sliding, T=eS. The pressure on a plane in any direc 
tion is then found. This pressure is compounded of a normal 
pressure, alike in all directions, and a variable oblique pressure, 
the expression for which contains the one unknown quantity e. 
If the fluid be supposed incompressible, and e constant, the 
equations which would be obtained by the method of M. Barre 
de Saint- Venant agree with those of M. Navier. It will be 
observed that this method does not require the consideration of 
ultimate molecules at all. 

When the motion of the fluid is very small, Poisson s equations 
agree with those given by M. Cauchy for the motion of a solid 
entirely destitute of elasticity*, except that the latter do not 
contain the pressure p. These equations have been obtained 
by M. Cauchy without the consideration of molecules. His 
method would apply, with very little change, to the case of 
fluids. 

In a paper read last year before the Cambridge Philosophical 
Society "f 1 , I have arrived at the equations of motion in a different 
manner. The method employed in this paper does not neces 
sarily require the consideration of ultimate molecules. Its prin 
cipal feature consists in eliminating from the relative motion 
of the fluid about any particular point the relative motion which 
corresponds to a certain motion of rotation, and examining the 
nature of the relative motion which remains. The equations 

* Exercices de Mathematiques, torn. iii. p. 187. 

f- Transactions of the Cambridge Philosophical Society, vol. viii. p. 287. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 185 

finally adopted in the cases of a homogeneous incompressible 
fluid, and of an elastic fluid in which the change of density is 
small, agree with those of Poisson, provided we suppose in the 
latter A ^B. It is shewn that this relation between A and B 
may be obtained on Poisson s own principles. 

The equations hitherto considered are those which must be 
satisfied at any point in the interior of the fluid mass ; but there 
is hardly any instance of the practical application of the equations, 
in which we do not want to know also the particular conditions 
which must be satisfied at the surface of the fluid. With respect 
to a free surface there can be little doubt : the condition is simply 
that there shall be no tangential force on a plane parallel to the 
surface, taken immediately within the fluid. As to the case 
of a fluid in contact with a solid, the condition at which Navier 
arrived comes to this : that if we conceive a small plane drawn 
within the fluid parallel to the surface of the solid, the tangential 
force on this plane, referred to a unit of surface, shall be in the 
same direction with, and proportional to the velocity with which 
the fluid flows past the surface of the solid. The condition ob 
tained by Poisson is essentially the same. 

Dubuat stated, as a result of his experiments, that when the 
velocity of water flowing through a pipe is less than a certain 
quantity, the water adjacent to the surface of the pipe is at rest*. 
This result agrees very well with an experiment of Coulomb s. 
Coulomb found that when a metallic disc was made to oscillate 
very slowly in water about an axis passing through its centre 
and perpendicular to its plane, the resistance was not altered 
when the disc was smeared with grease; and even when the 
grease was covered with powdered sandstone the resistance was 
hardly increased f. This is just what one would expect on the 
supposition that the water close to the disc is carried along with 
it, since in that case the resistance must depend on the internal 
friction of the fluid ; but the result appears very extraordinary on 
the supposition that the fluid in contact with the disc flows 
past it with a finite velocity. It should be observed, however, 
that this result is compatible with the supposition that a thin 
film of fluid remains adhering to the disc, in consequence of 
capillary attraction, and becomes as it were solid, and that the 

* See the Table given in torn. i. of his Principes d Hydr antique, p. 93. 
t Memoires de VInstitut, 1801, torn. iii. p. 286. 



186 REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 

fluid in contact with this film flows past it with a finite velocity. 
If we consider Dubuat s supposition to be correct, the condition 
to be assumed in the case of a fluid in contact with a solid is 
that the fluid does not move relatively to the solid. This con 
dition will be included in M. Navier s, if we suppose the coefficient 
of the velocity when M. Navier s condition is expressed analy 
tically, which he denotes by E, to become infinite. It seems 
probable from the experiments of M. Girard, that the condition to 
be satisfied at the surface of fluid in contact with a solid is 
different according as the fluid does or does not moisten the 
surface of the solid. 

M. Navier has applied his theory to the results of some ex 
periments of M. Girard s on the discharge of fluids through 
capillary tubes. His theory shews that if we suppose E to be 
finite, the discharge through extremely small tubes will depend 
only on E, and not on A. The law of discharge at which he 
arrives agrees with the experiments of M. Girard, at least when 
the tubes are extremely small. M. Navier explained the differ 
ence observed by M. Girard in the discharge of water through 
tubes of glass and tubes of copper of the same size by supposing 
the value of E different in the two cases. This difference was 
explained by M. Girard himself by supposing that a thin film 
of fluid remains adherent to the pipe, in consequence of molecular 
action, and that the thickness of this film differs with the sub 
stance of which the tube is composed, as well as with the liquid 
employed*. If we adopt Navier s explanation, we may reconcile 
it with the experiments of Coulomb by supposing that E is very 
large, so that unless the fluid is confined in a very narrow pipe, 
the results will depend mainly on A, being sensibly the same as 
they would be if E were infinite. 

There is one circumstance connected with the motion of a 
ball-pendulum oscillating in air, which has not yet been ac 
counted for, the explanation of which seems to depend on this 
theory. It is found by experiment that the correction for the 
inertia of the air is greater for small than for large spheres, 
that is to say, the mass which we must suppose added to that 
of the sphere bears a greater ratio to the mass of the fluid dis 
placed in the former, than in the latter case. According to the 
common theory of fluid motion, in which everything is supposed 

* M6moires de VAcademie des Sciences, torn i. pp. 203 and 234. 



REPORT ON RECENT RESEARCHES IN HYDRODYNAMICS. 187 

to be perfectly smooth, the ratio ought to be independent of the 
magnitude of the sphere. In the imperfect theory of friction in 
which the friction of the fluid on the sphere is taken into account, 
while the equal and opposite friction of the sphere on the fluid is 
neglected, it is shewn that the arc of oscillation is diminished, 
while the time of oscillation is sensibly the same as before. But 
when the tangential action of the sphere on the fluid, and the 
internal friction of the fluid itself are considered, it is clear that 
one consequence will be, to speak in a general way, that a portion 
of the fluid will be dragged along with the sphere. Thus the 
correction for the inertia of the fluid will be increased, since the 
same moving force has now to overcome the inertia of the fluid 
dragged along with the sphere, and not only, as in the former 
case, the inertia of the sphere itself, and of the fluid pushed away 
from before it, and drawn in behind it. Moreover the additional 
correction for inertia must depend, speaking approximately, on 
the surface of the sphere, whereas the first correction depended on 
its volume, and thus the effect of friction in altering the time of 
oscillation will be more conspicuous in the case of small, than in 
the case of large spheres, other circumstances being the same. 
The correction for inertia, when friction is taken into account, will 
not, however, depend solely on the magnitude of the sphere, but 
also on the time of oscillation. With a given sphere it will be 
greater for long, than for short oscillations. 



[From the Transactions of the Cambridge Philosophical Society, Vol. VIII. 

p. 409.] 

SUPPLEMENT TO A MEMOIR ON SOME CASES OF FLUID 
MOTION. 

Eead Nov. 3, 1846. 

IN a memoir which the Society did me the honour to publish 
in their Transactions*, I shewed that when a box whose interior 
is of the form of a rectangular parallelepiped is filled with fluid 
and made to perform small oscillations the motion of the box 
will be the same as if the fluid were replaced by a solid having 
the same mass, centre of gravity, and principal axes as the 
solidified fluid, but different moments of inertia about those axes. 
The box is supposed to be closed on all sides, and it is also 
supposed that the box itself and the fluid within it were both 
at rest at the beginning of the motion. The investigation was 
founded upon the ordinary equations of Hydrodynamics, which 
depend upon the hypothesis of the absence of any tangential 
force exerted between two adjacent portions of a fluid in motion, 
an hypothesis which entails as a necessary consequence the 
equality of pressure in all directions. The particular case of 
motion under consideration appears to be of some importance, 
because it affords an accurate means of comparing with experiment 
the common theory of fluid motion, which depends upon the 
hypothesis just mentioned. In my former paper, I gave a series 
by means of which the numerical values of the principal moments 
of the solid which may be substituted for the fluid might be 
calculated with facility. The present supplement contains a 
different series for the same purpose, which is more easy of 
numerical calculation than the former. The comparison of the 

* Vol. YIII. Part i. p. 105. (Ante, p. 17.) 



ON SOME CASES OF FLUID MOTION. 189 

two series may also be of some interest in an analytical point 
of view, since they appear under very different forms. I have 
taken the present opportunity of mentioning the results of some 
experiments which I have performed on the oscillations of a box, 
such as that under consideration. The experiments were not 
performed with sufficient accuracy to entitle them to be described 
in detail. 

The calculation of the motion of fluid in a rectangular box 
is given in the 13th article of my former paper. I shall not 
however in the first instance restrict myself to a rectangular 
parallelepiped, since the simplification which I am about to give 
applies more generally. Suppose then the problem to be solved 
to be the following. A vessel whose interior surface is composed 
of any cylindrical surface and of two planes perpendicular to the 
generating lines of the cylinder is filled with a homogeneous, 
incompressible fluid ; the vessel and the fluid within it having 
been at first at rest, the former is then moved in any manner ; 
required to determine the motion of the fluid at any instant, 
supposing that at that instant the vessel has no motion of rotation 
about an axis parallel to the generating lines of the cylinder. 

I shall adopt the notation of my former paper, u, v, w are 
the resolved parts of the velocity at any point along the rect 
angular axes of x, y, z. Since the motion begins from rest we 
shall have udx + vdy -f wdz an exact differential d$. Let the 
rectangular axes to which the fluid is referred be fixed relatively 
to the vessel, and let the axis of x be parallel to the generating 
lines of the cylindrical surface. The instantaneous motion of 
the vessel may be decomposed into a motion of translation, and 
two motions of rotation about the axes of y and z respectively ; 
for by hypothesis there is no motion of rotation about the axis 
of x. According to the principles of my former paper, the in 
stantaneous motion of the fluid will be the same as if it had 
been produced directly by impact, the impact being such as 
to give the vessel the velocity which it has at the instant con 
sidered. We may also consider separately the motion of trans 
lation of the vessel, and each of the motions of rotation ; the 
actual motion of the fluid will be compounded of those which 
correspond to each of the separate motions of the vessel. For 
my present purpose it will be sufficient to consider one of the 



190 . SUPPLEMENT TO A MEMOIR 

motions of rotation, that which takes place round the axis of 
z for instance. Let co be the angular velocity about the axis 
of z, co being considered positive when the vessel turns from 
the axis of x to that of y. It is easy to see that the instantaneous 
motion of the cylindrical surface is such as not to alter the volume 
of the interior of the vessel, supposing the plane ends fixed, 
and that the same is true of the instantaneous motion of the 
ends. Consequently we may consider separately the motion of 
the fluid due to the motion of the cylindrical surface, and to that 
of the ends. Let cf) c be the part of < due to the motion of the 
cylindrical surface, <j) e the part due to the motion of the ends. 
Then we shall have 

*=*.+*. .......................... a)- 

Consider now the motion corresponding to a value of <, wxy. 
It will be observed that wxy satisfies the equation, {(36) of my 
former paper,} which (f> is to satisfy. Corresponding to this value 
of </> we have 

u wy, v = cox, w = 0. 

Hence the velocity, corresponding to this motion, of a particle 
of fluid in contact with the cylindrical surface of the vessel, 
resolved in a direction perpendicular to the surface, is the same 
as the velocity of the surface itself resolved in the same direction, 
and therefore the fluid does not penetrate into, nor separate 
from the cylindrical surface. The velocity of a particle in contact 
with either of the plane ends, resolved in a direction perpendicular 
to the surface, is equal and opposite to the velocity of the surface 
itself resolved in the same direction. Hence we shall get the 
complete value of </> by adding the part already found, namely 
a>xy, to twice the part due to the motion of the plane ends. We 
have therefore, 

</> = nay + 2<^ = 2( c - vxy, by (1) ........... (2), 

and $ c $ e = axy ............................. (3). 



Hence whenever either <j> c or <f) e can be found, the complete 
solution of the problem will be given by (2). And even when 
both these functions can be obtained independently, (2) will 
enable us to dispense with the use of one of them, and (3) will 
give a relation between them. In this case (3) will express a 
theorem in pure analysis, a theorem which will sometimes be 



ON SOME CASES OF FLUID MOTION. 191 

very curious, since the analytical expressions for <f> c and </><, will 
generally be totally different in form. The problem admits of 
solution in the case of a circular cylinder terminated by planes 
perpendicular to its axis, and in the case of a rectangular paral 
lelepiped. In the former case, the numerical calculation of the 
moments of inertia of the solid by which the fluid may be re 
placed would probably be troublesome, in the latter it is extremely 
easy. I proceed to consider this case in particular. 

Let the rectangular axes to which the fluid is referred coincide 
with three adjacent edges of the parallelepiped, and let a, 6, c 
be the lengths of the edges. The motion which it is proposed 
to calculate is that which arises from a motion of rotation of the 
box about an axis parallel to that of z and passing through the 
centre of the parallelepiped. Consequently in applying (2) we 
must for a moment conceive the axis of z to pass through the 
centre of the parallelepiped, and then transfer the origin to the 
corner, and we must therefore write co (x -Ja) (y J b) for wxy. 
In the present case the cylindrical surface consists of the four 
faces which are parallel to the axis of x, and the remaining faces 
form the plane ends. The motion of the face xy and the opposite 
face has evidently no effect on the fluid, so that <j) G will be the 
part of cf> due to the motion of the face xz and the opposite face. 
The value of this quantity is given near the middle of page 62 in my 
former paper. We have then by the second of the formulae (2) 



(e ~ n7rb/a - 

~ COS 



the sign S denoting the sum corresponding to all odd integral 
values of n from 1 to oo . This value of expresses completely 
the motion of the fluid due to a motion of rotation of the box 
about an axis parallel to that of z, and passing through the centre 
of its interior. 

Suppose now the motion to be very small, so that the square 
of the velocity may be neglected. Then, p denoting the part of 
the pressure due to the motion, we shall have p = p d<f>/dt. 
Also in finding d$/dt we may suppose the axes to be fixed in 



192 SUPPLEMENT TO A MEMOIR 

space, since by taking account of their motion we should only 
introduce terms depending on the square of the velocity. In fact, 
if for the sake of distinction we denote the co-ordinates of a 
fluid particle referred to the moveable axes by x, y, while a?, y 
denote its co-ordinates referred to axes fixed in space, which 
after differentiation with respect to t we may suppose to coincide 
with the moveable axes at the instant considered, and if we 
denote the differential coefficient of </> with respect to t by (d(f>/dt) 
when x, y, t are the independent variables, and by d(f)/dt when 
x , y, t are the independent variables, we shall have 

(d(f)\ dcf> d(f> dx d(f> dy d(f> dx rfy * 
\dtj dt dx dt dy dt ~ dt dt dt 

for d<l>/dx, dcf)/dy mean absolutely the same as d<f>/dx, dcfr/dy, and 
are therefore equal to u, v respectively. Now dx/dt, dy /dt, de 
pending on the motion of the axes, are small quantities of the 
order co ; their values are in fact coy, cox ; so that, omitting 
small quantities of the order &> 2 , we have 



"dt 

We shall therefore find the value of p from that of </> by merely 
writing pdco/dt for o>. In order to determine the motion of 
the box it will be necessary to find the resultant of the fluid 
pressures on its several faces. As shewn in my former paper, 
these pressures will have no resultant force, but only a resultant 
couple, of which the axis will evidently be parallel to that of z. 
In calculating this couple, it is immaterial whether we take the 
moments about the axis of z, or about a line parallel to it passing 
through the centre of the parallelepiped : suppose that we adopt 
the latter plan. If we reckon the couple positive when it tends 
to turn the box from the axis of x to that of y we shall evidently 

have I I p y =Q ( x - ) dxdz for the part arising from the 

J o Jo \ 2t) 

* It may be very easily proved by means of this equation, combined with the 
general equation which determines^, that whether the velocity be great or small 
the fluid will have the same effect on the motion of the box as the solid of which the 
moment of inertia is determined in this paper on the supposition that the motion 
is small. 



ON SOME CASES OF FLUID MOTION. 193 

rb re / i\ 

pressure on the face xz, and p 9 -*[y - 5 J dydz for the part 

Jo J o \ &/ 

arising from the pressure on the face ya. It is easily seen from 
(4) that^ =a = -^ a . =0 , and py= b = -p y = , so that the couples due 
to the pressures on the faces xz t yz are equal to the couples due 
to the pressures on the opposite faces respectively. In order, 
therefore, to find the whole couple we have only got to double 
the part already found. As the integrations do not present the 
slightest difficulty, it will be sufficient to write down the result. 
It will be found that the whole couple is equal to Cdw/dt, 
where 



This expression has been simplified after integration by putting 
for S 1/n* its value 7r 4 /96. 

It appears then that the effect of the inertia of the fluid is 
to increase the moment of inertia of the box about an axis passing 
through its centre and parallel to the edge c by the quantity C. 
In equation (40) of my former paper, there is given an expression 
for C which is apparently very different from that given by (5), 
but the numerical values of the two expressions are necessarily 
the same. If we denote the moment of inertia of the fluid sup 
posed to be solidified by C,, we shall have C t = pabc (a 2 + 6 2 )/12 ; 
and if we put 



and treat (5) as equation (40) of my former paper was treated we 
shall find 

f(r) = (1 + r 2 )- 1 {1 - 3r 2 + 2r 3 (1.260497 - 1.254821 2 1 versin 26> n )} 

fi 

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

where tab. log tan 6 n = 10 - .6821882 - . 






The equation (6) is true, (except as regards the decimals 
omitted,) whatever be the value of r; but for convenience of 
calculation it will be proper to take r less than 1, that is, to 
choose for a the smaller of the two a, b. The value of/(r) given 
by (6) is apparently very different from that given at the bottom 

s - 13 



194 



SUPPLEMENT TO A MEMOIR 



of page 64 of my former paper, but any one may easily satisfy 
himself as to equivalence of the two expressions by assigning 
to r a value at random, and calculating the value of f(r) from 
the two expressions separately. The expression (6) is however 
preferable to the other, especially when we have to calculate the 
value of f (r) for small values of r. The infinite series contained 
in (6) converges with such rapidity that in the most unfavourable 
case, that is, when r = 1 nearly, the omission of all terms after the 
first would only introduce an error of about .000003 in the value 
of/(r). 

For the sake of shewing the manner in which f (r) alters 
with r, I have calculated the following values of the function. 
The expression (6) shews that / (r) = 0, when r = 0; and f (r) 

is also =0 when r l, since /(-] =f(r). 



r 


f(r) 


? 


f(r] 


0.0 


1 


0.6 


0.3374 


0.1 


0.9629 


0.7 


0.2521 


0.2 


0.8655 


0.8 


0.1958 


0.3 


0.7322 


0.9 


0.1655 


0.4 


0.5873 


1 


0.1565 


0.5 


0.4512 







The experiments to which I have alluded were made with a 
wooden box measuring inside 8 inches by 4 square. The box 
weighed not quite 1 lb., and contained about 4J Ibs. of water, 
so that the inertia of the water which had to be overcome was 
by no means small compared with that of the box. The box 
was suspended by two parallel threads 3 inches apart and between 
4 and 5 feet long : it was twisted a little, and then left to itself, 
so that it oscillated about a vertical axis midway between the 
threads. The points of attachment of the threads were in a line 
drawn through the centre of the upper face parallel to one of its 
sides, and were equidistant from the centre. The weight of the 
box when empty, the length and distance of the threads, the time 



ON SOME CASES OF FLUID MOTION. 195 

of oscillation, and the known length of the seconds pendulum 
are data sufficient for determining the moment of inertia of the 
box about a vertical axis passing through its centre. When the 
box is filled with water the same quantities determine the mo 
ment of inertia of the box and the water it contains, whence the 
moment of inertia of the water alone is obtained by subtraction. 
It is supposed here that the centre of gravity of the box coincides 
with the centre of gravity of its interior volume. In the following 
experiments a different face of the box was uppermost each time. 
In Nos. 1 and 2 the long edges of the box were vertical, in Nos. 3 
and 4 they were horizontal. In all -cases the inertia determined 
by experiment was a little greater than that resulting from 
theory : the difference will be given in fractional parts of the 
latter. The difference was 1/21 in No. 1, 1/13 in No. 2, 1/17 in 
No. 3, and 1/21 in No. 4. On referring to the table at the end 
of the last paragraph, it will be seen that the ratio of the moment 
of inertia of the fluid to what it would be if the fluid were solid 
is about three times as great in the last two experiments as in 
the first two. 

I had expected beforehand to find the inertia determined by 
experiment a little greater than that given by theory, for this 
reason. In the theory, it is supposed that both the fluid itself 
and the surface of the box are perfectly smooth. This however 
is not strictly true. The box by its roughness exerts a tangential 
force on the fluid immediately in contact with it, and this force 
produces an effect on the fluid at a small distance from the surface 
of the box, in consequence of the internal friction of the fluid 
itself. We may conceive the effect of this force on the time of 
oscillation in a general way by supposing a thin film of fluid 
close to the surface of the box to be dragged along with it. Con 
sequently, the moment of inertia determined by experiment will 
be a little greater than it would have been had the fluid and 
the surface of the box been perfectly smooth. 

These experiments are sufficient to shew that in the case of 
a vessel of about the size and shape of the one I used, filled 
with water, and performing small oscillations of the duration of 
about one second (as was the case in my experiments), the time 
of oscillation is not much increased by friction; at least, if we 
suppose, as there is reason for supposing, that the effect of friction 

13-2 



196 SUPPLEMENT TO A MEMOIR ON FLUID MOTION. 

does not depend on the nature of the surface of the box. They 
are not however sufficiently exact to allow us to place any reliance 
on -the accuracy of the small differences between the results of 
experiment, and of the common theory of fluid motion, and con 
sequently they are useless as tests of any theory of friction. 



[From the Transactions of the Cambridge Philosophical Society, 
Vol. vni. p. 441.] 

ON THE THEORY OF OSCILLATORY WAVES. 

[Read March 1, 1847.] 

IN the Report of the Fourteenth Meeting of the British 
Association for the Advancement of Science it is stated by Mr 
Russell, as a result of his experiments, that the velocity of pro 
pagation of a series of oscillatory waves does not depend on the 
height of the waves*. A series of oscillatory waves, such as that 
observed by Mr Russell, does not exactly agree with what it is 
most convenient, as regards theory, to take as the type of oscil 
latory waves. The extreme waves of such a series partake in 
some measure of the character of solitary waves, and their height 
decreases as they proceed. In fact it will presently appear that 
it is only an indefinite series of waves which possesses the pro 
perty of being propagated with a uniform velocity, and without 
change of form : at least this is the case when the waves are 
such as can be propagated along the surface of a fluid which was 
previously at rest. The middle waves, however, of a series such 
as that observed by Mr Russell agree very nearly with oscillatory 
waves of the standard form. Consequently, the velocity of pro 
pagation determined by the observation of a number of waves, 
according to Mr Russell s method, must be very nearly the same 
as the velocity of propagation of a series of oscillatory waves of 
the standard form, and whose length is equal to the mean length 
of the waves observed, which are supposed to differ from each 
other but slightly in length. 

* Page 369 (note), and page 370. 



198 ON THE THEORY OF OSCILLATORY WAVES. 

On this account I was induced to investigate the motion of 
oscillatory waves of the above form to a second approximation, 
that is, supposing the height of the waves finite, though small. 
I find that the expression for the velocity of propagation is in 
dependent of the height of the waves to a second approximation. 
With respect to the form of the waves, the elevations are no 
longer similar to the depressions, as is the case to a first ap 
proximation, but the elevations are narrower than the hollows, 
and the height of the former exceeds the depth of the latter. 
This is in accordance with Mr Russell s remarks at page 448 of 
his first Report*. I have proceeded to a third approximation 
in the particular case in which the depth of the fluid is very 
great, so as to find in this case the most important term, de 
pending on the height of the waves, in the expression for the 
velocity of propagation. This term gives an increase in the 
velocity of propagation depending on the square of the ratio of 
the height of the waves to their length. 

There is one result of a second approximation which may 
possibly be of practical importance. It appears that the forward 
motion of the particles is not altogether compensated by their 
backward motion ; so that, in addition to their motion of oscil 
lation, the particles have a progressive motion in the direction 
of propagation of the waves. In the case in which the depth of 
the fluid is very great, this progressive motion decreases rapidly 
as the depth of the particle considered increases. Now when a 
ship at sea is overtaken by a storm, and the sky remains overcast, 
so as to prevent astronomical observations, there, is nothing to 
trust to for finding the ship s place but the dead reckoning. But 
the estimated velocity and direction of motion of the ship are 
her velocity and direction of motion relatively to the water. If 
then the whole of the water near the surface be moving in the 
direction of the waves, it is evident that the ship s estimated 
place will be erroneous. If, however, the velocity of the water 
can be expressed in terms of the length and height of the waves, 
both which can be observed approximately from the ship, the 
motion of the water can be allowed for in the dead reckoning. 

As connected with this subject, I have also considered the 
motion of oscillatory waves propagated along the common surface 
of two liquids, of which one rests on the other, or along the upper 
* Reports of the British Association, Vol. vi. 



ON THE THEOKY OF OSCILLATORY WAVES. 199 

surface of the upper liquid. In this investigation there is no 
object in going beyond a first approximation. When the specific 
gravities of the two fluids are nearly equal, the waves at their 
common surface are propagated so slowly that there is time to 
observe the motions of the individual particles. The second case 
affords a means of comparing with theory the velocity of pro 
pagation of oscillatory waves in extremely shallow water. For by 
pouring a little water on the top of the mercury in a trough we 
can easily procure a sheet of water of a small, and strictly uniform 
depth, a depth, too, which can be measured with great accuracy 
by means of the area of the surface and the quantity of water 
poured in. Of course, the common formula for the velocity of 
propagation will not apply to this case, since the motion of the 
mercury must be taken into account. 



1. In the investigations which immediately follow, the fluid 
is supposed to be homogeneous and incompressible, and its depth 
uniform. The inertia of the air, and the pressure due to a column 
of air whose height is comparable with that of the waves are also 
neglected, so that the pressure at the upper surface of the fluid 
may be supposed to be zero, provided we afterwards" add the at 
mospheric pressure to the pressure so determined. The waves 
which it is proposed to investigate are those for which the motion 
is in two dimensions, and which are propagated with a constant 
velocity, and without change of form. It will also be supposed 
that the waves are such as admit of being excited, independently of 
friction, in a fluid which was previously at rest. It is by these 
characters of the waves that the problem will be rendered de 
terminate, and not by the initial disturbance of the fluid, supposed 
to be given. The common theory of fluid motion, in which the 
pressure is supposed equal in all directions, will also be em 
ployed. 

Let the fluid be referred to the rectangular axes of x, y, z, 
the plane xz being horizontal, and coinciding with the surface 
of the fluid when in equilibrium, the axis of y being directed 
downwards, and that of x taken in the direction of propagation 
of the waves, so that the expressions for the pressure, &c. do not 
contain z. Let p be the pressure, p the density, t the time, u, v 
the resolved parts of the velocity in the directions of the axes 



200 ON THE THEORY OF OSCILLATORY WAVES. 

of x, y ; g the force of gravity, h the depth of the fluid when in 
equilibrium. From the character of the waves which was men 
tioned last, it follows by a known theorem that udx + vdy is an 
exact differential d(p. The equations by which the motion is to 
be determined are well known. They are 







= 0, wheny = A .................... (3); 



_ 0)Wl 



where (3) expresses the condition that the particles in contact with 
the rigid plane on which the fluid rests remain in contact with 
it, and (4) expresses the condition that the same surface of par 
ticles continues to be the free surface throughout the motion, 
or, in other words, that there is no generation or destruction of 
fluid at the free surface. 

If c be the velocity of propagation, u, v and p will be by 
hypothesis functions of x ct and y. It follows then from the 
equations u dfy/dx, v = dfyjdy and (1), that the differential 
coefficients of (f> with respect to x, y and t will be functions of 
x ct and y ; and therefore <f> itself must be of the form 

f(x-ct, y)+Ct. 

The last term will introduce a constant into (1) ; and if this 
constant be expressed, we may suppose </> to be a function of 
x ct and y. Denoting x ct by x , we have 

dp _ dp dp _ dp 
dx~d^ ft ~dt~ da/ 

and similar equations hold good for ^>. On making these sub 
stitutions in (1) and (4), omitting the accent of x, and writing 
gk for (7, we have 



. c + = 0, wbenp-O (G). 

dx dy dy 



ON THE THEORY OF OSCILLATORY WAVES. 201 

Substituting in (6) the value of p given by (5), we have 



d$_ ff -(dff$ ,d d^\ 
J dy da? ~* {dot do? * dy dxdyl 






_ 

dxdx* dxdydxdy \dydf~ 

......... (8). 



The equations (7) and (8) are exact; but if we suppose the 
motion small, and proceed to the second order only of approxima 
tion, we may neglect the last three terms in (7), and we may 
easily eliminate y between (7) and (8). For putting < , <,, &c. 
for the values of d(f)/dx, dfy/dy, &c. when y = 0, the number of 
accents above marking the order of the differential coefficient 
with respect to x, and the number below its order with respect 
to y, and observing that & is a small quantity of the first order 
at least, we have from (8) 

g (y + fc) + c (f + </> - i[ (f + <#>/) = 0, 



whence y = -*-jU +%/(&+ -f\ + _L 
y y \ y * y 

Substituting the first approximate value of y in the first two 
terms of (7), putting # = in the next two, and reducing, we 
have 



+2c 

<f> will now have to be determined from the general equation (2) 
with the particular conditions (3) and (10). When $ is known, 
?/, the ordinate of the surface, will be got from (9), and k will 
then be determined by the condition that the mean value of y 
shall be zero. The value of p, if required, may then be obtained 
from (5). 

2. In proceeding to a first approximation we have the equa 
tions (2), (3) and the equation obtained by omitting the small 
terms in (10), namely, 



* The reader will observe that the y in this equation is the ordinate of the 
surface, whereas the y in (1) and (2) is the ordinate of any point in the fluid. The 
context will always shew in which sense y is employed. 



202 ON THE THEORY OF OSCILLATORY WAVES. 

The general integral of (2) is 



the sign S extending to all values of A, m and n, real or imagi 
nary, for which m 2 + n 2 = : the particular values of </>, Cx + G , 
Dy + D , corresponding respectively to n = Q, ra = 0, must also be 
included, but the constants C , D may be omitted. In the 
present case, the expression for < must not contain real ex 
ponentials in x, since a term containing such an exponential 
would become infinite either for x = GO , or for x = + oo , as well 
as its differential coefficients which would appear in the ex 
pressions for u and v; so that m must be wholly imaginary. 
Replacing then the exponentials in x by circular functions, we 
shall have for the part of < corresponding to any one value 

of m, 

(Ae mv +^ e~ wy ) sin mx + (Bt mv + B e~ mv ) cos mx, 

and the complete value of < will be found by taking the sum of 
all possible particular values of the above form and of the par 
ticular value Cx + Dy. When the value so formed is substituted 
in (3), which has to hold good for all values of x, the coefficients 
of the several sines and cosines, and the constant term must be 
separately equated to zero. We have therefore 

D = 0, A = e* mh A, B = &*B ; 
so that if we change the constants we shall have 

<p = Cue + S (e m ( h ~rt + e~ m ( h -rt] (A sin mx + B cos mx)...(12), 

the sign S extending to all real values of m, A and B, of which 
in may be supposed positive. 

3. To the term Cx in (12) corresponds a uniform velocity 
parallel to x, which may be supposed to be impressed on the 
fluid in addition to its other motions. If the velocity of pro 
pagation be defined merely as the velocity with which the wave 
form is propagated, it is evident that the velocity of propagation 
is perfectly arbitrary. For, for a given state of relative motion 
of the parts of the fluid, the velocity of propagation, as so defined, 
can be altered by altering the value of C. And in proceeding to 
the higher orders of approximation it becomes a question what 
we shall define the velocity of propagation to be. Thus, we might 
define it to be the velocity with which the wave form is propa- 



ON THE THEORY OF OSCILLATORY WAVES. 203 

gated when the mean horizontal velocity of a particle in the 
upper surface is zero, or the velocity of propagation of the wave 
form when the mean horizontal velocity of a particle at the 
bottom is zero, or in various other ways. The following two 
definitions appear chiefly to deserve attention. 

First, we may define the velocity of propagation to be the 
velocity with which the wave form is propagated in space, when 
the mean horizontal velocity at each point of space occupied by the 
fluid is zero. The term "mean" here refers to the variation of 
the time. This is the definition which it will be most convenient 
to employ in the investigation. I shall accordingly suppose (7=0 
in (12), and c will represent the velocity of propagation according 
to the above definition. 

Secondly, we may define the velocity of propagation to be the 
velocity of propagation of the wave form in space, when the mean 
horizontal velocity of the mass of fluid comprised between two 
very distant planes perpendicular to the axis of x is zero. The 
mean horizontal velocity of the mass means here the same thing 
as the horizontal velocity of its centre of gravity. This appears 
to be the most natural definition of the velocity of propagation, 
since in the case considered there is 110 current in the mass of 
fluid, taken as a whole. I shall denote the velocity of propagation 
according to this definition by c. In the most important case 
to consider, name]y, that in which the depth is infinite, it is 
easy to see that c = c, whatever be the order of approximation. 
For when the depth becomes infinite, the velocity of the centre 
of gravity of the mass comprised between any two planes parallel 
to the plane yz vanishes, provided the expression for u contain 
no constant term. 

4. We must now substitute in (11) the value of <. 

</> =2 ( m tt-0> + e--tf) (A siumx + I? cos m#)... (13); 

but since (11) has to hold good for all values of #, the coefficients 
of the several sines and cosines must be separately equal to zero : 
at least this must be true, provided the series contained in (11) 
are convergent. The coefficients will vanish for any one value 
of m, provided 



fj gm _ ~ 
t *-! 



204 ON THE THEORY OF OSCILLATORY WAVES. 

Putting for shortness 2mh = p, we have 
^logc a = _l 2 

dp fl e* - -* 

which is positive or negative, p being supposed positive, according 
as 



and is therefore necessarily negative. Hence the value of c given 
by (14) decreases as jj, or m increases, and therefore (11) cannot 
be satisfied, for a given value of c, by more than one positive 
value of m. Hence the expression for <f> must contain only one 
value of m. Either of the terms A cos mcc, B sin mx may be 
got rid of by altering the origin of sc. We may therefore take, 
for the most general value of <, 

< = ^(e w ^-^+e-^-2/>)sinra^ (15). 

Substituting in (8), we have for the ordinate of the surface 

D? A (* 

y = --( h + e- h )cosmx (16), 

u 

k being = 0, since the mean value of y must be zero. Thus 
everything is known in the result except A and m, which are 
arbitrary. 

5. It appears from the above, that of all waves for which 
the motion is in two dimensions, which are propagated in a fluid 
of uniform depth, and which are such as could be propagated 
into fluid previously at rest, so that udx + vdy is an exact differ 
ential, there is only one particular kind, namely, that just con 
sidered, which possesses the property of being propagated with 
a constant velocity, and without change of form ; so that a 
solitary wave cannot be propagated in this manner. Thus the 
degradation in the height of such waves, which Mr Russell ob 
served, is not to be attributed wholly, (nor I believe chiefly,) to 
the imperfect fluidity of the fluid, and its adhesion to the sides 
and bottom of the canal, but is an essential characteristic of a 
solitary wave. It is true that this conclusion depends on an 
investigation which applies strictly to indefinitely small motions 
only : but if it were true in general that a solitary wave could be 
propagated uniformly, without degradation, it would be true in 



ON THE THEORY OF OSCILLATORY WAVES. 205 

the limiting case of indefinitely small motions ; and to disprove a 
general proposition it is sufficient to disprove a particular case. 

6. In proceeding to a second approximation we must sub 
stitute the first approximate value of <j>, given by (15), in the 
small terms of (10). Observing that k = to a first approximation, 
and eliminating g from the small terms by means of (14), we 
find 

9<t>,-c 2 <j>" - 6A*m*c sin 2mx= ............ (17). 

The general value of < given by (13), which is derived from (2) 
and (3), must now be restricted to satisfy (17). It is evident that 
no new terms in < involving sin mx or cos mx need be introduced, 
since such terms may be included in the first approximate value, 
and the only other term which can enter is one of the form 



Substituting this term in (17), and simplifying by means of (14), 
we find 



~ C ( 

Moreover since the term in <p containing sin. ma must disappear 
from (17), the equation (14) will give c to a second approxi 
mation. 

If we denote the coefficient of cosmic in the first approximate 
value of y, the ordinate of the surface, by a, we shall have 

A _ go* ca 

me (e mh + e ~ mh ) ( 6 w&-_ e -mh) > 

and substituting this value of A in that of <j>, we have 



e m(h - y) _|_ 6 - m(h - y) 2m(h - y) _|_ e - 2m(h - y) 

= - - sin 



...... (18). 

The ordinate of the surface is given to a second approximation 
by (9). It will be found that 

I e -mh\ ( f 2mh i f -2mh _i_ /f\ 



ma 



7. The equation to the surface is of the form 

mx Ka? cos 2w# (20), 



206 ON THE THEORY OF OSCILLATORY WAVES. 

where K is necessarily positive, and a may be supposed to be 
positive, since the case in which it is negative may be reduced to 
that in which it is positive by altering the origin of x by the 
quantity TT/W or X/2, X being the length of the waves. On re 
ferring to (20) we see that the waves are symmetrical with respect 
to vertical planes drawn through their ridges, and also with 
respect to vertical planes drawn through their lowest lines. The 
greatest depression of the fluid occurs when % = or = + X, &c., 
and is equal to aa?K: the greatest elevation occurs when 
# = X/2 or = + 3X/2, &c., and is equal to a + a*K. Thus the 
greatest elevation exceeds the greatest depression by Za*K. When 
the surface cuts the plane of mean level, cos mx aKcos 2mx = 0. 
Putting in the small term in this equation the approximate value 
mx = 7T/2, we have cos mx - aK= cos (?r/2 + aK], whence 

x = + (x/4 + a/a/2<7r), = (5X/4 + aKX/Zir), &c. 

We see then that the breadth of each hollow, measured at the 
height of the plane of mean level, is X/2 + aK\/7r, while the 
breadth of each elevated portion of the fluid is X/2 - aK\/7r. 

It is easy to prove from the expression for K, which is given 
in (19), that for a given value of X or of m, K increases as h 
decreases. Hence the difference in form of the elevated and 
depressed portions of the fluid is more conspicuous in the case 
in which the fluid is moderately shallow than in the case in 
which its depth is very great compared with the length of the 
waves. 

8. When the depth of the fluid is very great Compared with 
the length of a wave, we may without sensible error suppose h to 
be infinite. This supposition greatly simplifies the expressions 
already obtained. W T e have in this case 

sin mx .................................... (21), 



y =a cos mx 

m TT 



(22), 



the y in (22) being the ordinate of the surface. 

It is hardly necessary to remark that the state of the fluid at 
any time will be expressed by merely writing x-ct in place of x 
in all the preceding expressions. 



ON THE THEORY OF OSCILLATORY WAVES. 207 

9. To find the nature of the motion of the individual par 
ticles, let x + f be written for x, y + 77 for y, and suppose x and y 
to be independent of t, so that they alter only in passing from one 
particle to another, while f and 77 are small quantities depending 
on the motion. Then taking the case in which the depth is in 
finite, we have 

~sau 7wace~ m ^ +1? )cosm(# + f ct) mace~ my cosm(# ct) 
dt 

+ m?ac~ my sin m(x ct).% + m*ace~ my cos m (x ct) . 77, nearly, 
-^ = v = mace~ m( - y+ ^ sin m (x + f ctf) = mace~ my sin 7?2(# c) 

+ m*ace~ my cos m(x ct).% m 2 ace~ my sin m(# c) . 77, nearly. 
To a first approximation 

% = ae~ my sin w (a? ct), rj = ae~ my cos m (x ct), 

the arbitrary constants being omitted. Substituting these values 
in the small terms of the preceding equations, and integrating 
again, we have 

= ae~ my sin m(x ct) + m*a?cte~ 2my , 
77 = ae~ my cos m(x ct). 

Hence the motion of the particles is the same as to a first 
approximation, with one important difference, which is that in 
addition to the motion of oscillation the particles are transferred 
forwards, that is, in the direction of propagation, with a constant 
velocity depending on the depth, and decreasing rapidly as the 
depth increases. If U be this velocity for a particle whose depth 
below the surface in equilibrium is y, we have 

re-?. (23). 



The motion of the individual particles may be determined in 
a similar manner when the depth is finite from (18). In this case 
the values of f and r t contain terms of the second order, involving 
respectively sin 2m (x ct) and cos %m(x ct), besides the term in 
f which is multiplied by t. The most important thing to consider 
is the value of U, which is 

-h) \ e -1m(y-li) 

_ mh (24). 



208 ON THE THEORY OF OSCILLATORY WAVES. 

Since U is a small quantity of the order a 2 , and in proceeding 
to a second approximation the velocity of propagation is given to 
the order a only, it is immaterial which of the definitions of velo 
city of propagation mentioned in Art. 3 we please to adopt. 

10. The waves produced by the action of the wind on the 
surface of the sea do not probably differ very widely from those 
which have just been considered, and which may be regarded as 
the typical form of oscillatory waves. On this supposition the 
particles, in addition to their motion of oscillation, will have a 
progressive motion in the direction of propagation of the waves, 
and consequently in the direction of the wind, supposing it not to 
have recently shifted, and this progressive motion will decrease 
rapidly as the depth of the particle considered increases. If the 
pressure of the air on the posterior parts of the waves is greater 
than on the anterior parts, in consequence of the wind, as un 
questionably it must be, it is easy to see that some such progres 
sive motion must be produced. If then the waves are not break 
ing, it is probable that equation (23), which is applicable to deep 
water, may give approximately the mean horizontal velocity of 
the particles ; but it is difficult to say how far the result may be 
modified by friction. If then we regard the ship as a mere parti 
cle, in the first instance, for the sake of simplicity, and put f/ for 
the value of U when y = 0, it is easy to see that after sailing for 
a time t, the ship must be a distance U Q t to the lee of her estimated 
place. It will not however be sufficient to regard the ship as a 
mere particle, on account of the variation of the factor e~ 2w % as y 
varies from to the greatest depth of the ship below the surface 
of the water. Let 8 be this depth, or rather a depth something 
less, in order to allow for the narrowing of the ship towards the 
keel, and suppose the effect of the progressive motion of the water 
on the motion of the ship to be the same as if the water were 
moving with a velocity the same as all depths, and equal to the 
mean value of the velocity U from y = to y = 8. If U l be this 
mean velocity, 



ma?c 



On this supposition, if a ship be steered so as to sail in a direc 
tion making an angle 6 with the direction of the wind, supposing 
the water to have no current, and if F be the velocity with which 



ON THE THEORY OF OSCILLATORY WAVES. 209 

the ship moves through the water, her actual velocity will be the 
resultant of a velocity V in the direction just mentioned, which, 
for shortness, I shall call the direction of steering, and of a velocity 
Z7 X in the direction of the wind. But the ship s velocity as esti 
mated by the log-line is her velocity relatively to the water at the 
surface, and is therefore the resultant of a velocity V in the direc 
tion of steering, and a velocity U U t in a direction opposite to 
that in which the wind is blowing. If then E be the estimated 
velocity, and if we neglect U 2 , 



But the ship s velocity is really the resultant of a velocity V+ 
in the direction of steering, and a velocity U l sin 6 in the perpen 
dicular direction, while her estimated velocity is E in the direction 
of steering. Hence, after a time t, the ship will be a distance 
U t cos 6 ahead of her estimated place, and a distance Uj sin 6 
aside of it, the latter distance being measured in a direction per 
pendicular to the direction of steering, and on the side towards 
which the wind is blowing. 

I do not suppose that the preceding formula can be employed 
in practice ; but I think it may not be altogether useless to call 
attention to the importance of having regard to the magnitude 
and direction of propagation of the waves, as well as to the wind, 
in making the allowance for lee-way. 

11. The formula of Art. 6 are perfectly general as regards the 
ratio of the length of the waves to the depth of the fluid, the only 
restriction being that the height of the waves must be sufficiently 
small to allow the series to be rapidly convergent. Consequently, 
they must apply to the limiting case, in which the waves are sup 
posed to be extremely long. Hence long waves, of the kind con 
sidered, are propagated without change of form, and the velocity 
of propagation is independent of the height of the waves to a 
second approximation. These conclusions might seem, at first 
sight, at variance with the results obtained by Mr Airy for the 
case of long waves *. On proceeding to a second approximation, 
Mr Airy finds that the form of long waves alters as they proceed, 
and that the expression for the velocity of propagation contains a 

* Encyclopedia Metropolitana, Tides and Waves, Articles 198, &c. 
S. 14 



210 ON THE THEORY OF OSCILLATORY WAVES. 

term depending on the height of the waves. But a little attention 
will remove this apparent discrepancy. If we suppose mh very 
small in (19), and expand, retaining only the most important 
terms, we shall find for the equation to the surface 

3a 2 
a cos mx -. r cos zmx. 



Now, in order that the method of approximation adopted may be 
legitimate, it is necessary that the coefficient of cos Zmx in this 
equation be small compared with a. Hence a/m z h 3 , and therefore 
X 2 a/A 3 , must be small, and therefore a/h must be small compared 
with (h/\y. But the investigation of Mr Airy is applicable to the 
case in which \/h is very large ; so that in that investigation a/h 
is large compared with (/t/\) 2 . Thus the difference in the results 
obtained corresponds to a difference in the physical circumstances 
of the motion. 

12. There is no difficulty in proceeding to the higher orders 
of approximation, except what arises from the length of the for 
mulas. In the particular case in which the depth is considered 
infinite, the formulae are very much simpler than in the general 
case. I shall proceed to the third order in the case of an infinite 
depth, so as to find in that case the most important term, depend 
ing on the height of the waves, in the expression for the velocity 
of propagation. 

For this purpose it will be necessary to retain the terms of 
the third order in the expansion of (7). Expanding this equation 
according to powers of y, and neglecting terms of the fourth, &c. 
orders, we have 

t - c 2 </>,") y + (g^r <ty,,") + 2c (f f 



(25). 



In the small terms of this equation we must put for <f> and y 
their values given by (21) and (22) respectively. Now since the 
value of <f> to a second approximation is the same as its value to a 
first approximation, the equation g$ C 2 <"=0 is satisfied to terms 
of the second order. But the coefficients of y and y 2 / 2 , ^ n tne 
first line of (25), are derived from the left-hand member of the 



ON THE THEORY OF OSCILLATORY WAVES. 211 

preceding equation by inserting the factor G~ m v, differentiating 
either once or twice with respect to y, and then putting y = 0. 
Consequently these coefficients contain no terms of the second 
order, and therefore the terms involving y in the first line of (25) 
are to be neglected. The next two terms are together equal to 
But 



which does not contain oc, so that these two terms disappear. The 
coefficient of y in the second line of (25) may be derived from the 
two terms last considered in the manner already indicated, and 
therefore the terms containing y will disappear from (25). The 
only small terms remaining are the last three, and it will easily 
be found that their sum is equal to raVc 3 sin mx, so that (25) be 
comes 

#(/>,- c 2 </> + ra Vc 3 sin m# = ..................... (26). 

The value of < will evidently be of the form Ae~ my sin mx. Sub 
stituting this value in (26), we have 

(mV - mg} A + mVc 3 = 0. 

Dividing by mA, and putting for A and c 2 their approximate values 
ac, g/m respectively in the small term, we have 

g + mV<7, 



The equation to the surface may be found without difficulty. It 
is 

y = a cos mx J ma 2 cos 2mx + f mV cos Smx* ......... (27) : 

we have also 

k = 0, < = ac (1 fmV) 



* It is remarkable that this equation coincides with that of the prolate cycloid, 
if the latter equation be expanded according to ascending powers of the distance of 
the tracing point from the centre of the rolling circle, and the terms of the fourth 
order be omitted. The prolate cycloid is the form assigned by Mr Russell to waves 
of the kind here considered. Reports of the British Association, Vol. vi. p. 448. 
When the depth of the fluid is not great compared with the length of a wave, the 
form of the surface does not agree with the prolate cycloid even to a second 
approximation. 

142 



212 ON THE THEORY OF OSCILLATORY WAVES. 

The following figure represents a vertical section of the waves 
propagated along the surface of deep water. The figure is drawn 



for the case in which a = . The term of the third order in (27) 

is retained, but it is almost insensible. The straight line represents 
a section of the plane of mean level. 

13. If we consider the manner in which the terms introduced 
by each successive approximation enter into equations (7) and (8), 
we shall see that, whatever be the order of approximation, the 
series expressing the ordinate of the surface will contain only 
cosines of mx and its multiples, while the expression for <f> will 
contain only sines. The manner in which y enters into the 
coefficient of cos rmx in the expression for <f> is determined in the 
case of a finite depth by equations (2) and (3). Moreover, the 
principal part of the coefficient of cos rmx or sin rmx will be of 
the order a r at least. We may therefore assume 

< = T>,"a r A r ("<*-> + e-rmUi-vY) sm rmXt 
y = a cos mx + 2 2 a r .Z? r cos rmx, 

and determine the arbitrary coefficients by means of equations 
(7) and (8), having previously expanded these equations according 
to ascending powers of y. The value of c 2 will be determined by 
equating to zero the coefficient of sin mx in (7). 

Since changing the sign of a comes to the same thing as 
altering the origin of x by \ X, it is plain that the expressions 
for A r , B r and c 2 will contain only even powers of a. Thus 
the values of each of these quantities will be of the form 

o o+ cx + cx + --- 

It appears also that, whatever be the order of approximation, 
the waves will be symmetrical with respect to vertical planes 
passing through their ridges, as also with respect to vertical planes 
passing through their lowest lines. 

14 Let us consider now the case of waves propagated at 
the common surface of two liquids, of which one rests on the 



ON THE THEORY OF OSCILLATORY WAVES. 213 

other. Suppose as before that the motion is in two dimensions, 
that the fluids extend indefinitely in all horizontal directions, 
or else that they are bounded by two vertical planes parallel to 
the direction of propagation of the waves, that the waves are 
propagated with a constant velocity, and without change of form, 
and that they are such as can be propagated into, or excited 
in, the fluids supposed to have been previously at rest. Suppose 
first that the fluids are bounded by two horizontal rigid planes. 
Then taking the common surface of the fluids when at rest for 
the plane xz, and employing the same notation as before, we 
have for the under fluid 



= wheny=a ................ (29), 



neglecting the squares of small quantities. Let h / be the depth 
of the upper fluid when in equilibrium, and let p t , p,, <f>,, C, be 
the quantities referring to the upper fluid which correspond to 
P, p> & referring to the under : then we have for the upper 
fluid 



d* df - ( 3 )> 

^P = when y = -h, (31), 



We have also, for the condition that the two fluids shall not 
penetrate into, nor separate from each other, 



Lastly, the condition answering to (11) is 



-he 



214 ON THE THEORY OF OSCILLATORY WAVES. 

Since C C is evidently a small quantity of the first order at 
least, the condition is that (33) shall be satisfied when # = 0. 
Equation (34) will then give the ordinate of the common surface 
of the two liquids when y is put = in the last two terms. 

The general value of <j> suitable to the present case, which 
is derived from (28) subject to the condition (29), is given by (13) 
if we suppose that the fluid is free from a uniform horizontal 
motion compounded with the oscillatory motion expressed by (18). 
Since the equations of the present investigation are linear, in 
consequence of the omission of the squares of small quantities, 
it will be sufficient to consider one of the terms in (13). Let 
then 

m(h -^smmx ......... (35). 



The general value of <f> t will be derived from (13) by merely 
writing h l for h. But in order that (32) may be satisfied, the 
value of </> y must reduce itself to a single term of the same form 
as the second side of (35). We may take then for the value 
offc 

y = A,(e t(h + rt + e- m V t + v } )&in.mx ............ (36). 

Putting for shortness 



and taking $,, D, to denote the quantities derived from 8, D by 
writing A, for h, we have from (32) 

DA + D 4 A t = ...................... (37), 

and from (33) 

P (gD-mc*S)Ai-p l (gD, + mc*S)A, = ........ (38). 

Eliminating A and A t from (37) and (38), we have 



The equation to the common surface of the liquids will be 
obtained from (34). Since the mean value of y is zero, we have 
in the first place 

C = C .................................. (40). 

We have then, for the value of y, 

mx ............................. (41), 



ON THE THEORY OF OSCILLATORY WAVES. 215 

where 



_ 

- 



g p-p, 



Substituting in (35) and (36) the values of A and A f derived from 
(37) and (42), we have 

= -^ (*- > +--*>) sin wa? ....... .,...(43), 

rt * 

(44). 



Equations (39), (40), (41), (43) and (44) contain the solution 
of the problem. It is evident that C remains arbitrary. The 
values of p and p t may be easily found if required. 

If we differentiate the logarithm of c 2 with respect to m, and 
multiply the result by the product of the denominators, which 
are necessarily positive, we shall find a quantity of the form 
Pp+P t p,, where P and P t do not contain p or p r It may be 
proved in nearly the same manner as in Art. 4, that each of the 
quantities P, P t is necessarily negative. Consequently c will 
decrease as m increases, or will increase with X. It follows from 
this that the value of </> cannot contain more than two terms, 
one of the form (35), and the other derived from (35) by replacing 
sin mx by cos mx, and changing the constant A : but the latter 
term may be got rid of by altering the origin of x. 

The simplest case to consider is that in which both h and ti 
are regarded as infinite compared with X. In this case we have 

<j> = - ace ~ y sin mx, <, = ace my sin mx, 

P P Q 
c 2 = r \LL <L. y a cos mx, 

p + p,m 
the latter being the equation to the surface. 

15. The preceding investigation applies to two incompressible 
fluids, but the results are applicable to the case of the waves 
propagated along the surface of a liquid exposed to the air, pro 
vided that in considering the effect of the air we neglect terms 
which, in comparison with those retained, are of the order of 
the ratio of the length of the waves considered to the length of 



216 ON THE THEORY OF OSCILLATORY WAVES. 

a wave of sound of the same period in air. Taking then p for 
the density of the liquid, p t for that of the air at the time, and 
supposing h t = oo , we have 



If we had considered the buoyancy only of the air, we should 
have had to replace g in the formula (14) by - g* We should 
have obtained in this manner 



t_s_(?-PJV- 



m pS mS\ 



/A 

p) 



Hence, in order to allow for the inertia of the air, the correction 
for buoyancy must be increased in the ratio of 1 to 1 + D/S. 
The whole correction therefore increases as the ratio of the length 
of a wave to the depth of the fluid decreases. For very long 
waves the correction is that due to buoyancy alone, while in 
the case of very short waves the correction for buoyancy is 
doubled. Even in this case the velocity of propagation is altered 
by only the fractional part pjp of the whole ; and as this quantity 
is much less than the unavoidable errors of observation, the effect 
of the air in altering the velocity of propagation may be neglected. 

16. There is a discontinuity in the density of the fluid mass 
considered in Art. 14, in passing from one fluid into the other; 
and it is easy to shew that there is a corresponding discontinuity 
in the velocity. If we consider two fluid particles in contact 
with each other, and situated on opposite sides of the surface 
of junction of the two fluids, we see that the velocities of these 
particles resolved in a direction normal to that surface are the 
same ; but their velocities resolved in a direction tangential to 
the surface are different. These velocities are, to the order of 
approximation employed in the investigation, the values of d<j>/dx 
and dfyjdcc when y = 0. We have then from (43) and (44), 
for the velocity with which the upper fluid slides along the 
under, 

8 S\ 
mac I -W- -f -=: cos moc. 



ON THE THEORY OF OSCILLATORY WAVES. 21? 

17. When the upper surface of the upper fluid is free, the 
equations by which the problem is to be solved are the same 
as those of Art. 14, except that the condition (31) is replaced by 



= - h > .......... (45); 

and to determine the ordinate of the upper surface, we have 



where y is to be replaced by h t in the last term. Let us con 
sider the motion corresponding to the value of $ given by (35). 
We must evidently have 

<, = (A f e m v -f B t e- m y) sin ma?, 

where A t and B t have to be determined. The conditions (32), 
(33) and (45) give 



p (gD - mc*S) A+p,(g + mc z ) A, -p,(g- me 2 ) B t = 0, 
(g + me 2 ) e~ mh A t -(g- me 2 ) e"*>B t = 0. 

Eliminating A, A f and B t from these equations, and putting 



m 
we find 



The equilibrium of the fluid being supposed to be stable, we 
must have p, < p. This being the case, it is easy to prove that 
the two roots of (46) are real and positive. These two roots 
correspond to two systems of waves of the same length, which 
are propagated with the same velocity. 

In the limiting case in which p/p t = oo , (46) becomes 

SSf - (8D t + SD) +DD t = 0, 

the roots of which are D/8 and D]S t , as they evidently ought 
to be, since in this case the motion of the under fluid will not 
be affected by that of the upper, and the upper fluid can be in 
motion by itself. 

When p, = p one root of (46) vanishes, and the other becomes 



_ f -m(h+h t ) 

_ ! . The former of these roots cor- 
88 



218 ON THE THEORY OF OSCILLATORY WAVES. 

responds to the waves propagated at the common surface of the 
fluids, while the latter gives the velocity of propagation belonging 
to a single fluid having a depth equal to the sum of the depths 
of the two considered. 

When the depth of the upper fluid is considered infinite, 
we must put DJS, = \ in (46). The two roots of the equation 

f f \ 7~) 

so transformed are 1 and ^ITTJ-J) > the former corresponding to 

waves propagated at the upper surface of the upper fluid, and the 
latter agreeing with Art. 15. 

When the depth of the under fluid is considered infinite, and 
that of the upper finite, we must put D/S=l in (46). The two 

roots will then become 1 and ^~^ . The value of the 



former root shews that whatever be the depth of the upper fluid, 
one of the two systems of waves will always be propagated with 
the same velocity as waves of the same length at the surface of a 
single fluid of infinite depth. This result is true even when the 
motion is in three dimensions, and the form of the waves changes 
with the time, the waves being still supposed to be such as could 
be excited in the fluids, supposed to have been previously at rest, 
by means of forces applied at the upper surface. For the most 
general small motion of the fluids in this case may be regarded 
as the resultant of an infinite number of systems of waves of the 
kind considered in this paper. It is remarkable that when the 
depth of the upper fluid is very great, the root f = 1 is that which 
corresponds to the waves for which the upper fluid is disturbed, 
while the under is sensibly at rest; whereas, when the depth of 
the upper fluid is very small, it is the other root which corresponds 
to those waves which are analogous to the waves which would 
be propagated in the upper fluid if it rested on a rigid plane. 

When the depth of the upper fluid is very small compared 
with the length of a wave, one of the roots of (46) will be very 
small ; and if we neglect squares and products of mh i and f , the 
equation becomes %pD 2 (p />,) mhfl = 0, whence 



(47). 



These formulae will not hold good if mh be very small as well as 
inh /t and comparable with it, since in that case all the terms of 



ON THE THEORY OF OSCILLATORY WAVES. 219 

(46) will be small quantities of the second order, mh, being re 
garded as a small quantity of the first order. In this case, if we 
neglect small quantities of the third order in (46), it becomes 

4pf 2 - 4mp (h + \ ) f + 4 (p - Pt ) ra 2 M, = 0, 
whence 



(48). 



Of these values of c 2 , that in which the radical has the negative 
sign belongs to that system of waves to which the formula (47) 
apply when Ji t is very small compared with h. 

If the two fluids are water and mercury, p/p, is equal to about 
13*57. If the depth of the water be very small compared both 
with the length of the waves and with the depth of the mercury, 
it appears from (47) that the velocity of propagation will be less 
than it would have been, if the water had rested on a rigid plane, 
in the ratio of 9624 to 1, or 26 to 27 nearly. 



APPENDIX. 

[A. On the relation of the preceding investigation to a case of wave 
motion of the oscillatory kind in which the disturbance can be 
expressed in finite terms. 

In the Philosophical Transactions for 1863, p. 127, is a paper 
by the late Professor Rankine in which he has shewn that it is 
possible to express in finite terms, without any approximation, 
the motion of a particular class of waves of the oscillatory kind. 
It is remarkable that the results for waves of this kind were 
given as long ago as in 1802, by Gerstner*, whose investigation 
however seems to have been but little noticed for a long time. 
This case of motion has latterly attracted a good deal of atten* 
tion, partly no doubt from the facility of dealing with it, but 
partly, it would seem, from misconceptions as to its intrinsic 
importance. 

* See Weber s Wcllenlehrc auf Experimente gcgriindet, p. 338, 



220 ON THE THEORY OF OSCILLATORY WAVES. 

The investigation may be presented in very short compass in 
the following manner. 

Let us confine our attention to the case of a mass of liquid, re 
garded as a perfect fluid of a depth practically infinite, in which 
an indefinite series of regular periodic waves is propagated along 
the surface, the motion being in two dimensions, and vanishing at 
an infinite depth. Taking the plane of motion for the plane of xy, 
y being measured vertically downwards, let us seek to express the 
actual co-ordinates x, y of any particle in terms of two parameters 
h, k particularising that particle, and of the time t. Let us assume 
for trial 

x = h + Ksm m(h- ct), y = k + Kcos m(h ct) (49), 

where m, c are two constants, and K a function of k only. It 
will be easily seen that these equations, regarded merely as 
expressing the geometrical motion of points, and apart from the 
physical possibility of the motion, represent a wave disturbance 
of periodic character travelling in the direction of OX with a 
velocity of propagation c. 

As the disturbance is in two dimensions, we may speak of areas 
as representing volumes. Let us consider first the condition of 
constancy of the mass. The four loci corresponding to constant 
values h, h + dh, k, k + dk, of the two parameters respectively en 
close a quadrangular figure which is ultimately a parallelogram, 
the area of which must be independent of the time. Now the 
area is Sdhdk where 

* 
~ _ dx dy dx dy 

dh dk dk dh 



On performing the differentiations we find 

S=l + (mk + K )cQ$m(h-ct) + mKK (51), 

where K stands for dKfdk. In order that this may be indepen 
dent of the time it is necessary and sufficient that 

mK + K = (52), 

whence 

K= ae -mk (52 ^ 

and 

S= 1 -m 2 # 2 = 1 -mVe- 2 ^ (53). 



ON THE THEORY OF OSCILLATORY WAVES. 221 



The dynamical equations give 
dp , (d*x d 



gdy + m*c*K (sin in (h ct) dx -f cos m (h ct) dy} 
= gdy + wV {(x-h)d(a)-h) + (y-k)d(y- k)} 
+ m*c 2 {(x -h)dh+(y- k) dk}. 

The last line becomes by (49) and (52), 

mc 2 {mKsm m(h ct) dh K cos m(h ct) dk}, 

or mc*d . K cos m (h ct). 

The dynamical equations are therefore satisfied, the expression for 

dp being a perfect differential, and we have 

V {(as - h)* + (y - k) 2 } - mc*K cos m(h-ct) + C 
- mc z ) K cos m(h- ct) + G. 



It remains to consider the equations of condition at the boun 
daries of the fluid. The expression for K satisfies the condition of 
giving a disturbance which decreases indefinitely as the depth in 
creases, and we have only to see if it be possible to satisfy the 
condition at the free surface. Now the particles at the free sur 
face differ only by the value of the parameter h, as follows from 
the fundamental conception of wave motion, and therefore for some 
one value of k we must have p = independently of the time. 
This requires that 

*=_=&. 

m 2-7T 

and if we please to take k = at the surface, and determine C 
accordingly, we have 

(I --**) ......... (54). 



Since p is independent of the time, not merely for k = 0, but 
for any constant value of k, it follows that when the wave motion 
is converted into steady motion by superposing a velocity equal 
and opposite to that of propagation, it is not merely the line of 
motion or stream-line which forms the surface but all the stream 
lines that are lines of constant pressure. This is undoubtedly no 
necessary property of wave-motion converted into steady motion, 
which only requires that the particular stream-line at the surface 



222 ON THE THEORY OF OSCILLATORY WAVES. 

shall be one for which the pressure is constant, though Gerstner 
has expressed himself as if he supposed it necessarily true ; it is 
merely a character of the special case investigated by Gerstner 
and Kankine. Nevertheless in the case of deep water it must be 
very approximately true. For in the first place it is strictly true 
at the surface, and in the second place, it must be sensibly true 
at a very moderate depth and for all greater depths, since the 
disturbance very rapidly diminishes on passing from the surface 
downwards; so that unless the amount of disturbance be excessive 
the supposition that all the stream-lines are lines of constant 
pressure will not be much in error. 

In the case investigated by the mathematicians just mentioned, 
each particle returns periodically to the position it had at a given 
instant ; there is no progressive motion combined with a periodic 
disturbance, such as was found in the case investigated in the pre 
sent paper : and for deep water the absence of progressive motion 
is doubtless peculiar to the former case, as will presently more 
clearly appear. 

If we suppose a regular periodic wave motion to be going on, 
and then suppose small suitable pressures applied to the surface in 
such a manner as to check the motion, we may evidently produce 
a secular subsidence of the wave disturbance while still leaving it 
at any moment regular and periodic, save as to secular change, 
provided the opposing pressures are suitably chosen. The wave 
length will be left unchanged, but not so, in general, the periodic 
time. If the amount of disturbance in one wave period be insen 
sible, the particles which at one time have a common mean depth 
must at any future time have a common mean depth, and must 
ultimately lie in a horizontal plane when the wave motion has 
wholly subsided. In this condition therefore there can be no 
mption except a horizontal flow with a velocity which is some 
function of the depth. By a converse process we may imagine a 
regular periodic wave motion of given wave-length excited in a 
fluid in which there previously was none; and according to the 
nature of the arbitrary flow with which we start, we shall obtain 
as the result a wave motion of such or such a kind*. 

In any given case of wave motion, the flow which remains 

* To prevent possible misconception I may observe that I am not here con 
templating the actual mode of excitement of waves by wind, which in some respects 
is essentially different. 



ON THE THEORY OF OSCILLATORY WAVES. 223 

when the waves have been caused to subside in the manner above 
explained is easily determined, since we know that in the motion 
of a liquid in two dimensions the angular velocity is not affected 
by forces applied to thd surface. If a) be the angular velocity 

dv du _ 1 (dy dv dy dv dx du dx du} 
~ dx dy~ S (dk dh dh dk dk dh dh dk) 

S being denned by (50). In Gerstner and Rankine s solution 
u = mace~ mk cos m(h ct), v = mace~ mk sin m(h ct), 

and on effecting the differentiations and substituting for S from 
(53) we find 



Let y be the depth and u the horizontal velocity, after the 
wave-motion has been destroyed as above explained, of the line of 
particles which had k for a parameter ; then we must have 



,. 

(oC) - 



Since in a horizontal length which may be deemed infinite com 
pared with X the area between the ordinates y , y + dy must 
be the same as between the lines of particles which have k } k + dk 
for their ^-parameter 

dy = Sdk, 

S being defined by (50). Putting for S its value given by (53) 
we have 

dy =(l-m*a?-* mk )dk ..................... (57), 

y f = k-lma?(l-e-* mk } .................. (58). 

We have then from (56) by (55) and (57), 

u = 2wVc Je- 2m *cta = - wiVce" 21 "* ............ (59), 

since u vanishes when k oo . 

It appears then that in order that it should be possible to 
excite these waves in deep water previously free from wave dis 
turbance, by means of pressures applied to the surface, a prepara 
tion must be laid in the shape of a horizontal velocity decreasing 
from the surface downwards according to the value of e~ 2mk , where 
k is a function of the depth y determined by the transcendental 
equation in k (58), and moreover a velocity decreasing downwards 
according to this law will serve for waves of the present kind of 



224 ON THE THEORY OF OSCILLATORY WAVES. 

only one particular height depending on the coefficient of the ex 
ponential in the expression for the flow. Under these conditions 
the horizontal velocity depending (when we adopt approximations) 
on the square and higher powers of the elevation, which belongs to 
the wave-motion, is exactly neutralized by the pre-existing hori 
zontal velocity in a contrary direction, pre-existing, that is, when 
we think of the waves as having been excited in a fluid previously 
destitute of wave-motion, not as having gone on as they are from 
a time indefinitely remote. The absence of any forward horizontal 
motion of the individual particles in waves of this kind, though 
attractive at first sight, is not of any real physical import, 
because we are not concerned with the biographies so to speak of 
the individual particles. 

The oscillatory waves which most naturally present themselves 
to our attention are those which are excited in the ocean or on 
a lake by the action of the wind, or those which having been so 
excited are propagated into (practically, though not in a rigorous 
mathematical sense) still water. Of the latter kind are the surf 
which breaks upon our western coasts as a result of storms out in 
the Atlantic, or the grand rollers which are occasionally observed 
at St Helena and Ascension Island. The motion in these cases 
having been produced from rest, by forces applied to the surface, 
there is no molecular rotation, and therefore the investigation of 
the present paper strictly applies. Moreover, if we conceive the 
waves gradually produced by suitable forces applied to the surface, 
in the manner explained at p. 222, the investigation applies to the 
waves (secular change apart) at any period of their growth, and 
not merely when they have attained one particular height. 

There can be no question, it seems to me, that this is the class 
of oscillatory waves which on merely physical grounds we should 
naturally select for investigation. The interest of the solution first 
given by Gerstner, and it is of great interest, arises not from any 
physical pre-eminence of the class of waves to which it relates, but 
from the imperfection of our analysis, which renders it important 
to discuss a case in which all the circumstances of the motion can 
be simply expressed in mathematical terms without any approxima 
tion. And though this motion is not exactly that which on purely 
physical grounds we should prefer to investigate, namely, that in 
which the molecular rotation is nil, yet unless the height of the 



ON THE THEORY OF OSCILLATORY WAVES. 225 

waves be extravagant, it agrees so nearly with it that for many 
purposes the simpler expressions of Rankine may be used without 
material error, even when we are investigating wave motion of 
the irrotational kind. 



B. Considerations relative to the greatest height of oscillatory 
irrotational waves which can be propagated without change 
of form. 

In a paper published in the Philosophical Magazine, Vol. xxix. 
(1865), p. 25, Rankine gave an investigation which led him to 
the conclusion that in the steepest possible oscillatory waves of 
the irrotational kind, the crests become at the vertex infinitely 
curved in such a manner that a section of the crest by the plane 
of motion presents two branches of a curve which meet at a right 
angle*. 

In this investigati6n it is assumed in the first place that the 
steepness may be pushed to the limit of an infinite curvature 
at a particular point, and in the second place that the variations 



* It is not quite clear whether Rankine supposed his proposition, that "all 
waves in which molecular rotation is null, begin to break when the two slopes of 
the crest meet at right angles," to apply only to free waves, or to forced waves as 
well. One would have supposed the former, were it not that a figure is referred to 
representing forced waves of one particular kind. It is readily shewn that the 
contour of a forced wave is arbitrary, even though the motion be restricted to be 
irrotational. Let U=C (p. 4) be the general equation of the stream lines when the 
wave motion is converted into steady motion. Then in the general case of a finite 
depth, which includes as a limiting and therefore particular case that of an infinite 
depth, the parameter C has one constant value at the upper surface, and another at 
the bottom, and it satisfies the partial differential equation (5) of p. 4. Hence the 
problem of finding U is the same as that of determining the permanent tem 
perature, varying in two dimensions only, of a homogeneous isotropic solid the 
section of which is bounded below by a horizontal line at a finite or infinite depth, 
and above by a given arbitrary contour, the bounding surfaces being at two given 
constant temperatures. The latter problem is evidently determinate, and therefore 
also the former, so that forced waves may present in their contour sharp angles, 
not merely of 90, but of any value we please to take. 

s. 15 



226 ON THE THEORY OF OSCILLATOKY WAVES. 

of the components of the velocity, in passing from the crest to 
a point infinitely close to it, may be obtained by differentiation, 
or in other words from the second terms of the expansion by 
Taylor s Theorem applied to infinitely small increments of the 
variables. 

The first assumption might perhaps be called in question, 
but it would appear likely to give at any rate a superior limit 
to the steepest form possible, if not the steepest form itself. 
But as regards the second it would seem a priori very likely 
that the crest might just be one of those singular points where 
Taylor s Theorem fails; and that such must actually be the case 
may be shewn by simple considerations. 

Let us suppose that a fluid of either finite or infinite depth 
is disturbed by a wave motion which is propagated uniformly 
without change, the motion of the fluid being either rotational 
or not, and let us suppose further that the crests are perfectly 
sharp, so that a crest is formed by two branches of a curve which 
either meet at a finite angle (their prolongations belonging to the 
region of space where the fluid is not), or else touch, forming 
a cusp. 

Reduce the wave motion to steady motion by superposing 
a velocity equal and opposite to that of propagation. Then 
a particle at the surface may be thought of as gliding along a 
fixed smooth curve: this follows directly from physical considera 
tions, or from the ordinary equation of steady motion. On 
arriving at a crest the particle must be momentarily at rest, and 
on passing it must be ultimately in the condition of a particle 
starting from rest down an inclined or vertical plane. Hence the 
velocity must vary ultimately as the square root of the distance 
from the crest. 

Hitherto the motion has been rotational or not, Jet us now 
confine ourselves to the case of irrotational motion. Place the 
origin at the crest, refer the function $ to polar co-ordinates r, 6 ; 
6 being measured from the vertical, and consider the value of </> 
very near the origin, where </> may be supposed to vanish, as the 
arbitrary constant may be omitted. In general <j> will be of the 
form ^A n r n s\unO + ^B n cosn0. In the present case < must con 
tain sines only on account of the symmetry of the motion, as 



ON THE THEORY OF OSCILLATORY WAVES. 227 

already shewn (p. 212), so that retaining only the most important 
term we may take (j> = Ar n sin n6. Now for a point in the section 
of the profile we must have dfy/dO = 0, and dfyjdr varying 
ultimately as ?A This requires that n = %, and for the profile 
that \Q ^TT, so that the two branches are inclined at angles of 
60 to the vertical, and at an angle of 120 to each other, not 
of 90 as supposed by Rankine. 

This however leaves untouched the question whether the 
disturbance can actually be pushed to the extent of yielding crests 
with sharp edges, or whether on the other hand there exists 
a limit, for which the outline is still a smooth curve, beyond which 
no waves of the oscillatory irrotational kind can be propagated 
without change of form. 

After careful consideration I feel satisfied that there is no 
such earlier limit, but that we may actually approach as near 
as we please to the form in which the curvature at the vertex 
becomes infinite, and the vertex becomes a multiple point where 
the two branches with which alone we are concerned enclose an 
angle of 120. But whether in the limiting form the inclination 
of the wave to the horizon continually increases from the trough 
to the summit, and is consequently limited to 30, or whether on 
the other hand the points of inflexion which the profile presents 
in the general case remain at a finite distance from the summit 
when the limiting form is reached, so that 011 passing from the 
trough to the summit the inclination attains a maximum from 
which it begins to decrease before the summit is reached, is a 
question which I cannot certainly decide, though I feel little doubt 
that the former alternative represents the truth. 

In Rankine s case of wave motion the limiting form presents 
crests which are cusped. For the maximum wave ma = 1 or 
a = A/27T. We see from (55) that in this case the angular velo 
city becomes infinite at the surface, where 7c vanishes; and if 
we suppose such waves excited in the manner already explained 
in a fluid initially destitute of wave motion, the horizontal velocity 
u r which must exist in preparation for the waves must be such that 
dujdy becomes infinite at the surface. It appears to be this cir 
cumstance which renders it possible for even rotational waves to 
attain in the limit to an infinite thinness of crest without losing 
the property of uniform propagation. 

152 



228 ON THE THEORY OF OSCILLATORY WAVES. 

When swells are propagated towards a smooth, very gently 
shelving shore, the height increases when the finiteness of depth 
begins to take effect. Presently the limiting height for uniformly 
propagated irrotational waves is passed, and then the form of the 
wave changes independently of the mere secular change due to 
diminishing depth. The tendency is now for the high parts to 
overtake the less high in front of them, and thereby to become 
higher still, until at last the crest topples over and the wave 
finally breaks. The breaking is no doubt influenced by friction 
against the bottom (denoting by " friction" the effect of the eddies 
produced), but I do not believe that it is wholly or even mainly 
due to this cause. Before the wave breaks altogether the top 
gets very thin, but the maximum height for uniform propaga 
tion is probably already passed by a good deal, so that we must 
guard against being misled by this observation as to the character 
of the limiting form. 

In watching many years ago a grand surf which came rolling 
in on a sandy beach near the Giant s Causeway, without any storm 
at the place itself, I recollect being struck with the blunt wedge- 
like form of the waves where they first lost their flowing outline, 
and began to show a little broken water at the very summit. It is 
only I imagine on an oceanic coast, and even there on somewhat 
rare occasions, that the form of waves of this kind, of nearly the 
maximum height, can be studied to full advantage. The observer 
must be stationed nearly in a line with the ridges of the waves 
where they begin to break. 



C. Remark on the method of Art. I. 

There appears to be a slight advantage in employing the 
function U or ^ (= j(udy - vdx) ) instead of <, the wave motion 
having been reduced to steady motion as is virtually done in 
Art. 1. The general equation for i/r is the same as for </>, (2), and 
the general expression for ty answering to that given for < on 
p. 212 is 

C e rm h -ri - -(k-ti cos rmx. 



The expression for p in terms of ty is almost identical with that in 
terms of c/>. So far there is nothing to choose between the two. But 



ON THE THEORY OF OSCILLATORY WAVES. 229 

for the two equations which have to be satisfied simultaneously at 
the surface, instead of p = and the somewhat complicated equa 
tion (7), we have ^ = and ^r = const., which constant we may 
take =0 if we leave open the origin of y. The substitution of 
this equation of simpler form for (7) is a gain in proceeding to 
higher orders of approximation. I remember however thinking 
as I was working at the paper that as far as the approximation 
there went the gain was not such as to render it worth while to 
make the change. 

But while these sheets were going through the press I devised 
a totally different method of conducting the approximation, which 
I find possesses very substantial advantages in proceeding to 
higher orders of approximation. The reader will find this new 
method after the paper "on the critical values of the sums of 
periodic series."] 



[From the Report of the British Association for 1847, Part n. p. 6.] 

ON THE RESISTANCE OF A FLUID TO Two OSCILLATING 
SPHERES. 

THE object of this communication was to shew the application 
of Professor Thomson s method of images to the solution of certain 
problems in hydrodynamics. Suppose that there exists in an in 
finite mass of incompressible fluid a point from which, or to which 
the fluid is flowing with a velocity alike in all directions. Con 
ceive now two such points, of intensities equal in magnitude and 
opposite in sign, to coexist in the fluid ; and then suppose these 
points to approach, and ultimately coalesce, their intensities varying 
inversely as the distance between them. Let the resulting point be 
called a singular point of the second order. The motion of a fluid 
about a solid, oscillating sphere is the same as if the solid sphere 
were replaced by fluid, in the centre of which existed such a point. 
It is easy to shew that the motion of the fluid due to a point of 
this kind, when the fluid is interrupted by a sphere having its 
centre in the axis of the singular point, is the same as if the 
sphere s place were occupied by fluid containing one singular point 
of the second order. By the application of this principle may be 
found the resistance experienced by a sphere oscillating in presence 
of a fixed sphere or plane, or within a spherical envelope, the 
oscillation taking place in the line joining the centres, or perpen 
dicular to the plane. In a similar manner may be found the resist 
ance to two spheres which touch, or are connected by a rod, or to 
the solid made up of two spheres which cut, provided the exterior 
angle of the surfaces be a submultiple of two right angles, the 
oscillation in these cases also taking place in the line joining the 
centres. The numerical calculation is very simple, and may be 
carried to any degree of accuracy. 



RESISTANCE OF A FLUID TO TWO OSCILLATING SPHERES. 231 

The investigation mentioned in the preceding paper arose out 
of the communication to me by Sir William Thomson of his 
beautiful method of electrical images before he had published it. 
Having myself paid more attention to the motion of fluids than 
to electricity, I endeavoured to find if it would in any manner 
apply to the solution of problems in the motion of fluids. I found 
that what is called above a singular point of the second order had 
a perfect image in a sphere when its axis was in the direction of 
a radius, which led to a complete solution of the problem men 
tioned in the paper when one sphere lay wholly outside or inside 
the other. I shewed this to Professor Thomson, who pointed out 
to me that a solution was also attainable, and that in finite 
terms, when the spheres intersected, provided the angle of inter 
section was a submultiple of two right angles. He saw that the 
property of a singular point of the second order of giving a perfect 
image in the case mentioned, admitted of an application to the 
theory of magnetism, which he has published in a short paper in 
the second volume of the Cambridge and Dublin Mathematical 
Journal, (1847) p. 240. 

Although the mathematical result is contained in the paper 
just mentioned, I subjoin the process by which I found it out. 

The expression (see p. 41) for the function < around a sphere 
which moves in a perfect fluid previously at rest may be thought 
of as applying to the whole of an infinite mass of fluid, provided 
we conceive what has here been called a singular point of the 
second order to exist at the origin. Let us conceive a spherical 
surface S with its centre at and having a radius a to exist in 
the fluid ; let P be the singular point, lying either within or with 
out the sphere S, and having its axis in the line OP. Let /, & be 
polar co-ordinates originating at P, & being measured from OP 
produced, and let r, 6 be polar co-ordinates originating at ; let 
m be a constant, and OP c, then </> being the function due to the 
singular point we have 

mcos# m.r cos0 rcos# c 



Now if e be less than 1, 



m-j- (r 2 2cr cos 6 + c 2 / 
etc 



232 ON THE RESISTANCE OF A FLUID 

where P , P x , P 2 ... are Laplace s, or in this case more properly 
Legendre s, coefficients*. Hence by expanding and differentiating 
with respect to c, we have 



(1), 



We are not of course concerned with the constant term in the 
latter of these two expressions. For the normal velocity (v) at 
the surface of the sphere we get by differentiating with respect 
to r, and then putting r a 

/1 . 2P t 2 . 3cP 2 3 . 4c 2 P 8 , \ . , , Q , 

v = m( ^ L -] 1 H 5 -+ ... ), if a> c (3), 

\ a a CL / 




First suppose the point P outside the sphere, let the sphere be 
thought of as a solid sphere, and consider the motion "reflected " 
(p. 28) from it. The reflected motion being symmetrical about 
the axis, we must have for it 



where Q , Q lf Q t ... are Laplace s functions involving 6 only. This 
gives for the normal velocity (v) in the reflected motion at the 
surface of the sphere 



and since we must have v = v we get from (4) and (6) 

la P. 2a 5 P 2 3a 7 P 3 

Q = 0, Q^wj H, Q 2 = m-^- 2 , Q 9 = n*gp 

which reduces (5) to 

a 3 /!P 2a 2 P 3a 4 P 



* The functions which in Art. 9 of the paper " On some Cases of Fluid Motion" 
(p. 38) I called "Laplace s coefficients," following the nomenclature of Pratt s 
Mechanical Philosophy, are more properly called " Laplace s functions ;" the term 
"Laplace s coefficients" being used to mean the coefficients in the expansion of 

[1 - 2e { cos 6 cos & + sin sin ff cos (w - w ) } + c 2 ]"^, 
to be understood according to the usual notation and not as in the text. 



TO TWO OSCILLATING SPHERES. 233 

This is identical with what (1) becomes on writing w , c for m, C 

provided that 

a 3 , a 2 

m =- m - 3) c=-. 

Hence the reflected motion is perfectly represented by sup 
posing the sphere s place occupied by fluid within which, at the 
point P in the line OP determined by OP = c, there exists a 
singular point of the same character as P, but of opposite sign, 
and of intensity less in the ratio of a 3 to c 3 . 

The case of a spherical mass of fluid within a rigid enclosure 
and containing a singular point of the second order with its axis 
in a radial direction might be treated in a manner precisely similar, 
by supposing the space exterior to the sphere filled with fluid, 
taking to represent the reflected motion in this case, instead of (5), 
the corresponding expression according to ascending powers of r, 
and comparing the resulting normal velocity at the surface of the 
sphere with (3) instead of (4). This is however unnecessary, since 
we see that the relation between the two singular points P, P is 
reciprocal, so that either may be regarded as the image of the 
other. 

Suppose now that we have two solid spheres, S, S , exterior to 
each other, immersed in a fluid. Suppose that S is at rest, and 
that S moves in the direction of the line joining the centres, the 
fluid being at rest except as depends on the motion of S. The 
motion of the fluid may be determined by the method of successive 
reflections (p. 28), which in this case becomes greatly simplified 
in consequence of the existence of a perfect image representing 
each reflected motion, so that the process is identical with that of 
Thomson s method of images, except that the decrease of intensity 
of the successive images takes place according to the cubes of the 
ratios of the successive quantities such as a, c, instead of the first 
powers. 

If a sphere move inside a spherical envelope, in the direction 
of the line joining the centres, the space between being filled with 
fluid which is otherwise at rest, the motion may be determined in 
a precisely similar manner. 

If two spheres outside each other, or just touching, be con 
nected by an infinitely thin rod, and move in a fluid in the direction 
of the line joining their centres, we have only to find the motion 



234 ON THE RESISTANCE OF A FLUID 

due to the motion of each sphere supposing the other at rest, and 
to superpose the results. 

I should probably not have thought of applying the method 
to the solid bounded by the outer portions of two intersect 
ing spheres, had not Professor Thomson shewn me that it was 
not limited to the cases in which each sphere is complete ; and 
that although it fails from non-convergence when the spheres 
intersect, yet when the exterior angle of intersection is a sub- 
multiple of two right angles the places of the successive images 
recur in a cycle, and a solution of the problem may be obtained 
in finite terms by placing singular points of the second order at 
the places of the images in a complete cycle. 

The simplest case is that in which the spheres are generated 
by the revolution round their common axis of two circles which 
intersect at right angles. In this case if $, S f are the spheres, 
0, their centres, O l the middle point of the common chord of 
the circles, the image of in $ will be at O lt and the image 
of O in S will be at . 



Let a, b be the radii of the spheres ; c the distance V( 2 
of their centres ; e, f the distances a?/c, tf/c of O l from 0, ; 
C the velocity of the spheres ; r, 6 the polar co-ordinates of any 
point measured from 0; r\, t the co-ordinates measured from 4 ; 
r, & the co-ordinates measured from ; 0, 6 lt & being all 
measured from the line 00 . If S were away, we should have 
for the fluid exterior to 8 

n 3 cos 6 

+ -<&-&- 

For the image of this in S we have a singular point at 6) x for 

which 

, Ca*b* cosfl, 

* = ~<T ^> 

and for the image of this again in 8 we have a singular point at 
for which 

3COS0 7 
9 = - oo -gpr > 

which is precisely what is required to give the right normal 
velocity at the surface of S . Moreover all the singular points 
lie inside the space bounded by the exterior portions of the inter- 



TO TWO OSCILLATING SPHERES. 235 

secting spheres. Hence the three motions together satisfy all the 
conditions of the problem, so that for the complete solution we 
have 

. , (a* cos d a s b 3 cos O l cos ff\ 
0= -*V\?r- -ff*- -?r-} 

Just as in the case of a sphere, if a force act on the solid in 
the direction of its axis, causing a change in the velocity C, the 
only part of the expression for the resistance of the fluid which 
will have a resultant will be that depending upon dC/dt. This 
follows at once, as at pp. 50, 51, from the consideration that when 
there is no change of C the vis viva is constant, and therefore the 
resultant pressure is nil. If we denote by M dC/dt the resultant 
pressure acting backwards, we get for the part of M due to the 
pressure of the fluid on the exposed portion of the surface of S t 



72 ffa 3 cos0 cWcosfl, , ,,/! . 
vrpb I j -- 2 --- 3-72 + # cos f cos # sin 

taken between proper limits. Putting b cos & x t we have 
r cos 6 = c + x, ?\ cos A =/ + x, 



Expressing cos 0, cos0 t , cos & in terms of x and r, x and r lt x, 
and changing the independent variable, first to x y and then in the 
first term to r and in the second to r lt we have for the indefinite 
integral with sign changed 



which is to be taken between the limits r = a to r = c + b, r^ ab/c 
to f+b, x f to 6. The part of M due to the integral over 
the exposed part of the surface of 8 will be got from the above 
by interchanging; and on adding the two expressions together, 
and putting /= b 2 /c, c = V( 2 + & 2 )> we get for the final result 

- 26 6 }. 

When one of the radii, as b, vanishes, we get M = f Trpa 3 as 
it ought to be. 



[From the Transaction* of the Cambridge Philosophical Society, 
Vol. vin. p. 533.] 



ON THE CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES. 

[Read December 6, 1847.] 

THERE are a great many problems in Heat, Electricity, Fluid 
Motion, &c., the solution of which is effected by developing an 
arbitrary function, either in a series or in an integral, by means of 
functions of known form. The first example of the systematic 
employment of this method is to be found in Fourier s Theory 
of Heat. The theory of such developements has since become an 
important branch of pure mathematics. 

Among the various series by which an arbitrary function f(x) 
can be expressed within certain limits, as and a, of the variable 
#, may particularly be mentioned the series which proceeds accord 
ing to sines of TTX/O, and its multiples, and that which proceeds 
according to cosines of the same angles. It has been rigorously 
demonstrated that an arbitrary, but finite function of #, /(#), may 
be expanded in either of these series. The function is not 
restricted to be continuous in the interval, that is to say, it may 
pass abruptly from one finite value to another ; nor is either the 
function or its derivative restricted to vanish at the limits and a. 
Although however the possibility of the expansion of an arbitrary 
function in a series of sines, for instance, when the function does 
not vanish at the limits and a, cannot but have been contem 
plated, the utility of this form of expansion has hitherto, so far as 
I am aware, been considered to depend on the actual evanescence 
of the function at those limits. In fact, if the conditions of the 
problem require that /(O) and f(a) be equal to zero, it has been 



CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES. 237 

considered that these conditions were satisfied by selecting the 
form of expansion referred to. The chief object of the following 
paper is to develope the principles according to which the expan 
sion of an arbitrary function is to be treated when the conditions 
at the limits which determine the particular form of the expansion 
are apparently violated ; and to shew, by examples, the advantage 
that frequently results from the employment of the series in such 
cases. 

In Section I. I have begun by proving the possibility of the 
expansion of an arbitrary function in a series of sines. Two 
methods have been principally employed, at least in the simpler 
cases, in demonstrating the possibility of such expansions. One, 
which is that employed by Poisson, consists in considering the 
series as the limit of another formed from it by multiplying its 
terms by the ascending powers of a quantity infinitely little less 
than 1 ; the other consists in summing the series to n terms, that 
is, expressing the sum by a definite integral, and then considering 
the limit to which the sum tends when n becomes infinite. The 
latter method certainly appears the more direct, whenever the 
summation to n terms can be effected, which however is not always 
the case; but the former has this in its favour, that it is thus 
that the series present themselves in physical problems. The 
former is the method which I have followed, as being that which 
I employed when I first began the following investigations, and 
accordingly that which best harmonizes with the rest of the paper. 
I should hardly have ventured to bring a somewhat modified 
proof of a well-known theorem before the notice of this Society, 
were it not for the doubts which some mathematicians seem to 
feel on this subject, and because there are some points which 
Poisson does not seem to have treated with sufficient detail. 

I have next shewn how the existence and nature of the dis 
continuity of /(& ) and its derivatives may be ascertained merely 
from the series, whether of sines or cosines, in which f(x) is 
developed, even though the summation of the series cannot be 
effected. I have also given formulae for obtaining the develope - 
ments of the derivatives of f(x) from that of f(x) itself. These 
developements cannot in general be obtained by the immediate 
differentiation of the several terms of the developement of f(x), 
or in other words by differentiating under the sign of summa 
tion. 



238 ON THE CRITICAL VALUES OF 

It is usual to restrict the expanded function to be finite. This 
restriction however is not necessary, as is shewn towards the end 
of the section. It is sufficient that the integral of the function be 
finite. 

Section II. contains formulae applicable to the integrals which 
replace the series considered in Section I. when the extent a of 
the variable throughout which the function is considered is sup 
posed to become infinite. 

Section III. contains some general considerations respecting 
series and integrals, with reference especially to the discontinuity 
of the functions which they express. Some of the results obtained 
in this section are referred to by anticipation in Sections I. and II. 
They could not well be introduced in their place without too much 
interrupting the continuity of the subject. 

Section IV. consists of examples of the application of the pre 
ceding results. These examples are all taken from physical 
problems, which in fact afford the best illustrations of the applica 
tion of periodic series and integrals. Some of the problems 
considered are interesting on their own account, others, only as 
applications of mathematical processes. It would be unnecessary 
here to enumerate these problems, which will be found in their 
proper place. It will be sufficient to make one or two remarks. 

The problem considered in Art. 52, which is that of determin 
ing the potential due to an electrical point in the interior of a 
hollow conducting rectangular parallelepiped, and to the electricity 
induced on the surface, is given more for the sake of the artifice 
by which it is solved than as illustrating the methods of this paper. 
The more obvious mode of solving this problem would lead to a 
very complicated result. 

The problem solved in Art. 54 affords perhaps the best example 
of the utility of the methods given in this paper. The problem 
consists in determining the motion of a fluid within the sector of a 
cylinder, which is made to oscillate about its axis, or a line parallel 
to its axis. The expression for the moment of inertia of the 
fluid which would be obtained by the methods generally employed 
in the solution of such problems is a definite integral, the numeri 
cal calculation of which would be very laborious; whereas the 
expression obtained by the method of this paper is an infinite series 
which may be summed, to a sufficient degree of approximation, 
without much trouble. 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 239 

The series for the developement of an arbitrary function con 
sidered in this paper are two, a series of sines and a series of 
cosines, together with the corresponding integrals ; but similar 
methods may be applied in other cases. I believe that the follow 
ing statement will be found to embrace the cases to which the 
method will apply. 

Let u be a continuous function of any number of independent 
variables, which is considered for values of the variables lying 
within certain limits. For facility of explanation, suppose u a 
function of the rectangular co-ordinates x, y, z, or of x, y, z and t, 
where t is the time, and suppose that u is considered for values of 
x, y, z, t lying between and a, and b, and c, and T, respec 
tively. For such values suppose that u satisfies a linear partial 
differential equation, and suppose it to satisfy certain linear equa 
tions of condition for the limiting values of the variables. Let 
Z7=0, U = Q be two of the equations of condition, corresponding 
to the two limiting values of one of the variables, as x. Then 
the expansion of u to which these equations lead may be applied 
to the more general problem which leads to the corresponding 
equations of condition U F, U = F t where F and F are any 
functions of all the variables except x, or of any number of 
them. 



SECTION I. 

Mode of ascertaining the nature of the discontinuity of a function 
which is expanded in a series of sines or cosines, and of obtain 
ing the developements of the derived functions. 

1. By the term, function I understand in this paper a quantity 
whose value depends in any manner on the value of the variable, 
or on the values of the several variables of which it is composed. 
Thus the functions considered need not be such as admit of being 
expressed by any combination of algebraical symbols, even between 
limits of the variables ever so close. I shall assume the ordinary 
rules of the differential and integral calculus as applicable to such 
functions, supposing those rules to have been established by the 
method of limits, which does not in the least require the 
possibility of the algebraical expression of the functions con 
sidered. 



240 ON THE CRITICAL VALUES OF 

The term discontinuous, as applied to a function of a single 
variable, has been used in two totally different senses. Sometimes 
a function is called discontinuous when its algebraical expression 
for values of the variable lying between certain limits is different 
from its algebraical expression for values of the variable lying 
between other limits. Sometimes a function of x, f(x), is called 
continuous when, for all values of #, the difference between f (x) 
and f(xh) can be made smaller than any assignable quantity by 
sufficiently diminishing h, and in the contrary case discontinuous. 
If / (x) can become infinite for a finite value of x, it will be con 
venient to consider it as discontinuous according to the second 
definition. It is easy to see that a function may be discontinuous 
in the first sense and continuous in the second, and vice versa. 
The second is the sense in which the term discontinuous is I 
believe generally employed in treatises on the differential calculus 
which proceed according to the method of limits, and is the sense 
in which I shall use the term in this paper. The terms continuous 
and discontinuous might be applied in either of the above senses 
to functions of two or more independent variables. If I have 
occasion to employ them as applied to such a function, I shall 
employ them in the second sense; but for the present I shall 
consider only functions of one independent variable. 

In the case of the functions considered in this paper, the value 
of the variable is usually supposed to be restricted to lie within 
certain limits, as will presently appear. I exclude from considera 
tion all functions which either become infinite themselves, or have 
any of their differential coefficients of the orders considered 
becoming infinite, within the limits of the variable within which 
the function is considered, or at the limits themselves, except 
when the contrary is expressly stated. Thus in an investigation 
into which / (x) and its first n differential coefficients enter, and 
in which f(x) is considered between the limits x=0 and a? = a, 
those functions are excluded, at least at first, which are such that 
any one of the quantities /(#), / (x) ...f n (x) is infinite for a 
value of x lying between and a, or for x = or x = a ; but the 
differential coefficients of the higher orders may become infinite. 
The quantities /(#), / (#) ..-/ n (x) may however alter discon- 
tinuously between the limits x = and x = a, but I exclude 
from consideration all functions which are such that any one of 
the above quantities alters discontinuously an infinite number 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 241 

of times between the limits within which x is supposed to 
lie. 

The terms convergent and divergent, as applied to infinite series, 
will be used in this paper in their usual sense ; that is to say, a 
series will be called convergent when the sum to n terms 
approaches a finite and unique limit as n increases beyond all 
limit, and divergent in the contrary case. Series such as 

1 1 + 1 . . . , sin x + sin Zx + sin 3 ic + . . . , 

(where x is supposed not to be or a multiple of TT,) will come 
under the class divergent ; for, although the sum to n terms does 
not increase beyond all limit, it does not approach a unique limit 
as n increases beyond all limit. Of course the first n terms of a 
divergent series may be the limits of those of a convergent series : 
nor does it appear possible to invent a series so rapidly divergent 
that it shall not be possible to find a convergent series which shall 
have for the limits of its first n terms the first n terms respectively 
of the divergent series. Of course we may employ a divergent 
series merely as an abbreviated mode of expressing the limit of 
the sum of a convergent series. Whenever a divergent series is 
employed in this way in the present paper, it will be expressly 
stated that the series is so regarded. 

Convergent series may be divided into two classes, according 
as the series resulting from taking all the terms of the given 
series positively is convergent or divergent. It will be convenient 
for the purposes of the present paper to have names for these two 
classes. I shall accordingly call series belonging to the first class 
essentially convergent, and series belonging to the second acci 
dentally convergent, while the term convergent, simply, will be used 
to include both classes. Thus, according to the definitions which 
will be employed in this paper, the series 

x + J z* 4- J x 9 + ... 

is essentially convergent so long as x 2 < 1 ; it is divergent when 
as* > 1, and when x = 1 ; and it is accidentally convergent when 
x = -l. 

The same definitions may be applied to integrals, when one at 

/oo 

least of the limits of integration is oo . Thus, if a > 0, / of 2 djc 

J a 

s. 16 



242 ON THE CRITICAL VALUES OF 

t* 00 

is essentially convergent at the limit oo , while I ^-^ dx is only 

J a & 

accidentally convergent, andjsina;^, not being convergent, 

J a 

comes under the class of divergent integrals. These definitions 
may be applied also to integrals taken between finite limits, when the 
quantity under the integral sign becomes infinite within the limits 

f a 
of integration, or at one of the limits. Thus I log # dx is conver- 

J o 

[ a dx 

gent, but I - divergent, at the limit 0. 
J o ^ 

2. Let f (x) be a function of x which is only considered 
between the limits x = and x = a, and which can be expanded 
between those limits in a convergent series of sines of irx/a and 
its multiples, so that 



To determine A n , multiply both sides of (1) by smmrx/a.dx and 
integrate from x = to x a. Since the series in (1) is conver 
gent, and sin mrx/a does not become infinite for any real value 
of x, we may first multiply each term by sin mrx/a . dx and 
integrate, and then sum, instead of first summing and then inte 
grating*. But each term of the series in (1) except the n th will 
produce in the new series a term equal to zero, and the n th will 
produce J aA n . Hence 

A 2 [ a , . . nirx 7 
A - fix) sin - dx, 
n aJ 

and therefore 

*f \ % \? f a ^/ \ n7rx 7 7?7r< ^ /o\ 

fW = -% /Wsin-- doe. am-- ............ (2). 

Uj J o tl U 

3. Hence, whenever f(x) can be expanded in the convergent 
series which forms the right-hand side of (1), the value of A n can 
be very readily found, and the expansion performed. But this 
leaves us quite in the dark as to the degree of generality that a 
function which can be so expanded admits of. In considering this 

* Moigno, Lecons de Cdlcul DifferenUel, &c. Tom. n. p. 70. 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 243 

question it will be convenient, instead of endeavouring to develope 
f(x), to seek the value of the infinite series 



2<C f // \ n7rx j i n7r % /o\ 

- 2, /() sm ax . sm - (3), 

a J a a 

provided the series be convergent ; for it is only in that case that 
we can, without further definition, speak of the sum of the series 
at all. Now if we had only a finite number n of terms in the 
series (3) we might of course replace the series by 

- I f(x ) |sin sin + sin - sin . . , 

aJo [a a a a 

. njrx . nirx] -, , , . N 

4- sin sin \ dx (4). 

a a j 

As it is however this transformation cannot be made, because, the 
series within brackets in the expression which would replace (4) 
not being convergent, the expression would be a mere symbol 
without any meaning. If however the series (3) is essentially con 
vergent, its sum is equal to the limit of the sum of the following 
essentially convergent series 

- 2<7 n / f (V) sin dx .sin (5), 

a y Jo a a 

when g from having been less than 1 becomes in the limit 1. It 
will be observed that if (3) were only accidentally convergent, we 
could not with certainty affirm the sum of (3) to be the limit of 
the sum of (5). For it is conceivable, or at least not at present 
proved to be impossible, that the mode of the mutual destruction 
of the terms of (3) in the infinitely remote part of the series 
should be altered by the introduction of the factor g n , however 
little^ might differ from 1. Let us now, instead of seeking the 
sum of (3) in those cases in which the series is convergent, seek 
the limit to which the sum of (5) approaches as g approaches to 1 
as its limit. 

4. The transformation already referred to, which could not be 
effected on the series (3), may be effected on (5), that is to say, 
instead of first integrating the several terms and then summing, 
we may first sum and then integrate. We have thus, for the value 
of the series, 

9 Ca ( -v/ ^\ 

(6). 

162 



244 ON THE CRITICAL VALUES OF 

The convergent series within brackets can easily be summed. The 
expression (6) thus becomes 



(7). 



1 2# cos TT (se -f a?) /a + # 

Now since the quantity under the integral sign vanishes when 
g = 1, provided cos TT (a? + a?) /a be not = 1, the limit of (7) when 
g 1 will not be altered if we replace the limits and a of x by 
any other limits or groups of limits as close as we please, provided 
they contain the values of x which render x x equal to zero or 
any multiple of 2a. Let us first suppose that we are considering 
a value of x lying between and a, and in the neighbourhood of 
which /(#) alters continuously. Then, since x + x never becomes 
equal to zero or any multiple of 2a within the limits of integra 
tion, we may omit the second term within brackets in (7) ; and 
since x x never becomes equal to any multiple of 2a, and 
vanishes only when x = x } we may take for the limits of x two 
quantities lying as close as we please to x, and therefore so close 
as to exclude all values of x for which f(x) alters discontinuously. 
Let g=~Lh, x = x + f , expand cos irf/a by the ordinary formula, 
and put f(x) =/ (x) + R. Then the limit of (7) will be the same 
as that of 



the limits of f being as small as we please, the first negative and 
the second positive. Let now 



so that d/dj; is ultimately equal to a/ir, that is to say when g is 
first made equal to 1, and then the limits of f, and therefore those 
of f, are made to coalesce. Let now G, L be respectively the 
greatest and least values of (1 - |A) a~ l dg/di; {/(#) + R] within 
the limits of integration. Then if we observe that 



W + * 
were tan" 1 denotes an angle lying between ?r/2 and ?r/2, putting 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 245 

- f p 2 for the limits of , we shall see that the value of the inte 
gral (8) lies between 

G (tan 1 fjh + tan- 2 //t) and L (tan 1 gjh + tan" 1 gjh) : 

but in the limit, that is to say, when we first suppose h to vanish 
and then and f g , 6r and Z become equal to each other and to 
ir~ l f(x) t and tan" 1 f x /& + tan" 1 f 2 //i becomes equal to TT. Hence, 
/(#) is the limit of (7). 

Next, suppose that the value of x which we are considering 
lies between and a, and that as x passes through it /(# ) alters 
suddenly from M to N. Then the reasoning will be exactly as 
before, except that we must integrate separately for positive and 
negative values of f , replacing f(x) + R by M + R in the latter 
case, and by N + R in the former. Hence, the limit of (7) will be 



Lastly, if we are considering the extreme values as = and 
sc = a, it follows at once from the form of (7) that its limiting value 
is zero. 

Hence the limit to which the sum of the convergent series (5) 
tends as g tends to 1 as its limit is f(x) for values of x lying 
between and a, for which f(x) alters continuously, it is J (M+N) 
for values of as for which f (x) alters suddenly from M to N, and it 
is zero for the extreme values and a. 

5. Of course the limiting value of the series (5) is f(fty and 
not zero, if we suppose that g first becomes 1 and then x passes 
from a positive value to zero. In the same way, if/(#) alters 
abruptly from M to N as x increases through x lt the limiting value 
of (5) will be M if we suppose that g first becomes 1 and then x 
increases to x^ and it will be N if we suppose that g first becomes 
1 and then x decreases to x lt It would be futile to argue that the 
limiting value of (5) for # = is zero rather than /(O), or /(O) 
rather than zero, since that entirely depends on the sense in which 
we employ the expression limiting value. Whichever sense we 
please to adopt, no error can possibly result, provided we are only 
consistent, and do not in the course of the same investigation 
change the meaning of our words. 

It is a principle of great importance in these investigations, 
that a function of two independent variables which becomes 



246 ON THE CRITICAL VALUES OF 

indeterminate for particular values of the variables may have 
different limiting values according to the order in which we 
suppose the variables to assume their particular values, or ac 
cording to the nature of the arbitrary relation which we conceive 
imposed on them as they approach those values together. 

I would here make one remark on the subject of consistency. 
We may speak of the sum of an infinite series which is not con 
vergent, if we define it to mean the limit of the sum of a con 
vergent series of which the first n terms become in the limit 
the same as those of the divergent series. According to this 
definition, it appears quite conceivable that the same divergent 
series should have a different sum according as it is regarded 
as the limit of one convergent series or of another. If however 
we are careful in the same investigation always to regard the 
same divergent series, and the series derived from it, as the 
limits of the same convergent series and the series derived from 
it, it does not appear possible to fall into error, assuming of course 
that we always reason correctly. For example, we may employ 
the series (3), and the series derived from it by differentiation, &c., 
without fear, provided we always regard these series when di 
vergent, or only accidentally convergent, as the limits of the 
particular convergent series formed by multiplying their n th terms 

6. We may now consider the convergency of the series (3), in 
order to find whether we may employ it directly, or whether we 
must regard it as the limit of (5). 

By integrating by parts in the ri b term of (3), we have 

mrx 7 , 2 -, , N nirx 
sin dx = -f (x ) cos 



a WIT a 



Suppose that / (x) does not necessarily vanish at the limits 
x = and x = a, and that it alters discontinuously any finite 
number of times between those limits, passing abruptly from 
M^ to N t when x increases through a lt from M 2 to JV 2 when x 
increases through 2 , and so on. Then,. if we put 8 for the sign of 
summation referring to the discontinuous values of /(a? ), on 
taking the integrals in (9) from # = to x = a, we shall get for 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 247 

the part of the integral corresponding to the first term at the 
right-hand side of the equation 



^ {/(O) -(-)"/() + S(N-M) cos - ........... (10). 

It is easily seen that the last two terms in (9) will give a 
part of the integral taken from to a, which is numerically 
inferior to L/n*, where L is a constant properly chosen. As far 
as regards these terms therefore the series (3) will be essentially 
convergent, and its sum will therefore be the limit of the sum 
of (5). 

Hence, in examining the convergency or divergency of the 
series (3), we have only got to consider the part of the coefficient 
of sin mrx/a of which (10) is the expression. The terms /(O), 
/(a) in this expression may be included under the sign 8 if we 
put for the first a = 0, M 0, -ZV=/(0), and for the second a = a, 
M=f(a), N=Q. We have thus got a set of series to consider 
of which the type is 

2 ., r ,,, ^ 1 mrx . niroc ..,,. 

- (N M)2, - cos - sm - ............ (11). 

TT n a a 

If we replace the product of the sine and cosine in this ex 
pression by the sum of two sines, by means of the ordinary 
formula, and omit unnecessary constants, we shall have for the 
series to consider 



Let now 



2-sinw* (12). 

n ^ } 



u sin^ + Jsin 2z ... -f - sin nz (13), 

Tl 



then 



du sin (n 4- ] z 

-T- = cos z -f cos 2z . . . + cos nz = ,\ V 

dz 2 sm z 



and since u vanishes with z, in which case m \ l ^ z J s finite, 

sm \z 

we shall have, supposing z to lie between 2?r and + 2?r, so 
that the quantity under the integral sign does not become infinite 
within the limits of integration, 



*H;=EH^*-i W> 



248 ON THE CRITICAL VALUES OF 

and we have to find whether the integral contained in this 
equation approaches a finite limit as n increases beyond all limit, 
and if so what that limit is. Since u changes sign with z, we 
need not consider the negative values of z. 

First suppose the superior limit z to lie between and 2?r ; 
and to simplify the integral write 2z for z, n for 2n + 1, so that 
the superior limit of the new integral lies between and TT ; 
then the integral 

[ z sinnz , f sOnnz z 7 f^sinnz,., ^ . 7 
= . dz = - . dz \ (1 + Ez\ dz, 

J o sin z Jo z sin z J o z 

where R = ; , a quantity which does not become infinite 

z sin z 

within the limits of integration. Hence, as is known, the limit of 
sin nz . Rdz when n increases beyond all limit is zero. Hence, 



o 
if / be the limit of the integral, 



,.,..-- [*smnz 7 r ., f C 
I = limit of -dz- limit of 

JQ Z JO 



b 

Now, z being given, the limit of nz is oo , and therefore 

7T 

2 

Secondly, suppose z in (14) to be equal to 0. Then it follows 
directly from this equation, or in fact at once from (13), that 
u = 0, and consequently the limit of u = 0. 

The value of u in all other cases, if required, may be at once 
obtained from the consideration that the values of u recur when z 
is increased or diminished by 2?r, 

Hence, the series (12) is in all cases convergent, and has for 
its sum when z = 0, and \ (TT - z} when z lies between 
and 27T. 

Now, if in the theorem of Article 4, we write z for x, and 
put a = TT, f (z) = | (TT - z), we find, for values of z lying between 
and TT, and for z = TT, 



limit of 2, - g n sin nz = J (TT z) ; 



and evidently 



limit of 2 - # M sin n^ = 0, when a = 0, 



THE SUMS OF PERIODIC SERIES. (SECT, I. SERIES.) 249 

that is of course supposing z first to vanish and then gio become 1. 
Also the limit of ^n~ l g n sin nz changes sign with z, and recurs 
when z is increased or diminished by 2?r. Hence, the series (12), 
which has been proved to be convergent, is in all cases the limit 
to which the sum of the convergent series %n~ l g n sin nz tends 
as g tends to 1 as its limit. Now the series (11) may be de 
composed into two series of the form just discussed, whence it 
follows that the series (3) is always convergent, and its sum for 
all values of x, critical as well as general, is the limit of the 
sum of the series (5), when g becomes equal to 1. 

The examination of the convergency of the series (3) in the 
only doubtful case, that is to say, the case in which / (x) is dis 
continuous, or does not vanish for x = and for x = a, is more 
curious than important. For in the analytical applications of 
the series (3) it would be sufficient to regard it as the limit of 
the series (5) ; and in the case in which (3) is only accidentally 
convergent, we should hardly think of employing it in the 
numerical computation of f(x) if we could possibly help it, and 
it will immediately appear that in all the cases which are most 
important to consider we can get rid of the troublesome terms 
without knowing the sum of the series. 

The proof of the convergency of the series (3) which has just 
been given, though in some respects I believe new, is certainly 
rather circuitous, and it has the disadvantage of not applying 
to the case in which f (x) is infinite*, an objection which does 
not apply to the proof given by M. Dirichletf. It has been 
supposed moreover that f"(x) is not infinite. The latter re 
striction however may easily be removed, as in the end of the 
next article. 

7. Let f(x) be a function of x which is expanded between 
the limits x = and x a in the series (3). Let the series be 



A . TTOC A . 7rx . . mrx , . 

^sin -- h^4 2 sm --- ... + A H sin --- K.., ..,.(15), 
CL a a 

and suppose that we have given the coefficients A } , A 2 ... , but 
do not know the sum of the series f(x). We may for all that 
find the values of /(O) and /(a), and likewise the values of x 

* This restriction may however be dispensed with by what is proved in Art. 20. 
t Crelle s Journal, Tom. iv. p. 157. 



250 ON THE CRITICAL VALUES OF 

for which f(x) is discontinuous, and the quantity by which f(x) is 
increased as x increases through each of these critical values. 
For from (9) and (10) 



E being a quantity which does not become infinite with n. If 
then we use the term limit in an extended sense, so as to include 
quantities of the form (7 cos ny, [of course C ( l) n is a particular 
case,] or the sum of any finite number of such quantities, we 
shall have for n = oo , 

limit of ^= 

Let then the limit of nA n be found. It will appear under the 
form 



y ...... . ........... (17). 

Comparing this expression with (16), we shall have 



and for each term of the series denoted by 8 we shall have 



In particular, if f(x) is continuous, and if the limit of nA n is 
L Q or L e according as n is odd or even, we shall have 



+/()}, A = = !/(0) -/()); 

whence 

If f(x) were discontinuous for an infinite number of values 
of x lying between and a, it is conceivable that the infinite 
series coming under the sign 8 might be divergent, or if con 
vergent might have a sum from which n might wholly or partially 
disappear, in which case the limit of nA n might not come out 
under the form (17). It was for this reason among others, that 
in Art. 1, I excluded such functions from consideration. 

If f(x) be expressible algebraically between the limits x = Q 
and x = a, or if it admit of different algebraical expressions within 



THE SUMS OF PEEIODIC SERIES. (SECT. I. SERIES.) 251 

different portions into which that interval may be divided, A n 
will be an algebraical function of n, and the limit of nA n may 
be found by the ordinary methods. Under the term algebraical 
function, I here include transcendental functions, using the term 
algebraical function in opposition to what has been sometimes 
called an empirical function, or a general function, that is, a function 
in the sense in which the ordinate of a curve traced liberd manu 
is a function of the abscissa. Of course, in applying the theorem 
in this article to general functions, it must be taken as a postulate 
that the limit of nA n can be found, and put under the form (17). 

The theorem in question has been proved by means of equation 
(9), in which it is supposed that f"(sc] does not become infinite 
within the limits of integration. The theorem is however true 
independently of this restriction. To prove it we have only got to 
integrate by parts once instead of twice, and we thus get for 

T> 

the quantity which replaces the integral 



n 



9 /* 

I/. 



nirx , , 
cos - - ax , 



which by the principle of fluctuation* vanishes when n becomes 
infinite. There is however this difference between the two cases. 
When the series (15) has been cleared of the part for which the 
limit of n A n is finite, by the method which will be explained in 
the next article, the part which remains will be at least as con 
vergent in the former case as the series ^2 + -^... + -^+..., 

-L .Z n 

whereas we cannot affirm this to be true, and in fact it may 
be proved that it is not true, in the case in which f"(x) becomes 
infinite. Observing that the same remark will apply when we 
come to consider the critical values of the differential coefficients 



* I borrow this term from a paper by Sir William E. Hamilton On Fluctuating 
Functions, (Transactions of the Royal Irish Academy, Vol. xix. p. 264.) Had I 
been earlier acquainted with this paper, and that of M. Dirichlet already referred 
to, I would probably have adopted the second of the methods mentioned in the 
introduction for establishing equation (2) for any function, or rather, would have 
begun with Art. 7, taking that equation as established. I have retained Arts (2) 
(6), first, because I thought the reader would enter more readily into the spirit of 
the paper if these articles w r ere retained, and secondly, because I thought that 
Section in, which is adapted to this mode of viewing the subject, might be found 
useful. 



252 ON THE CRITICAL VALUES OF 

of f(x\ I shall suppose the functions and derived functions 
employed in each investigation not to become infinite, according 
to what has already been stated in Art. 1. 

8. After having found the several values of a, and the cor 
responding values of N M, we may subtract the expression (10) 
from A n , provided we subtract from the sum of the series (15) the 
sums of the several series such as (11). Now if X be the sum of 
the series (11), 



X = (N-M) s S in +2 gm-... (19). 

JT \ n a n a ) 

But it has been already shewn that 2 - sin nz = \ (TT z) when z 

n 

lies between and 2?r, = when z = 0, and = J (IT + z) when z 
lies between and 2?r. Now when x lies between and a, 
TT (x + a) /a lies between and 2?r, and TT (x a) /a lies between 
2?r and ; and when x lies between a. and a, TT (x + a) /a still lies 
between and 2?r, and TT (a? a) /a now lies between the same 
limits. Hence 

x 
X = (N- M) - , when x lies between and a 

...(20). 

/- _ / 

(N M) - , when x lies between a and a 
a 

We need not trouble ourselves with the singular values of the 
sum of the series (15), since we have seen that a singular value is 
always the arithmetic mean of the values of the sum for values 
of x immediately above and below the critical Value. This rule 
will apply to the extreme cases in which x = and x = a, if we 
consider the sum of the series for values of x lying beyond those 
limits. The rule applies to the series in (19), which is only a 
particular case of (15), and consequently will apply to any combi 
nation of series having this property, formed by way of addition 
or subtraction; since, when we increase or diminish any two. 
quantities M Q) N by any other two M, N respectively, we increase 
or diminish the arithmetic mean of the two former by the arith 
metic mean of the two latter. 

It has been already stated that we may, with a certain conven 
tion, include quantities referring to the limits x = and x = a 
under the sign of summation S. If we do so, and put H for the 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 253 



sum of the series (15), and B n for the remainder arising from sub 
tracting the expression (10) from A n , we shall have 



i n TT ^ r, . 

s, SX = zB n sm - 

CL 

and the sum of the series forming the right-hand side of this equa 
tion will be a continuous function of x. As to SX, the value of 
each series contained in it is given by equation (20). 

To illustrate this, suppose H the ordinate of a curve of which 
x is the abscissa. Let OGr be the axis of x\ OA, MB, ND, Ob 
right lines perpendicular to it, and let OG = a. Let the curve of 




Q 




which % is the ordinate be the discontinuous curve AB, CD, 
EFG. Take Gb equal to BC, and on the positive or negative 
side of the axis of x according as the ordinate decreases or increases 
as x increases through OM, and from measure an equal length 
Oc on the opposite side of the axis. Take Gd, Oe, each equal to 
DE, and draw the right lines AG, Ob b, cc G, Od d, ee G. Then 
it will be easily seen that if X v X lt X z be the values of X cor 
responding to the critical values of x, x = 0, x OM, x = ON, 
respectively, X will be represented by the right line AG ; X l by 



254 ON THE CRITICAL VALUES OF 

the discontinuous right line 06 , c G\ and X^ by the discontinuous 
right line Od , e G. Take MP equal to the sum of the ordinates 
of the points in which the right lines lying between OA and c B 
cut the latter line; M Q equal to the sum of the ordinates of the 
points in which the right lines lying between c B and d E cut the 
former, and so on, the ordinates being taken with their proper 
signs. Let P, Q, R, S be the points thus found, and join AP, 
QR, SO. Then SX will be represented by the discontinuous right 
line AP, QR, 8G. Let the ordinates of the discontinuous curve 
be diminished by those of the discontinuous right line last men 
tioned, and let the dotted curve be the result. Then H SX will 
be represented by the continuous, dotted curve. It will be 
observed that the two portions of the dotted curve which meet in 
each of the ordinates MB, NE may meet at a finite angle. If 
there should be a point in one of the continuous portions, such as 
AB, of the discontinuous curve where two tangents meet at a 
finite angle, there will of course be a corresponding point in the 
dotted curve. 

If we choose to take account of the conjugate points of 
the curve of which SX is the ordinate, it will be observed that 
they are situated at 0, and midway between P and Q, and between 
R and S. 

9. There are a great many series, similar to (3), in which f (x) 
may be expanded within certain limits of x. I shall consider one 
other, which as well as (3) is of great use, observing that almost 
exactly the same methods and the same reasoning will apply in 
other cases. 

The limit of the sum of the series 

1 f%/ N J 2 V n f%/ N nrjTX J U7rX /-M \ 

- f (x) dx + - 2.g H I /(a?) cos dx .cos - ...(21), 

d J o & J M & 

when g from having been less than 1 becomes 1, is f (#), x being 
supposed not to lie beyond the limits and a. For values, how 
ever, of x for which f (x) alters discontinuously, the limit of the 
sum is the arithmetic mean of the values of f (x) for values of x 
immediately above and below the critical value. I assume this as 
being well known, observing that it may be demonstrated jusfc as a 
similar theorem has been demonstrated in Art. 4. 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 255 

10. Let us now consider the series 

1 [ a /./ ,x 7 , 2 -, [ a , . , N n-irx , , mrx /c .<*\ 

- f(x) dx +- 2, /UP) cos das . cos .... (22). 

Q>JO a Jo a a 

We have by integration by parts 



mrx 7 , 
cos dx 



and now, taking the limits properly, and employing the letters 
M, N, a. and S in the same sense as before, we have 

2 [ a 
-jj-(x)a 



mr a n 2 " 

E being a quantity which does not become infinite with n. It 
follows from (23), that the series (22) is in all cases convergent, and 
its sum for all values of x } critical as well as general, is the limit of 
the sum of (21). 

It will be observed that if f (x) is a continuous function the 
series (22) is at least as convergent as the series 2 l/n*. This is 
not the case with the series (3), unless /(O) =/() = 0. 

If the constant term and the coefficient of cos mrx fa in the 
general term of (22) are given, f (x) itself not being known, 
except by its developement, we may as before find the values of x 
for which /(a?) is discontinuous, and the quantity by which f (x) 
is suddenly increased as x increases through each critical value. 
We may also, if we please, clear the series (22) of the slowly con 
vergent part corresponding to the discontinuous values of f (x}. 

11. Since the series (3) is convergent, if we have occasion to 
integrate /(a?) we may, instead of first summing the series and 
then integrating, first integrate the general term and then sum. 
More generally, if < (x) be any function of x which does not 
become infinite between the limits x = and x = a, we shall 
have 



f * ^ / N i / \ 7 2 _^ [ a . . ,. . UTTX , , f* , . U7TX -, 

f(x] <l>(x)dx = -2 f(x ) am - - dot. <#> (x) sm - - dx, 

Jo ft J o &Jo a 

the superior limit x of the integrals being supposed not to lie 
beyond the limits and a ; and the series at the second side of the 



250 ON THE CRITICAL VALUES OF 

above equation will be convergent. In fact, even in the case in 
which f (x) is discontinuous the series will be as convergent as the 
series 2 l//i 2 . A second integration would give a series still 
more rapidly convergent, and so on. Hence, the resulting series 
may be employed directly, and not merely when regarded as limits 
of converging series. The same remarks apply in all respects to 
the series (22) as to the series (3). 

12. But the series resulting from differentiating (3) or (22) 
once, twice, or any number of times would not in general be conver 
gent, and could not be employed directly, but only as limits of the 
convergent series which would be formed by inserting the factor y n 
in the general term. This mode of treating the subject however 
appears very inconvenient, except in the case in which the series 
are only temporarily divergent, being rendered convergent again 
by new integrations ; and even then it requires great caution. 
The series in question may however be rendered convergent by 
means of transformations to which I now proceed, and which, 
with their applications, form the principal object of this paper. 

The most important case to consider is that in which / (a?) and 
its derivatives are continuous, so that the divergency arises from 
what takes place at the limits and a. I shall suppose then, for 
the present, that f (x) and its derivatives of the orders considered 
are continuous, except the last, which will only appear under the 
sign of integration, and which may be discontinuous. 

Consider first the series of sines. Suppose that / (x) is not 
given in finite terms, but only by its developement 



where A n is supposed to be given, and where the developement of 
f(x) is supposed to be that which would result from the formula 
(3). I shall call the expansions of / (.*) which are obtained, or 
which are to be looked on as obtained from the formulae (3) and 
(22) direct expansions, as distinguished from other expansions 
which may be obtained by differentiation, and which, being diver 
gent, cannot be directly employed. Let us consider first the even 
differential coefficients of/(0), and let A\ A* ... be the coeffi 
cients of smmrx/a in the direct expansions of /" (x), f 4 (x) ... 
The coefficient of sin mrx/a in the series which would be obtained 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 257 

by differentiating twice the several terms in the series in (24) 
would be - nV/a* . A n . Now 

2 r a mrx , 



and we have by integrating by parts 

mrx , 2w7r ,, , x mrx 2 ,,, ,. . mrx 
d*=^f(*)<--f(*)* a 



Taking now the limits, remembering the expression for A^ , and 
transposing, we get 



. 



Any even differential coefficient may be treated in the same 
way. We thus get, /ut, being even, 



13. In the applications of these equations which I have 
principally in view, /(O), f(a), /"(O)... are given, and A lt A 2 , 
A 3 ... are indeterminate coefficients. If however A lt A z ... A n ... 
are given, and /(O), f (a) ... unknown, we must first find /(O), 
f(a)..., and then we shall be able to substitute in (25) and (26). 
This may be effected in the following manner. 

We get by integrating by parts 

r ,. , . mrx , ">/ * n7rsc> 
/ sm dx =--f(x) cos - 



a m 

. O / 

mrx 



+ f(cc } sm + if (x cos 



Multiplying now both sides by 2/a, and taking the limits of the 
integrals, we get 

s. 17 



258 ON THE CRITICAL VALUES OF 



Hence, if n be always odd or always even, A n can be expanded, 
at least to a certain number of terms, in a series according to 
descending powers of n, the powers being odd, and the first of 
them 1. The number of terms to which the expansion in this 
form is possible will depend on the number of differential coef 
ficients of f(x) which remain finite and continuous between the 
limits # = and x = a. Let the expansion be performed, and 
let the result be 

A n = -+ a l+0 4 l,+ ... when n is odd, 1 

; I i [-.(28). 

A n E - + E 2 -3 4- E 4 5 4- . . . when n is even 

W Yl 111 J 

Comparing (27) and (28), we shall have 



\.(0.+E& /(a) = I (0 fl 

7T 3 , 



..(29), 



and so on. The first two of these equations agree with (18). 

If we conceive the value of A n given by (27) substituted 
in (26), we shall arrive at a very simple rule for finding the direct 
expansion off*(x). It will only be necessary to expand A n as 
far as l/w 4 " 1 , admitting ( l) n into the expansion as if it were 
a constant coefficient, and then, subtracting from A n the sum 
of the terms thus found, employ the series which would be ob 
tained by differentiating the equation (24) /z, times. It will be 
necessary to assure ourselves that the term in I/nf- vanishes in 
the expansion of A n , since otherwise f*(x) might be infinite, 
or/**" 1 ^) discontinuous without our being aware of it. It will 
be seen however presently (Art. 20) that the former circumstance 
would not vitiate the result, nor introduce a term involving 1/n*. 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 259 

Should A n already appear under such a form as 
l +c r; (-lyJ + nV, &c., 

/ It 

where c 2 < 1, it will be sufficient to differentiate equation (24) 
//, times, and leave out the part of the series which becomes 
divergent. For it will be observed that the terms c n , nV, in 
the examples chosen, decrease with I/n faster than any inverse 
power of n. 

14. Let us now consider the odd differential coefficients 
of /(#), supposing f(x) to be expanded in a series of cosines, so 
that 



/(*) =1*0 + 21?. cos .................. (30). 

\AJ 

Let A f n , A " n ... be the coefficients of smnirx/a in the direct 
expansions of f (x\ / "(#)... in series of sines. If we were to 
differentiate (30) once we should have mr/a . B n for the coef 
ficient of sin n7rx/a. Now 

nir 2 f - , ,. mrx , . 
-- . - f(x) cos - dx 

a aj j ^ a 

2 ., ,. . UTTX 2 /",,,. ,. . H7TX , , 

= fix ) sm --- f- - / (x ) sm - dx ; 
u j a a] J a 

and taking the limits of the integrals, and introducing B n and 
A n , we get 

A n=-*n ....................... (31). 

Hence, the series arising from differentiating (30) once gives 
the direct expansion of/ (x) in a series of sines. 

The coefficient of sin n7rx/a in the series which would be 
obtained by differentiating (30) p times, /u. being odd, would 
be (- 1)0* +1 )/ 2 (nw/aYB n . By proceeding just as in the last article 
we obtain 



~!/"- 2 W-(-i)"/"- 2 ()!. 

172 



260 ON THE CRITICAL VALUES OF 

When / (O), f (a), &c., are known, this series enables us to develope 
f*(x) in a direct series of sines, the direct developement of f(x) 
in a series of cosines being given. 

15. If we treat the expression for B n by integration by 
parts, just as the expression for A n was treated, going on till 
we arrive at the integral which gives Af, and observe that the 
very same process is used in deducing the value of Af from 
that of B n as in expanding the latter according to inverse powers 
of n, and that the index of n in the coefficient of Af is yu-, 
and that A n vanishes when n becomes infinite, we shall see that 
in order to obtain the direct expansion of f^(x) we have only 
got to expand B n as far as l/n M , (the coefficient of 1/n* will 
vanish,) and subtract from B n these terms of the expansion, and 
then differentiate (30) //< times. 

The expansion of B n , at least to a certain number of terms, 
will proceed according to even powers of 1/w, beginning with 1/Vi 2 . 
If we suppose that 

B. = ^ + 8 i + U . . . when n is odd, 

I \ \ 

B E. -2 + E 3 - 4 + E. -3 + . . . when n is even I 
1 ri* 3 n* 5 n" J 

and compare these expansions with that given by integration by 
parts, we shall have 



...(34), 



and so on, the signs of the coefficients being alternately + and , 
and the index of vr/a increasing by 2 each time. 

16. The values of / (O) and /**(a) when / M (#) is expanded 
in a series of sines and p is odd, or when f(x) is expanded in a 
series of cosines and JJL is even, will be expressed by infinite series. 
To find these values we should first have to obtain the direct ex 
pansion of /^(.T), which would be got by differentiating the equa 
tion (24) or (30) p times, expanding A n or B n according to powers 
of 1/w, and rejecting the terms which would render the series 
contained in the /A th derived equation divergent. The reason of 
this is the same as before. 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 261 

17. The direct expansions of the derivatives of f(x) may 
be obtained in a similar manner in the cases in which f(x} itself, 
or any one of its derivatives is discontinuous. In what follows, 
cc will be taken to denote a value of x for which f(x) or any one 
of its derivatives of the orders considered is discontinuous; Q, 
Qi> -Qi*. will denote the quantities by which f(x], f (x), ...f*(x) 
are suddenly increased as x increases through a; S will be used 
for the sign of summation relative to the different values of a, 
and will be supposed to include the extreme values and a, under 
the convention already mentioned in Art. 6. Of course f(x) may 
be discontinuous for a particular value of x while f*(x) is con 
tinuous, and vice versa. In this case one of the two Q, Q^ will 
be zero while the other is finite. 

The method of proceeding is precisely the same as before, 
except that each term such as f(x) cos mrx/a in the indefinite 
integral arising from the integration by parts will give rise to a 
series such as SQ cos mra/a in the integral taken between limits. 
We thus get in the case of the even derivatives of f(x), when f(x) 
is expanded in a series of sines, 

*\~ l SQco8 * 
/ a 



nira. 



2 f^Y~ 2 S Q sin !^ + 2 (^Y~ 2 S Q : 
a \a J a a \ a J 

In the case of the odd derivatives of f(x), when f(x) is ex 
panded in a series of cosines, we get 



UTTOL 

a 



When the several values of a, Q, Q^... are given, these equa 
tions enable us to find the direct expansion of f* (x). The series 
corresponding to the odd derivatives in the first case and the even 
in the second might easily be found. 

If we wish to find the direct expansion of f^(x) in the case 
in which A n or B n is given, we have only to expand A n or B n in a 
series according to descending powers of n, regarding cos ny or 



262 .ON THE CRITICAL VALUES OF 

sin ny, as well as ( 1)", as constant coefficients, and then reject from 
the series obtained by the immediate differentiation of (24) or (30) 
those terms which would render it divergent. This readily follows 
as in Art. 15, from the consideration of the mode in which A n * 
is obtained from A n or B n . 

The equations (85) and (36) contain as particular cases (26) 
and (32) respectively. It was convenient however to have the 
latter equations, on account of their utility, expressed in a form 
which requires no transformation. 

18. If we transform A n and B n by integration by parts, we 
get 



A 2 

A= - 



n7r" a 



(37), 



n 2 ~ . mra 2a ar . 

B n = --- SQ sin ------ a --g SQ. cos 

mr a nV a 

2a? ,-y^ . UTTO. /oox 

+ n ,^SQ tS m +..., ............. (38), 

where the law of the series is evident, if we only observe that 
two signs of the same kind are always followed by two of the 
opposite kind. The equations (37), (38) may be at once obtained 
from (35), (36). The former equations give the true expansions 
of A n and B n according to powers of 1/n; because when we stop 
after any number of integrations by parts the last integral with 
its proper coefficient always vanishes compared with the coefficient 
of the preceding term. 

Hence A n and B n admit of expansion according to powers 
of 1/n, if we regard 003717 or sin ny as a constant coefficient in 
the expansion. Moreover quantities such as cos ny, smny will 
occur alternately in each expansion, the one kind going along 
with odd powers of 1/n and the other along with even. If we 
suppose the value of A n or B n , as the case may be, given, and 
the expansion performed, so that 



y. + SFt sin ny., + SF, cos W7 .- 8 + ..., ...(39), 
B H = SG sin ny . - + 0, cos ny . \ + #. sin ny . - 3 + . , . , ... (40), 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 263 

and compare these expansions with (37) or (38), we shall get the 
several values of a, and the corresponding values of Q, Q v Q 2 ... 
We may thus, without being able to sum the series in equation 
(24) or (30), find the values of x for which f(x) itself or any one 
of its derivatives is discontinuous, and likewise the quantity by 
which the function or derivative is suddenly increased. This 
remark will apply to the extreme values and a of x if we con 
tinue to denote the sum of the series by f(x) when x is outside 
of the limits and a. 

19. Having found the values of a, Q, Q^..., we may if we 
please clear the series in (24) or (30) of the terms which render 
f(ss) itself, or any one of its derivatives, discontinuous. If we 
wish the function which remains expressed by an infinite series 
and its first //, derivatives to be continuous, we have only to sub 
tract from A n or B n the terms at the commencement of its ex 
pansion, ending with the term containing I/nf +1 , and from f(x) 
itself the sums of the series corresponding to the terms subtracted 
from A n or B n . These sums will be obtained by transforming 
products of sines and cosines into sums or differences, and then 
employing known formulae such as 

COS Z COS 3.3 7T 2 7T ~ _ , . _ . 

-Br + T- + = ^---jT> from 2; = to z = TT (41), 

_L O o "r 

which are obtained by integrating several times the equation 
sin z -f- J sin 2z -f l sin 3.Z + . . . = | (IT z\ from z = to z ZTT, 

or the equation deduced from it by writing TT z for z, and taking 
the semi-sum of the results. It will be observed that in the 
several series to be summed we shall always have sines coming 
with odd powers of n and cosines with even. Of course, by clear 
ing the series in (24) or (30) in the way just mentioned we shall 
increase the convergency of the infinite series in which a part of 
f(x) still remains developed. 

When A n or B n decreases faster than any inverse power of n 
as n increases, (as is the case for instance when it is the w th term of 
a geometric series with a ratio less than 1,) all the terms of its 
expansion in a series according to inverse powers of n vanish. In 
this case, then, f(x) and its derivatives of all orders are con 
tinuous. 



204 ON THE CRITICAL VALUES OF 

20. In establishing the several theorems contained in this 
Section, it has been supposed that none of the derivatives of / (x) 
which enter into the investigation are infinite. It should be 
observed, however, that if/ 4 (x) is the last derivative employed, 
which only appears under the sign of integration, it is allowable to 
suppose that /> (so) becomes infinite any finite number of times 
within the limits of integration. To shew this, we have only got 
to prove that 

ra ra 

/> (x) sin vx dx or /> (#) cos vx dx 

Jo Jo 

approaches zero as its limit as v increases beyond all limit. Let us 
consider the former of these integrals, and suppose that /** (x) 
becomes infinite only once, namely, when x = a, within the limits of 
integration. Let the interval from to a be divided into these 
four intervals to a f, a - f to a, a to a + f , a + f to a, where 
f and f are supposed to be taken sufficiently small to exclude all 
values of x lying between the limits a f and a + f for which 
/ /x ~ 1 (#) alters discontinuously, or for which f*(x) changes sign, 
unless it be the value a. For the first and fourth intervals /** (x) 
is not infinite, and therefore, as it is known, the corresponding parts 
of the integral vanish for v = oo . Since sin vx cannot lie beyond 
the limits + 1 and 1, and is only equal to either limit for parti 
cular values of x, it is evident that the second and third portions 
of the integral are together numerically inferior to /, where 



the symbol A ~B denoting the arithmetical difference of A and B, 
and e being an infinitely small quantity, so that /(a e), f(a + e) 
denote the limits to which f(x) tends as x tends to the limit a by 
increasing and decreasing respectively. Hence the limit of the 
integral first considered, for v = oo , must be less than I. But /may 
be made as small as we please by diminishing f and f , and there 
fore the limit required is zero. 

The same proof applies to the integral containing cos vx, and 
there is no difficulty in extending it to the case in which f* (x) is 
infinite more than once within the limits of integration, or at one 
of the limits. 

21. It has hitherto been supposed that the function expanded 
in the series (3) or (22) does not become infinite ; but the expan 
sions will still be correct even if / (x} becomes infinite any finite 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 265 

number of times, provided that jf(x)dx be essentially convergent. 
Suppose that f (x) becomes infinite only when x = a. Then it is 
evident that we may find a function of x, F (x), which shall be 
equal to f(x) except when x lies between the limits a ? and 
a + f, which shall remain finite from x a. % to a? = a + f , and 

/+? /+ 

which shall be such that F(x)dx=\ f(x}dx. Suppose 

J a- J a-f 

that we are considering the series (3). Then, if G n be the coeffi 
cient of sin mrx/a in the expansion of F (x) in a series of the 
form (3), it is evident that C n will approach the finite limit A n 

2 C a mrx 

when t and ? vanish, where A -- I /(a?) sin - cZar. But so 

n 



long as f and f differ from zero the series 2) (7 n sin mrx I a is conver 
gent, and has jP T (a?) for its sum, and F (x) becomes equal to / (as) 
when f and f vanish, for any value of # except a. We might 
therefore be disposed to conclude at once that the series (3) is 
convergent, and has f(x) for its sum, unless it be for the particular 
value x = a. but this point will require examination, since we 
might conceive that the series (3) became divergent, or if it 
remained convergent that it had a sum different from f (x), when 
f and f were made to vanish before the summation was performed. 
If we agree not to consider the series (3) directly, but only the limit 
of the series (5) when g becomes 1, it follows at once from (7) that 
for values of x different from a that limit is the same as in Art. 4. 
For x = a the limit required is that of \ {/(a e) +/ (a + e)} when 
evanishes. IS. f(x) does not change sign as x passes through a 
the limit required is therefore positive or negative infinity, accord 
ing as f(x) is positive or negative; but if f (x) changes sign in 
passing through oo the limit required may be zero, a finite quantity, 
or infinity. The expression just given for the limit may be 
proved without difficulty. In fact, according to the method of 
Art. 4, we are led to examine an integral of the form 



where f is a constant quantity which may be taken as small as we 
please, and supposed to vanish after h. Now by a known property 
of integrals the above integral is equal to 



\. {/(-) +/( + &} , where f, lies between and 



2G6 OX THE CRITICAL VALUES OF 



But JTT% > which is equal to tan" 1 - t } becomes equal to ^ when 
J o M ~r 5 ft 2t 

h vanishes, and the limit of , when h vanishes must be zero, since 
it cannot be greater than f, and may be made to vanish 
after h. 

22. The same thing may be proved by the method which con 
sists in summing the series S sin nirx/a . sin mrx /a to n terms. 
If we adopt this method, then so long as we are considering a value 
of x different from a it will be found that the only peculiarity in 
the investigation is, that the quantity under the integral sign in 
the integrals we have to consider becomes infinite for one value of 
the variable ; and it may be proved just as in Art. 20, that this 
circumstance has no effect on the result. If we are considering the 
value x = a, it will be found that the integral we shall have to con 
sider will be 



where v is first to be made infinite, and then f may be supposed to 
vanish. If /(a + e) +/(a e) approaches a finite limit, or zero, 
when e vanishes, as may be the case if f(x) changes sign in passing 
through oo, it may be proved, just as in the case in which f (x) 
does not become infinite, that the above integral approaches the 
same limit as \ {/( + e) + /( e)]. In all cases however in 
which / (x) does not change sign in passing through oo , and in 
some cases in which it does change sign, f(a + e) +/(% e) becomes 
infinite when e vanishes. f 

In such cases put for shortness 



and let the numerical values of the integral I "*""* dj: taken from 

to TT/V, from ir/v to ZTT/V ... or which is the same those of 

1 S1 di; taken from to TT, from TT to 2?r . . . be denoted by / t , / 2 . .. 

Then evidently J 1 >/ 2 >/3... Also, if f be sufficiently small, 
.F (f ) will decrease from = to f = f, if we suppose, as we may, 
F(%) to be positive. Hence the integral (42), which is equal to 

...} (43), 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 267 

where , 2 ... are quantities lying between and TT/V, TT/V and 
27T/z/ ... is greater than 



if we neglect the incomplete pair of terms which may occur at the 
end of the series (43), and which need not be considered, since they 
vanish when v = oo . Hence, the integral (42) is a fortiori 
>T~ 1 (/ 1 -/ 8 )^(? 1 ). But vanishes and F(%^ becomes infinite 
when v becomes infinite ; and therefore for the particular value 
x = a the sum of the first n terms of the series (3) increases 
indefinitely with n. 

If a coincides with one of the extreme values and a of x, the 
sum of the series (3) vanishes for x = a. This comes under the 
formula given above if we consider the sum of the series for values 
of x lying beyond the limits and a. The same proof as that 
given in the present and last article will evidently apply if / (x) 
become infinite for several values of x, or if the series considered 
be (22) instead of (3). In this case, the sum of the series becomes 
infinite for x = a when a = or = a. 

23. Hence it appears that f(x] may be expanded in a series 
of the form (3) or (22), provided only If (as) dx be continuous. It 
should be observed however that functions like (sin c/x)~% , which 
become infinite or discontinuous an infinite number of times within 
the limits of the variable within which they are considered, have 
been excluded from the previous reasoning. 

Hence, we may employ the formulae such as (26), (35), &c., to 
obtain the direct developement of/** (x), without enquiring whether 
it becomes infinite or not within the limits of the variable for 
which it is considered. All that is necessary is that f(x) and its 
derivatives up to the (/-t l) th inclusive should not be infinite within 
those limits, although they may be discontinuous. 

24. In obtaining the formulae of Arts. 7 and 13, and generally 
the formulae which apply to the case in which A n or B n is given, 
and f(x) is unknown, it has hitherto been supposed that we knew 
a priori that f (x) was a function of the class proposed in Art. 1 for 
consideration, or at least of that class with the extension mentioned 



ON THE CRITICAL VALUES OF 

in the preceding article. Suppose now that we have simply pre 
sented to us the series (3) or (22), namely 

-, . . UTTX ^ mrx 

zA n sm - , or B Q + 2<B n cos -- , 

Oj d 

where A n or B n is supposed given and want to know, first, whether 
the series is convergent, secondly, whether if it be convergent it is 
the direct developement of its sum f(x), and thirdly, whether we 
may directly employ the formulae already obtained, trusting to the 
formulae themselves to give notice of the cases to which they do 
not apply by leading to processes which cannot be effected. 



25. If the series 2-4 n or ^B n is essentially convergent, it is 
evident a fortiori that the series (3) or (22) is convergent. 

If A n = S - cos ny + G n , or if B n = 8 - sin ny + C n , where 2(7 

11 n 

is essentially convergent, the given series will be convergent, as is 
proved in Art. 6. 

In either of these cases let / (as) be the sum of the given series. 
Suppose that it is the series of sines which we are considering. 
Let E n be the coefficient of sin mrx/a in the direct developement of 
f(x). Then we have 

* / x ~ A . niTX ^ -r, . H7TX 

f(x) = 2A n sm = ^E n sm - - 

U/ (Jj 

and since both series are convergent, if we multiply by any finite 
function of vc, <f> (x), and integrate, we may first integrate each 
term, and then sum, instead of first summing and then integrating. 
Taking < (x) = sin nirx/a, and integrating from,, a = to x = a, we 
get E n =.A n , so that the given series is the direct developement 
of its sum f(x). The proof is the same for the series of cosines. 

26. Consider now the more general case in which the series 
2 I/n . A n is essentially convergent. The reasoning which is about 
to be offered can hardly be regarded as absolutely rigorous ; never-* 
theless the proposition which it is endeavoured to establish seems 
worthy of attention. Let u n be the sum of the first n terms of the 
given series, and F (n, x) the sum of the first n terms of the series 
2 a/mr . A n cos mrx I a. Then we have 

5( u n+ m ~ u n) dx = F (n + ra, x) - F (n, x} = ^r (n, x), suppose. . .(44). 
Now by hypothesis the series S I/n . A H is essentially convergent, 
and therefore a fortiori the series 2 a/HTr . A n cos mrx/a is con- 



THE SUMS OF PERIODIC SERIES. (SECT. I. SERIES.) 269 

vergent, and therefore ^ (co , x) = 0, whatever be the value of m. 
Let the limits of x in (44) be x and x + A#, and divide by A#, and 
we get 



and as we have seen the limit of the second side of this equation 
when we suppose n first to become infinite and then Ace to vanish 
is zero. But for general values of x the limit will remain the same 
if we first suppose A# to vanish and then n to become infinite ; and 
on this supposition we have 

limit of (w n+m u n ) = 0, for n = oo ; 

so that for general values of x the series considered is convergent. 

To illustrate the assumption here made that for general values 
of x the order in which n and A# assume their limiting values is 
immaterial, let ty (y, x) be a continuous function of x which 
becomes equal to ^r (n, x) when y is a positive integer ; and con 
sider the surface whose equation is z = ty (y, x). Since 

^ (x , x) = 

for integral values of y, the surface approaches indefinitely to the 
plane xy when y becomes infinite ; or rather, among the infinite 
number of admissible forms of ^r (y, x) we may evidently choose 
an infinite number for which that is the case. Now the assertion 
made comes to this ; that if we cut the surface by a plane parallel 
to the plane xz, and at a distance n from it, the tangent at the 
point of the section corresponding to any given value of x will 
ultimately lie in the plane xy when n becomes infinite, except in 
the case of singular, isolated values of x, whose number is finite 
between x = and x = a. For such values the sum / (x) of the 
infinite series may become infinite, while $f(x)dx remains finite. 
The assumption just made appears evident unless A n be a function 
of n whose complexity increases indefinitely with its rank, i. e. with 
the value of n. 

Since the integral of /(a?) is continuous, f(x] may be expanded 
by the formula in a series of sines. Let E n be the coefficient 
of smn7rx/a in its direct expansion ; so that, 



*i \ ^ A - 
/= S^sm - 

(45), 



i 



270 ON THE CRITICAL VALUES OF 

where both series are convergent, except it be for isolated values 
of x. Consequently, we have, in a series which is convergent, 
at least for general values of x, 



(46). 



The series (45) may become divergent for isolated values of x y 
and are in fact divergent for values of x which render f (x] infinite. 
But the first side of (46) being constantly zero, and the series 
at the second side being convergent for general values of x, it 
does not seem that it can become divergent for isolated values. 
Hence according to the preceding article the second side of the 
equation is the direct developement of the first side, i.e. of zero; 
and therefore E n A n , or the given series is the direct develope 
ment of its sum, which is what it was required to prove. The 
same reasoning applies to the series of cosines. 

It may be observed that the well known series, 

J + cos x + cos 2x + cos 3# ........ , ........ (47), 

forms no exception to the preceding observation. This series 
is in fact divergent for general values of x, that is to say not 
convergent, and in that respect it totally differs from the series 
in (46). When it is asserted that the sum of the series (47) 
is zero except for x = or any multiple of 2?r, when it is infinite, 
all that is meant is that the limit to which the sum of the 
convergent series \ + %g n cos nx approaches when g becomes 1 
is zero, except for x or any multiple of 2?r, in which case it is 
infinity. 

27. It follows from the preceding article that even without 
knowing a priori the nature of the function f(x) we may employ 
the formulae such as (35), provided that if n~* be the highest 
power of I/n required by the formula, and n~>* C n the remainder in 
the expansion of A n) the series ^n~ l C n be essentially convergent. 
For let G n be the sum of the terms as far as that containing n~* in 
the expansion of A n , those terms having the form assigned by (35), 
that is to say cosines like cos ny coming along with odd powers 
of 1/n, and sines along with even powers. Then 



Let 2 sin 



THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 271 



then f(x)-F(x) = 2w* C n sin - ............ (48). 

Now if <(#) = Sw n , where the series 2w n , %du n /dx are both 
convergent, we may find < (a?) by differentiating under the sign of 
summation. This is evident, since by the theorem referred to 

in Art. 2 (note), we may find \t-~ dx by integrating under the 
sign of summation. Consequently we have from (48) 

..... (49); 



and since the series 2 ri~ l C n is essentially convergent, the con- 
vergency of the series forming the right-hand side of (49) cannot 
become infinitely slow (see Sect. III.), and therefore, the n ih term 
being a continuous function of x, the sum is also a continuous 
function of x, and therefore /**(#) F ti -(x} is a function which 
by Art. 23 can be expanded in a series of sines or cosines. But 
Fp (x) is also such a function, being in fact a constant, and 
therefore /**(#) is a function of the kind considered in Art. 23, 
which is what is assumed in obtaining the formula (35). 

It may be observed that these results do not require the 
assumptions of Art. 26 in the case in which the series 2<7 n is 
essentially convergent, or composed of an essentially convergent 
series and of a series of the form S/Stan" 1 sin wy or "ZScrT 1 cosny, 
according as C n is the coefficient of a cosine or of a sine. 



SECTION II. 

Mode of ascertaining the nature of the discontinuity of the 
integrals which are analogous to the series considered in Section I., 
and of obtaining the developements of the derivatives of the expanded 
functions. 

28. Let us consider the following integral, which is analogous 
to the series in (1), 

I 00S)sin#eZ (50), 

J o 

where <f>(j3) = - f" f(x) sinfa dx (51). 



272 ON THE CEITICAL VALUES OF 

Although the integral (50) may be written as a double in 
tegral, 

2 f 00 [ a 

I f(x)$mjBxsmBxdBc!x (52), 

7T J Jo 

the integration with respect to x must be performed first, because, 
the integral of sin fix sin fix df3 not being convergent at the 

,00 

limit oo , I sin /3x sin fBx d/3 would have no meaning. Suppose, 

Jo 

however, that instead of (52) we consider the integral, 
2 f C a 

7TJO Jo 

where h is a positive constant, and e is the base of the Napierian 
logarithms. It is easy to see that at least in the case in which the 
integral (50) is essentially convergent its value is also the limit to 
which the integral (53) tends when h tends to zero as its limit. 
It is well known that the limit of (53) when h vanishes is in 
general / (x) ; but when x = the limit is zero ; when x a the 
limit is J/(a); and when f(x) is discontinuous it is the arith 
metic mean of the values of f (x) for two values of x infinitely 
little greater and less respectively than the critical value. When 
x > a it is zero, and in all cases it is the same, except as to sign, for 
negative as for positive values of x. 

We may always speak of (53), but we cannot speak of the 
integral (50) till we assure ourselves that it is convergent. Now we 
get by integration by parts, , 

/(# ) sin fix dx - ~~ o/ ( x } cos fa 



~T ry>j (x ) sin px ~ a I j (x j sin px dx \^^V* 

When this integral is taken between limits, the first term will 
furnish a set of terms of the form G/@ . cos fix, where a may be 
zero, and the last two terms will give a result numerically less than 
Z//3 2 , where L is a constant properly chosen. Now whether a. be 
zero or not, /cos /3a sin /3x . /3" 1 d/3 is convergent at the limit oo , and 
moreover its value taken from any finite value of /3 to /3 = oo is 
the limit to which the integral deduced from it by inserting the 
factor e~ 7 ^ tends when h vanishes. The remaining part of the 
integral (50) is essentially convergent at the limit oo . Hence the 



THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 273 

integral (50) is convergent, and its value for all values of x, both 
critical and general, is the limit to which the value of the integral 
(53) tends when h vanishes. 

29. Suppose that we want to find f" (x), knowing nothing 
about f(x), at least for general values of x, except that it is the 
value of the integral (50), and that it is not a function of the class 
excluded from consideration in Art. 1. We cannot differentiate 
under the integral sign, because the resulting integral would, 
usually at least, be divergent at the limit oo , We may however 
find /" (x) provided we know the values of x for which f(x) and 
/ (x) are discontinuous, and the quantities by which f(x) and / (a;) 
are suddenly increased as x increases through each critical value, 
supposing the extreme values included among those for which 

f(x) or f (x) is discontinuous, under the same convention as in 
Art. 6. Let a be any one of the critical values of x ; Q, Q t the 
quantities by which f(x), f (x) are suddenly increased as x 
increases through a ; $ the sign of summation referring to the 
critical values of #; fa(ft) the coefficient of sin fix in the direct 
developement of /" (x) in a definite integral of the form (50). Then 
taking the integrals in (54) between limits, and applying the 
formula (51) to/" (x), we get 

2 9 

(j) (fi) = - /3* 6(P) + - /3SQ cos fa - - SO sin 82. 

7T 7T 

We may find fa(/3) in a similar manner. We get thus when 
yu, is even 

- 2 2 

(- I) 2 fa () - 0ty(/9) - - 0*~ 1 8Q cos fiy. + - 0*-*8Q. sin fa + ... 

7T 7T 

where sines and cosines occur alternately, and two signs of the 
same kind are always followed by two of the opposite. The expres 
sion for fr (@). when //, is odd might be found in a similar manner. 
These formulae enable us to express f^ (x) when <f> (/3) is an 
arbitrary function which has to be determined, and /(O), &c. are 
given. 

30. If however <p ((3) should be given, and /(O), &c. be 
unknown, <f> (fi) will admit of expansion according to powers of pi 1 , 
beginning with the first, provided we treat sin/3a or cos fa as if it 

s. 18 



274 ON THE CRITICAL VALUES OF 

were a constant coefficient ; and sin fix, cos /3a will occur with even 
and odd powers of ft respectively. The possibility of the expan 
sion of </> (ft) in this form depends of course on the circumstance 
that (f> (x) is a function of the class which it is proposed in Art. 1 
to consider, or at least with the extension mentioned in Art. 23. 
It appears from (55) that in order to express /** (x) as a definite 
integral of the form (50) we have only got to expand c/> (ft), to 
differentiate (50) p times with respect to x, differentiating under 
the integral sign, and to reject those terms which appear under the 
integral sign with positive powers of ft or with the power 0. The 
same rule applies whether //, be odd or even. 

31. If we have given < (a), but are not able to evaluate the 
integral (50), we may notwithstanding that find the values of x 
which render f(x) or any of its derivatives discontinuous, and the 
quantities by which the function considered is suddenly increased. 
For this purpose it is only necessary to compare the expansion of 
</> (ft) with the expansion 



jftBQmfr-p8Q l mnl*- ......... (5G), 

given by (55), just as in the case of series. 

We may easily if we please clear the function <f> (ft) of the part 
for which f(x) or any one of its derivatives is discontinuous, or 
does not vanish for x = and x = a. For this purpose it will be 
sufficient to take any function F (x) at pleasure, which as well as 
its derivatives of the orders considered has got the same discon 
tinuity as f(x) and its derivatives, to develope F (x) in a definite 

f 
integral of the form <J> (ft) sin fix dft by the formula (51), and to 

Jo 

subtract F (x) from f(x) and 4> (ft) from $ (ft). It will be 
convenient to choose such simple functions as I + mx -f nx 2 ; 
I sin x + m cos x ; le~ x + we **, &c. for the algebraical expressions of 
F(x) for the several intervals throughout which it is continuous, 
the functions chosen being such as admit of easy integration 
when multiplied by sin fixdx, and which furnish a sufficient number 
of indeterminate coefficients to allow of the requisite conditions 
as to discontinuity being satisfied. These conditions are that 
the several values of Q, Q lt &c. shall be the same for F(x) as for 

/<*) 



THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 275 


32. Whenever / f(x) dx is essentially convergent, we may at 

once put a = oo in the preceding formulae. For, first, it may be 
easily proved that in this case, (though not in this case only,) the 
limit of (53) when h vanishes is f(x) ; secondly, the limit of (53) 
is also the value of (52); and, lastly, all the derivatives of f(x) 
have their integrals, (which are the preceding derivatives,) essentially 
convergent, and therefore oo may be put for a in the developements 
of the derivatives in definite integrals. 

When f(x} tends to zero as its limit as x becomes infinite, arid 
moreover after a finite value of x does not change from decreasing 
to increasing nor from increasing to decreasing, 



f e-** f (x) sin flat dx 

J o 



/.oo 

will be more convergent than I f(x)sm$x dx t and the latter 

Jo 

integral will be convergent, and its convergency will remain finite* 
when j3 vanishes. In this case also we may put a = oo . 

Thus if /(a?) = sin Ix (b* + a 2 ) 1 , we may put a = oo because f(x) 
has its integral essentially convergent : if / (x) = (b 4- x)~*, we may 
put o- = co because f(x) is always decreasing to zero as its limit. 
But if f(x) = smlx (b + x)~*, the preceding rules will not apply, 
because f(x), though it has zero for its limit, is sometimes increas 
ing and sometimes decreasing. And in fact in this case the 
integral in equation (51) will be divergent when/3 = /, and $(J3) 
will become infinite for that value of @. It is true that/(#) is still 
the limit to which the integral (53) tends when h vanishes ; but I 
do not intend to enter into the consideration of such cases in 
this paper. 

33. When oo may be put for a, and/(#) is continuous, we get 
from (55) 



+ (-l)/3/>- 2 (0) ............ (57). 

In this case $ (fi) will admit of expansion, at least to a certain 
* See next Section. 

182 



276 ON THE CRITICAL VALUES OF 

number of terms, according to odd negative powers of (3. If 
we suppose < (/3) known, and the expansion performed, so that 



and compare the result (49), we shall get 
/(0)=|//o; /"(0)=-|tf 2 ; / 4 (0) = |# 4 ; &c ......... (58). 

34. The integral 

[ t(/3)cos/3^/3 ......................... (59), 

Jo 

where 

(^>os/3^ ............ (60), 



which is analogous to the series (22), is another in which it is some 
times useful to develope a function or conceive it developed. For 
positive values of x the value of (59) is the same as that of (50). 
When x = the value is /(O); and for negative values of x it is 
the same as for positive. It is supposed here that the integral (59) 
is convergent, which it may be proved to be in the same manner 
as the integral (50) was proved to be convergent. 

Suppose that we wish to find, in terms of ^r (/3), the develope- 
ment of /> (x) in a definite integral of the form (50) or (59), 
according as //, is odd or even. We cannot differentiate under the 
integral sign, because the resulting integral would be divergent. 
We may however obtain the required developemepit by transform 
ing the expression ^ (@) by integration by parts, just as before. We 
thus get for the case in which //, is odd 

Mi 9 9 

(-1) fa(&) 



* 



* ............ (61), 

7T 

where </y (/3) is the value of </> (/3) in the direct developement of 
/* (x) in the integral (50). In the same way we may get the value 
of.^(/3) when //, is even, ^(/3) being the value of i|r (/3) in the 
direct developement of/ 1 (a?) by the formulae (59), (60). 

The equation (61) is applicable to the case in which -^(/3) is an 
arbitrary function, and a, Q, &c., are given. If however 



THE SUMS OF PERIODIC SERIES. (SECT. II. INTEGRALS.) 277 

should be given, we may find ^(/5) or ^(/3) by the same rule as 
before. 

In the case in which ^ (/3) is given, we may find the values of 
a, Q, &c., without being able to evaluate the integral (59). For 
this purpose it is sufficient to expand ty (/3) according to negative 
powers of /3, and compare the expansion with that furnished by 
equation (61). 

35. The same remarks as to the cases in which we are at 
liberty to put oo for a apply to (60) as to (51), with one exception. 
In the case in which f(x) approaches zero as its limit, and is at 
last always decreasing numerically, or at least never increasing, as 
x increases, while $f(x) dx is divergent at the limit oo , it has been 
observed that </>(#) remains finite when /5 vanishes. This however 

is not the case with ^(/8), at least in general. I say in general, 

rx 
because, although I f(x] dx increases indefinitely with its superior 

limit, we are not entitled at once to conclude from thence that 



I cos @xf(x) dx becomes infinite when /3 vanishes, as will appear 

in Section III. It may be shewn from the known value of 

) 

x~ n cos jSxdx, where 1 > n > 0, that if f(x) = F(x) + Cx~ n , where 





F(x) is such that fF(x) dx is convergent at the limit oo , ty (ft) be 
comes infinite when /3 vanishes; and the same would be true if 
there were any finite number of terms of the form Cx~ n . There is 
no occasion however to enquire whether ^(/3) always becomes in 
finite : the point to consider is whether the integral (59) is always 
convergent at the limit zero. 

In considering this question, we may evidently begin the inte 
gration relative to x at any value X Q that we please. Suppose first 
\ve integrate from x =x to x = X, and let r($) be the result 
so that 

2 t x 
*r(/9) = - fix ) cos fix dx. 

TJ- J * 

Let ix i (/3) be the indefinite integral of -or (/3) dp : then, c being a 
positive quantity, we get from the above equation 

w / (8) C 7 / (c) = - I f(x } {sin fix sin cx \ -^- . 
T . jr e 



278 ON THE CRITICAL VALUES OF 

Now put JT=oo. Then since I /(a/) Sm , dx is a convergent 

J Xo & 

integral, and its convergency remains finite (Art. 39) when @ 
vanishes, as may be proved without much difficulty, its value can 
not become infinite, and therefore vr,(/3) does not become infinite 
when @ vanishes. Now 

/() cos fad/3 = w,(/8) cos fa + xfa, (0) sin pxdfi ......... (62), 

when x is positive ; and when x = 0, 



hence in either case JW (/3) cos fa d/3 is convergent at the limit 
zero. Now the quantity by which vr({3) differs from ty(@) evi 
dently cannot render (59) divergent, and therefore in the case con 
sidered the integral (59) is convergent at the limit zero. 

-00 

By treating I -sy(/3) e~*0cos (3%d{3 in the manner in which 

J o 
JW (/S) cos /3x d/3 is treated in (62), it may be shewn that the con 

vergency of the former integral remains finite when h vanishes. 
Hence, not only is the integral (59) convergent, but its value is 
the limit to which the integral similar to (53) tends when h 
vanishes. 

When f(x) is continuous, and oo may be put for a, we have 
from (61), /// being odd, 

9 9 



/"(O) 

7T " * *T ^ V 






...... (63). 



If -v/r(yS) be given we can find the values of/ (0), / "(O) ... just 
as before. 

36. The integral 

cos/3(x -x)f(x )d/3dx .............. (64), 



in which the integration with respect to x is supposed to be per 
formed before that with respect to fi, so that the integral has the 
form 



*eo ,. 

X(P)co*fad/3+l tr(P)&afad0 ......... (65), 

Jo Jo 



THE SUMS OF PERIODIC SERIES. (SECT. III. DISCONTINUITY.) 279 

may be treated just as the integral (59); and it may be shewn that 
in the same circumstances we may replace the limits a t and a by 
oo , + oo respectively. If we suppose %(/3) and cr(/3) known, we 
may find as before the values of x for which f(x), f (x) ... are 
discontinuous, and the quantities by which those functions are sud 
denly increased. We may also find the direct developement of 
f (x),f"(x) ... in two integrals of the form (65); and we may if we 
please clear the integrals (65) of the part which renders f(x), f (x)... 
discontinuous. 

37. In the developement of f(x) in an integral of the form (50) 
or (59), or in two integrals of the form (65), it has hitherto been 
supposed that f(x) is not infinite. It may be observed however 
that it is allowable to suppose f(x) to become infinite any finite 
number of times, provided ff(x) dx be essentially convergent about 
the values of x which render f(x) infinite. This may be shewn 
just as in the case of series. Hence, the formulae such as (55) which 
give the developement off t (x) are true even when/** (a?) is infinite, 
f*~ l (x) being finite. 



SECTION III. 

On the discontinuity of the sums of infinite series, and of the 
values of integrals taken between infinite limits. 

38. LET 

u l + u a ... + u n + (66), 

be a convergent infinite series having U for its sum. Let 

Wj+v,... +v n + (67), 

be another infinite series of which the general term v n is a function 
of the positive variable h, and becomes equal to u n when h vanishes. 
Suppose that for a sufficiently small value of h and all inferior 
values the series (67) is convergent, and has V for its sum. It 
might at first sight be supposed that the limit of V for h = was 
necessarily equal to U. This however is not true. For let the sum 
to n terms of the series (67) be denoted by f(n, h) : then the limit 
of V is the limit of f(n, h) when n first becomes infinite and then 
h vanishes, whereas Uis the limit of f(n, h) when h first vanishes 



280 ON THE CRITICAL VALUES OF 

and then n becomes infinite, and these limits may be different. 
Whenever a discontinuous function is developed in a periodic series 
like (15) or (30) we have an instance of this ; but it is easy to form 
two series, having nothing to to with periodic series, in which 
the same happens. For this purpose it is only requisite to take for 
f(n, h) U n , (U n being the sum of the first n terms of (66),) a 
quantity which has different limiting values according to the order 
in which n and h are supposed to assume their limiting values, and 
which has for its finite difference a quantity which vanishes when n 
becomes infinite, whether h be a positive quantity sufficiently small 
or be actually zero. 
For example, let 



(68), 
which vanishes when n = 0. Then 

A {/(, h) - U.} = V M - u, M = ^ ny |*L__ I5 . 
Assume 

U =1 -- 1 - r , sothatw =Af/ H = - r , 
n + 1 "- 1 n (n + 1) 

and we get the series 



5& 



2 (1 + h) " n (n + 1) {(w - 1; h + Ij (nh + 1) 

which are both convergent, and of which the general terms become 
the same when h vanishes. Yet the sum of the first is 1, whereas 
the sum of the second is 3. 

If the numerator of the fraction on the right-hand side of (68) 
had been pnh instead of 2nh, the sum of the series (70) would have 
been p + 1, and therefore the limit to which the sum approaches 
when h vanishes would have beenp + 1. Hence we can form as 
many series as we please like (67) having different quantities for 
the limits of their sums when h vanishes, and yet all having their 
n th terms becoming equal to u n when h vanishes. This is equally 
true whether the series (66) be convergent or divergent, the series 
like (67) of course being always supposed to be convergent for all 
positive values of h however small. 



THE SUMS OF PERIODIC SERIES. (SECT. III. DISCONTINUITY.) 281 

39. It is important for the purposes of the present paper to 
have a ready mode of ascertaining in what cases we may replace 
the limit of (67) by (66). Now it follows from the following 
theorem that this substitution may at once be made in an extensive 
class of cases. 

THEOREM. The limit of Fcan never differ from U unless the 
convergency of the series (67) becomes infinitely slow when h 
vanishes. 

The convergency of the series is here said to become infinitely 
slow when, if n be the number of terms which must be taken in 
order to render the sum of the neglected terms numerically less 
than a given quantity e which may be as small as we please, n 
increases beyond all limit as h decreases beyond all limit. 

DEMONSTRATION. If the convergency do not become infinitely 
slow, it will be possible to find a number n^ so great that for the 
value of h we begin with and for all inferior values greater than 
zero the sum of the neglected terms shall be numerically less than 
e. Now the limit of the sum of the first n^ terms of (67), when h 
vanishes is the sum of the first n^ terms of (66). Hence if e be the 
numerical value of the sum of the terms after the n^ of the series 
(66), Z7and the limit of V cannot differ by a quantity so great as 
e + e. But e and e may be made smaller than any assignable 
quantities, and therefore U is equal to the limit of V. 

COR. 1. If the series (66) is essentially convergent, and if, 
either from the very beginning, or after a certain term whose rank 
does not depend upon h, the terms of (67) are numerically less than 
the corresponding terms of (66), the limit of Fis equal to U. 

For in this case the series (67) is more rapidly convergent than 
(66), and therefore its convergency remains finite. 

COR. 2. If the series (66) is essentially convergent, and if the 
terms of (67) are derived from those of (66) by multiplying them 
by the ascending powers of a quantity g which is less than 1, and 
which becomes 1 in the limit, the limit of V is equal to U. 

It may be observed that when the convergency of (67) does 
not become infinitely slow when h vanishes there is no occasion to 
prove the convergency of (66), since it follows from that of (67). 
In fact, let V H be the sum of the first n terms of (67), U n the 
same for (66), F the value of V for h = 0. Then by hypothesis 



282 ON THE CRITICAL VALUES OF 

we may find a finite value of n such that V V n shall be numeri 
cally less than e, however small h may be ; so that 

V V n + a quantity always numerically less than e. 
Now let h vanish : then V becomes F and V n becomes U n . Also 
e may be made as small as we please by taking n sufficiently great. 
Hence U n approaches a finite limit when n becomes infinite, and 
that limit is F . 

Conversely, if (66) is convergent, and if 7 = F , the convergency 
of the series (67) cannot become infinitely slow when h vanishes. 

For if U n , V n represent the sums of the terms after the n th in 
the series (66), (67) respectively, we have 

F=F n +F n , U U n +U u i 
whence FJ = F - U - ( V n - U n ) + tT. 

Now F U, V n U n vanish with h, and U n r vanishes when n 
becomes infinite. Hence for a sufficiently small value of h and all 
inferior values, together with a value of n sufficiently large, and 
independent of h, the value of F n may be made numerically less 
than any given quantity e however small; and therefore, by defini 
tion, the convergency of the series (67) does not become infinitely 
slow when h vanishes. 

On the whole, then, when the convergency of the series (67) 
does not become infinitely slow when h vanishes, the series (66) is 
necessarily convergent, and has F for its sum : but in the contrary 
case there must necessarily be a discontinuity of some kind. Either 
F must become infinite when h vanishes, or the, series (66) must 
be divergent, or, if (66) is convergent as well as (67), U must be 
different from F . 

When a finite function of x, f (#), which passes suddenly from 
M to N as x increases through a, where a > a > 0, is expanded in 
the series (15) or (30), we have seen that the series is always con 
vergent, and its sum for all values of x except critical values is 
f(x), and for x a its sum is J (M+ N). Hence the convergency 
of the series necessarily becomes infinitely slow when a x 
vanishes. In applying the preceding reasoning to this case it will 
be observed that h is a - x, F is M, and U is \ (M + N), if we 
are considering values of a? a little less than a ; but h is x a. 
and F is N, if we are considering values of x a little greater 
than a. 



THE SUMS OF PERIODIC SERIES, (SECT. III. DISCONTINUITY.) 283 

When the series (66) is convergent as well as (67), it may be 
easily proved that in all cases 



where L is the limit of V lt when h is first made to vanish and then 
n to become infinite. 

40. Reasoning exactly similar to that contained in the preced 
ing article may be applied to integrals, and the same definitions 

,00 

may be used. Thus if F (x, h) dx is a convergent integral, we 

J a 

may say that the convergency becomes infinitely slow when h 
vanishes, when, if X be the superior limit to which we must inte 
grate in order that the neglected part of the integral, or 

F(x } h)dx, 

may be numerically less than a given constant e which may be as 
small as we please, X increases beyond all limit when h vanishes. 

The reasoning of the preceding article leads to the following 
theorems. 

If F= [ F (x, h) dx, if F be the limit of F when h = 0, and if 

J a 

F (x, 0) =/ (x) ; then, if the convergency of the integral F do not 
become infinitely slow when h vanishes, / (x) dx must be con- 

J a 

vergent, and its value must be F . But in the contrary case either 
F must become infinite when h vanishes, or the integral 



J 



must be divergent, or if it be convergent its value must differ 



from F . 



When the integral I f(x)dx is convergent, if we denote its 

J a 

value by U, we shall have in all cases 

when L is the limit to which \ F (x, h) dx approaches when h is 

J x 
first made to vanish and then X to become infinite. 



284 ON THE CRITICAL VALUES OF 

The same remarks which have been made with reference to the 
convergency of series such as (15) or (30) for values of x near 
critical values will apply to the convergency of integrals such as 
(50), (59) or (65). 

The question of the convergency or divergency of an integral 
might arise, not from one of the limits of integration being oo , but 
from the circumstance that the quantity under the integral sign 
becomes infinite within the limits of integration. The reasoning 
of the preceding article will apply, with no material alteration, to 
this case also. 

41. It may not be uninteresting to consider the bearing of the 
-reasoning contained in this Section on a method frequently given 
of determining the values of two definite integrals, more especially 
as the values assigned to the integrals have recently been called 
into question, on account of their discontinuity. 

Consider first the integral 

[ x sin ax , 

u= -dx 71), 

JQ x 

where a is supposed positive. Consider also the integral 

f 30 , sina# 7 
v= ~ hx - - dx. 
Jo 

It is easy to prove that the integral v is convergent, and that its 
convergency does not become infinitely slow when h vanishes. 
Consequently the integral u is also convergent, (as might also be 
proved directly in the same way as in the case of v,) and its value 
is the limit of u for h = 0. But we have ,,< 



dv^_r 

dh Jo 



e -hx sm axdx = 



9 
+ h 



whence v = G tan J - ; 

a 

and since v evidently vanishes when h = oo , we have (7=7r/2> 
whence 

7T _i h 7T 

= ^- tan -, tt = 2- 

Also u = when a = 0, and u Tr/2 when a is negative, since u 
changes sign with a. By the value of u for a = 0, which is 

C si o air 

asserted to be 0, is of course meant the limit of - dx when 

Jo x 

a is first made to vanish and then X made infinite. 



THE SUMS OF PERIODIC SERIES. (SECT. III. DISCONTINUITY.) 285 

It is easily proved that the convergency of the integral u 
becomes infinitely slow when a vanishes. In fact if 

, f sin ax , 
u = dx, 

Jx % 

we get by changing the independent variable 

, [ sinx 7 
u = ax : 



JaX ^ 

but for any given value of X, however great, the value of u 

becomes when a vanishes I dx, an integral which might have 

Jo & 

been very easily proved to be greater than zero even had we been 
unable to find its value. It readily follows from the above that if 
u has to be less than e the value of X increases indefinitely as a 
approaches to zero. 

42. Consider next the integrals 

00 cos ax dx [ , cos ax dx 

y I ^-flX 



It is easily proved that the convergency of the integral v does not 
become infinitely slow when h vanishes, whatever be the value of a. 
Consequently u is in all cases the limit of v for h = 0. Now v 
satisfies the equation 

tfv 

da* 



f ii 

v = e -hx cos a&dx = -rj* - z (73). 

Jo ti + a 



It is not however necessary to find the general value of v ; for if 
we put h = we see that u satisfies the equation 



so long as a is kept always positive or always negative : but we 
cannot pass from the value of u found for positive values of a to 
the value which belongs to negative values of a by merely writing 
a for a in the algebraical expression obtained. For although u 
is a continuous function of a, it readily follows from (73) that 

-r- is discontinuous. In fact, we have from this equation 
da 

(dv\ (dv\ f x ^ o+_-i x 

h 



fdv\ fdv\ f x , 

I -p- 1 ~" ( j~ J vaa ~ * ^ an 



286 ON THE CRITICAL VALUES OF 

Now let h first vanish and then X. Then v becomes u, and 
vda vanishes, since v does not become infinite for a = 0, whether 



h be finite or be zero. Therefore du/da is suddenly decreased by 
TT as a increases through zero, as might have been easily proved 
from the expression for u by means of the known integral (71), 
even had we been unable to find the value of u in (72), The equa 
tion (74) gives, a being supposed positive, 



But u evidently does not increase indefinitely with a, and 

[ dx TT , 
u = I -g = when a ; 

J Q JL -f-tG L 



whence C = 0, G Tr/2,^ u = Tr/2 . e~ a . Also, since the numerical 
value u is unaltered when the sign of a is changed, we have 
u = 7T/2 . e a when a is negative. 

It may be observed that if the form of the integral u had been 
such that we could not have inferred its value for a negative from 
its value for a positive, nor even known that u is not infinite 
for a = oc , we might yet have found its value for a negative by 
means of the known continuity of u and discontinuity of du/da 
when a vanishes. For it follows from (74) that u = C7 X e a + C g e~ a 
for a negative ; and knowing already that u = Tr/2 . t~ for a positive, 
we have 

177 _ n n 7r _r< rt 

whence G l = Tr/2, <7 2 = 0, u = Tr/2 . e a , for a ne gative. 

Of course the easiest way of verifying the result u = TT/% . e~ a for 
a positive is to develope e~ x for x positive in a definite integral of 
the form (59), by means of the formula (60). 



SECTION IV. 

Examples of the application of the formulce proved in the preceding 

Sections. 

43. Before proceeding with the consideration of particular 
examples, it will be convenient to write down the formulae which 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 287 

will have to be employed. Some of these formulae have been proved, 
and others only alluded to, in the preceding Sections. 

In the following formulae, when series are considered, f(x) is 
supposed to be a function of x which, as well as each of its deriva 
tives up to the (fj, l) th order inclusive, is continuous between 
the limits x = and x = a, and which is expanded between those 
limits in a series either of sines or of cosines of irx/a and its multi 
ples. A n denotes the coefficient of smmrx/a when the series is 
one of sines, B n the coefficient of cos mrx/a when the series is one 
of cosines, Af or B* the coefficient of sin mrx/a or cos mrx/a in 
the expansion of the /t th derivative. When integrals are considered 
f (x) and its first //, 1 derivatives are supposed to be functions of 
the same nature as before, which are considered between the limits 
x = and # = oo ; and it is moreover supposed that / (x) decreases 
as x increases to oo , sufficiently fast to allow ff(x)dx to be 
essentially convergent at the limit oo , or else that f(x) vanishes 
when x oo , and after a finite value of x never changes from 
increasing to decreasing nor from decreasing to increasing. < (fi) 
or -\/r(/3) denotes the coefficient of sin fix or cos fix in the develope- 

/oo 

ment of f(x) in a definite integral of the form (f>(fi) siufixdxor 

J o 

cosfixdx, $ (fi) or ^ (J3) denotes the coefficient of sin 8x 



, 

J 



o 



or cos fix in the developement of the /Lt th derivative of /(#). The 
formulas are 



^7r- 

-) ( /() -(-DV(<)} + -0* even) ..... (5), 



288 ON THE CRITICAL VALUES OF 



+ ~ f^TV (O) - (-D"/) -..-0* even) ..... 
except when n = 0, in which case we have always 



JB being the constant term in the expansion of / M (V) in a series of 
cosines. In the formulae (-4), (B), (C), (D) we must stop when 
we have written the term containing the power 1 or 0, (as the 
case may be,) of n7r/a. 

The formulae for integrals are 

(- I/** f , 08) -0* * (0) - 0- 1 / (0) 

+ V-s/"(0) -...(/, odd) ............. (a), 



08) = 



- *- 8 /"(0) - - fa even ) .............. (&) 

o 

-/^- 2 / (0) 

- /^- 4 / "(0) + ... (P odd) ........... (c), 

7T t 



-- />-*/ " (0) + ... (/t even) 

7T 



where we must stop with the last term involving a positive power 
of ft or the power zero. 

44. As a first example of the application of the principles 
contained in Sections I. and II. suppose that we have to determine 
the value of < for values of x lying between and a, and b 
respectively, from the equation 

79 I 

(75), 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 289 

with the particular conditions 

T? = a>(x %a), when y = or =b .......... (76), 

^ = -co(y-%b), when x = or =a ......... (77). 

This is the problem in pure analysis to which we are led in 
seeking to determine the motion of a liquid within a closed 
rectangular box which is made to oscillate. 

For a given value of y, the value of <f> can be expanded in 
a convergent series of cosines of irx/a and its multiples; for 
another value of y, < can be expanded in a similar series with 
different coefficients, and so on. Hence, in general, <f> can be 
expanded in a convergent series of the form 



...... (78), 

where Y n is a certain function of y, which has to be determined. 

In the first place the value of < given by (78) must satisfy (75). 
Now the direct developement of dty/dy* in a series of cosines will 
be obtained from (78) by differentiating under the sign of sum 
mation ; the direct developement of cPfydar will be given by the 
formula (D). We thus get 

~d*Y nV 2o> 



and the left-hand member of this equation being the result of 
directly developing the right-hand member in a series of cosines, 
we have 

* 17 , 

^b) or =0, 

2 



o- 

dy* a* a v<y 

according as n is odd or even. This equation is easily integrated, 
and the integral contains two arbitrary constants, C n , D n , suppose. 
It only remains to satisfy (76). Now the direct developement 
of dYJdy will be obtained by differentiating under the sign of 
summation, and the direct developement of &> (x J a) is easily 
found to be 2 4&)a/7rV . cos mrxja, the sign S denoting that 
odd values only of n are to be taken. We have then, both for 
s. 19 



290 ON THE CKITICAL VALUES OF 

y = and for y = &, 



dY n 

-7- = -- j-, or =0, 
dy irn 

according as n is odd or even, which determines G n and D n . 

It is unnecessary to write down the result, because I have 
already given it in a former paper*, where it is obtained by 
considerations applicable to this particular problem. The result 
is contained in equation (4) of that paper. The only step of the 
process which I have just indicated which requires notice is, 
that the term containing (x J a) (y J 6) at first appears as an 
infinite series, which may be summed by the formula (41). The 
present example is a good one for shewing the utility of the 
methods contained in the present paper, inasmuch as in the 
Supplement referred to I have pointed out the advantage of the 
formula contained in equation (6), with respect to facility of nu 
merical calculation, over one which I had previously arrived at 
by using developements, in series of cosines, of functions whose 
derivatives vanish for the limiting values of the variable. 

45. Let it be required to determine the permanent state of 
temperature in a rectangle which has two of its opposite edges 
kept up to given temperatures, varying from point to point, while 
the other edges radiate into a space at a temperature zero. The 
rectangle is understood to be a section of a rectangular bar of 
infinite length, which has all the points situated in the same line 
parallel to the axis at the same temperature, so that the pro 
pagation of heat takes place in two dimensions. , 

Let the rectangle be referred to the rectangular axes of x, y, 
the axis of y coinciding with one of the edges whose temperature 
is given, and the origin being in the middle point of the edge. 
Let the unit of length be so chosen that the length of either 
edge parallel to the axis of x shall be TT, and let 2/3 be the length 
of each of the other edges. Let u be the temperature at the 
point (as, y), h the ratio of the exterior, to the interior conductivity. 
Then we have 



-hu = 0, when 2/ = -/3 ............... (80), 

u 

* Supplement to a Memoir On some Cases of Fluid Motion, p. 409 of the 
present Volume [Ante, p. 188]. 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 291 

^ + tt = 0, when y= .................. (81), 

u=f(y) ) when # = ................... (82), 

u =F(y} ) when x = a ..................... (83), 

/(?/), JP(y) being the given temperatures of two of the edges. 

According to the method by which Fourier has solved a similar 
problem, we should first take a particular function Fe^, where Y 
is a function of y, and restrict it to satisfy (79). This gives 
Y = A cos \y + B sin \y, A and B being arbitrary constants. We 
may of course take, still satisfying (79), the sum of any number 
of such functions. It will be convenient to take together the 
functions belonging to two values of X which differ only in sign. 
We may therefore take, by altering the arbitrary constants, 

u = 2 [A (^(*-$ - -*(*-tf) + B (<** - e-**)} cos Xy, 
+ 2{C(^-^-e-^-^)+D(e^- e -* x )}sm\y ..... (84), 

in which expression it will be sufficient to take only one of two 
values of X which differ only by sign, so that X, if real, may be 
taken positive. Substituting now in (80) and (81) the value of 
u given by (84), we get either (7=0, D = 0, and 

X/3.tanX/3 = A/3 ......................... (85), 

or else A = 0, B = 0, and 

X/3.cotX/3 = -fy3 ..................... (86). 

It is easy to prove that the equation (85), in which X/3 is 
regarded as the unknown quantity, has an infinite number of 
real positive roots lying between each even multiple of ?r/2, in 
cluding zero, and the next odd multiple. The equation (86) 
has also an infinite number of real positive roots lying between 
each odd multiple of Tr/2 and the next even multiple. The 
negative roots of (85) and (86) need not be considered, since the 
several negative roots have their numerical values equal to those 
of the positive roots; and it may be proved that the equations 
do not admit of imaginary roots. The values of X in (84) must 
now be restricted to be those given by (85) for the first line, and 
those given by (86) for the second. It remains to satisfy (82) 
and (83). Now let 



- y} = 2^(3,), F(y) -F(-y} = 

192 



292 ON THE CRITICAL VALUES OF 

then we must have for all values of y from to @, and therefore 
for all values from /3 to 0, 



F 1 (y) .......... (87), 

F % (y) .......... (88), 

where L = e Xv - e~ A7r , M = e* n - e~^ n } 

JJL denoting one of the roots of the equation 

(89), 



and the two signs S extending to all the positive roots of the 
equations (85), (89), respectively. To determine A and B, multiply 
both sides of each of the equations (87) by cos\ ydy, X being 
any root of (85), and integrate from y = to y = /3. The integral 
at the first side will vanish, by virtue of (85), except when X = X, 
in which case it will become l/4\ . (2X/3 + sin 2X/3), whence A 
and B will be known. G and D may be determined in a similar 
manner by multiplying both sides of each of the equations (88) 
by smpydy, fi being any root of (89), integrating from 2/ = 
to y /3, and employing (89). We shall thus have finally 

u = 42X (2X + sin 2X/3) 



cos \y dy + (e Xa5 "**)) F^ (y) cos X^ dy] cos \y, 

J o 

- sin 2/*)- 1 (d" - e-^)" 1 {(e^~^ -^ ( 

sin ^^ + (e^ - e-^) ( F t (y) sin pydy} bin M y. . .(90). 

JO 

46. Such is the solution obtained by a method similar to that 
employed by Fourier. A solution very different in appearance 
may be obtained by expanding u in a series 2 I^sin nx, and em 
ploying the formula (B). We thus get from the equation (79) 



which gives 
r= Ae" + Bf~ - \f(y ) - (- l)"F(,j)} (e^ - ."("I) dy ; 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 293 

whence, dujdy 2 Y sin nx, where 



The values of A and B are to be determined by (80) and (81), 
which require that 



dy 
We thus get 

(n + h) * A-(-h)"#B-^ l\f (y 1 ) - (- 1)" F (y)} 

{(n + h) <*&-& +(n-h) e~ n ^-^} dy = 0, 

and the equation derived from this by changing the signs of h and 
/3 ; whence the values of A and B may be found. We get 
finally 

u = 2Ysmnx ........................ (91), 

where 

Y= - {(n + h) e n ? - (n - h) e ^}- 1 (<* + e ^) I {(n + h) e(0-> 

7T J 

+ (n - h) e -W-1} {/, (y} - (- 1)" F, (y )} dy 



+ - {(n + A) e 1 ^ + (w - A) e -*J -^e"" - e "") [ {( + h) 
"" Jo 

+ (n - A) 6-tf- V>} {/ 2 (y f ) - (- 1)" 



47. The two expressions for w given, one by (90), and the 
other by (91) and (92), are necessarily equal for values of x and y 
lying between the limits and TT, /3 and /3 respectively. They 
are also equal for the limiting values y = [B and y = ft, but not 
for the limiting values x = and x = TT, since for these values (91) 
fails ; that is to say, in order to find from this series the value of u 
for x = or x = TT, we should have first to sum the series, and then 
put x = or x = TT. 

The comparison of these expressions leads to two remarkable 
formulae. In the first place it will be observed that the first and 



294 ON THE CRITICAL VALUES OF 

second portions of the right-hand side of (92) are unchanged when y 
changes sign, while the third and fourth portions change sign with y. 
This is obvious with respect to the first and third portions, and may 
be easily proved with respect to the second and fourth by taking y 
instead of y for the variable with respect to which the integration 
is performed, and remembering thatj^ (y), F^ (y) are unchanged, and 
/ 2 (y)t F z (y) change sign, when y changes sign. Consequently the 
part of u corresponding to the first two portions of (92) is equal to the 
part expressed by the first two lines in (90), and the part correspond 
ing to the last two portions of (92) equal to the part expressed by the 
last two lines in (90). Hence the equation obtained by equating the 
two expressions for u splits into two ; and each of the new equa 
tions will again split into two in consequence of the independence 
of the functions f, F, which are arbitrary from y = to y = /3. 
As far however as anything peculiar in the transformations is con 
cerned, it is evident that we may suppress one of the functions 
f, F, suppose F, and consider only an element of the integral by 
which f is developed, or, which is the same, suppose f t (y f ) or f 2 (y) 
to be zero except for values of the variable infinitely close 
to a particular value y, and divide both sides of the equa 
tion by 

//,(/) dy or J/ 2 (/)<% . 

We get thus from the first two lines of (90) and the first two 
portions of (92), supposing y and y positive, and y the greater 
of the two, 



-e 
e-^ 

where the first 2 refers to the positive roots of (85), and the second 
to positive integral values of n from 1 to GO . 

Of course, if y become greater than y, y and y will have to 
change places in the second side of (93). This is in accordance 
with the formula (92), since now the second line does not vanish; 
and it will easily be found that the first and second lines together 
give the same result as if we had at once made y and y change 
places. Although y has been supposed positive in (93), it is easily 
seen that it may be supposed negative, provided it be numerically 
less than y. 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 295 

The other formula above alluded to is obtained in a manner 
exactly similar by comparing the last two portions of (92) with 
the last two lines in (90). It is 



-sn 

(n+fceP-^ + (w + A)e-P-^ 

- ~ 



where the first 2) refers to the positive roots of (89), the second to 
positive integral values of n, and where x is supposed to lie 
between and TT, y between and /3, y between and y, or, it 
may be, between y and y. Although x has been supposed less 
than TT, it may be observed that the formulae (93), (94) hold good 
so long as x t being positive, is less than 2?r. 

48. Let it be required to determine the permanent state of 
temperature in a homogeneous rectangular parallelepiped, suppos 
ing the surface kept up to a given temperature, which varies from 
point to point. 

Let the origin be in one corner of the parallelepiped, and let 
the adjacent edges be taken for the axes of x, y, z. Let a, b, c be 
the lengths of the edges ; /j (y, z), F l (y, z), the given temperatures 
of the faces for which x and x = a respectively ; f 2 (z, x), 
F 9 (z, x) the same for the faces perpendicular to the axis of y ; 
fs ( x > 2/)> ^3 ( x > y) * ne same f r those perpendicular to the axis of z. 
Then if we put for shortness v to denote the operation otherwise 
denoted by 



as will be done in the rest of this paper, and write only the charac 
teristics of the functions, we shall have,, to determine the tempera 
ture u, the general equation ^u = with the particular condi 
tions 

u= fi> wnen # = 0; u = F l , when x a ...... (95); 

u= fz> when # = 0; u = F 2 , when y = b ...... (96); 

u=f 9 , when = 0; u~F 3 , when zc ...... (97); 

It is evident that u is the sum of three temperatures u lt u 2 , u 3 , 
where u : satisfies the conditions (95), and vanishes at the four 
remaining faces, and w 2 , u s are related to the axes of y, z as w t is 



296 ON THE CRITICAL VALUES OF 

related to that of x y each of the quantities u lt u. 2 , u 3 representing 
a possible permanent temperature. Now u 3 may be expanded in 
a double series 2S Z mn sin mTrx/a . sin njry/b, where Z mn is a function 
of z which has to be determined. Let for shortness 

rmr nir PTT 



then the substitution of the above value of u 3 in the equation 
V^ 3 = leads to the equation 



tf-a^-o. 

where (f $ + z> 2 , which gives Z mn = A mn e qz + B mn e~ 9 " ; and the con 
stants A mn , B^ are easily determined by the condition (97). We 
may find u t and w 2 in a similar manner, and the sum of the results 
gives u. It is thus that such problems are usually solved. 
We may, however, expand u in a series of the form 



on px sn vy, 

even though it does not vanish for x and x = a, and for y = 
and y = b } and the formula proved in Section I. enable us to 
make use of this expansion. 

Let then u = SS^sin fix sin vy, 

the suffixes of Z being omitted for the sake of simplicity. We 
have by the formula (B) 

vy + * [/,_(- 1J-J-J j. sin ^. . 

/i (2/j *0 ~~ ("" l) m -^i(3/j s ) ^ e expanded in the series 2Qsini>y 
by the formula (3), so that will be a known function of z, m, 
and w. Then 



Q h sin /Lta? sin i/? 



The value of d*u/dy* may be expressed in a similar manner, and 
that of d?u/dz 2 is found by direct differentiation. We have thus, 
for the direct developement of yw, the double series 

SS j^jf - (^ 2 + ^ ! ) ^+ y P + J Q } sm ^ sin v y, 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 297 

where P is for x what Q is for y. The above series being the 
direct developement of yw, and v^ being equal to zero, each co 
efficient must be equal to zero, which gives 

d*Z 2v 2a 

-j- T q i Z-t-- r P+-i-Q = Q m .. (98) 

dz b a -\J)> 

where q means the same as before. The integral of the equation 
(98) is 

Q > o f[ J o 

2T denoting the sum of the last two terms of (98). It only re 
mains to satisfy (97). If the known functions f B (x, y\ F 3 (x, y) be 
developed in the double series 22(7 sin px sin.z/y, 22//sin (JLX sin vy y 
we shall have from (97) 



- - e c F <T*Tdz + - e^ c l\*Tdz = H. 
2 Jo q. Jo 

A and B may be easily found from these equations, and we shall 
have finally 



~ e~ (c - z) ) l\e qz - -") T dz 
Jo 



T being the value of T when z = - z. It will be observed that the 
letters Z, P, Q, T, A, B, G, H ought properly to be affected with 
the double suffix inn. It would be useless to write down the 
expression for u in terms of the known quantities /j (y, z), &o. 

It will be observed that u might equally have been expressed by 
means of the double series SSZ^sin vysm >&z, or 22 F^sm^sin-nr^, 
where p is any integer. We should thus have threeTdifferent ex 
pressions for the same quantity u within the limits x = and x = a, 
y = and y = &, * = and z = c. The comparison of these three 
expressions when particular values are assigned to the known 
functions / x (y, s) & c . would lead to remarkable transformations. 
The expressions differ however in one respect which deserves 
notice. Their numerical values are the same for values of the 



298 ON THE CRITICAL VALUES OF 

variables lying within the limits and a, and b, and c. The 
first expression holds good for the extreme values of z, but fails for 
those of x and y : in other words, in order to find from the series 
the value of u for the face considered, instead of first giving x or y 
its extreme value and then summing, which would lead to a result 
zero, we should first have to sum with respect to m or n, or con 
ceive the summation performed, and then give x or y its extreme 
value. The same remarks apply, mutatis mutandis, to the second 
and third expressions ; so that the three expressions are not equi 
valent if we take in the extreme values of the variables. 

49. Many other remarkable transformations might be obtained 
from those already referred to by differentiation and integration. 

We might for instance compare the three expressions which would 

ra rb re 

be obtained for I udxdydz. and we should thus have three 

JoJoJo 

different expressions for the same function of the three independent 
variables a, b, c, which are supposed to be positive, but may be of 
any magnitudes. Some examples of the results of transformations 
of this kind may be seen by comparing the formulae obtained in 
the Supplement alluded to in Art. 44 with the corresponding for 
mulae contained in the Memoir itself to which the Supplement has 
been added. Such transformations, however, when separated from 
physical problems, are more curious than useful. Nevertheless, it 
may be worth while to exhibit in its simplest shape the formula 
from which they all flow, so long as we restrict ourselves to a func 
tion u satisfying the equation VM = 0, and expanded between the 
limits x = and x a, &c. in a double series of sines. 

The functions /^y, z] &c., which are supposed known, are arbi 
trary, and enter into the expression for u under the sign of double 
integration. Consequently we shall not lose generality, so far as 
anything peculiar in the transformations is concerned, by consider 
ing only one element of the integrals by which one of the functions 
is developed. Let then all the functions be zero except / 3 ; and 
since in the process / 3 has to be developed in the double series 

4r52 If f a (, y } s i n P sm vy dx dy . sin JJLX sin vy, 

cio Jo Jo 

consider only the element / 8 (# , y ) sin pat sin vy dx dy of the double 
integral, omit the dx dy, and put/ 8 (# , y ) = 1 for the sake of sim- 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 299 

plicity. If we adopt the first expansion of u, and put <f for fi + z/ 2 , 
we shall have 

Z= A (e 9(c - 0) - e- 9(c " 2) ), (e 9C - <T qc ) A=~ b sin i*x sin vy ; 
whence 

4 ?(C~3) _ 6 -?(c-2) 

u = r SS - -- sin fjix sin i/y sin /z# sin vy. . . (99). 

O/U ~~ 

By expanding u in the double series 2S Fsin //,# sin nrz we should 
get 

2 __> cr (* - e-w) (e*( 6 -^ - e-^-tf) . 

w = 2,2, - - ; - ^ - - sm LLX sin ua; sin txz 
ac s e sb ~ s 

...... (100), 

where s 2 = ^ + &*, and y is the greater of the two y, y. The 
third expansion would be derived from the second by inter 
changing the requisite quantities. In these formulsB z may have 
any positive value less than 2c. 

We should get in a similar manner in the case of two variables 
x,y 



2 ^ e v(a-x] _ e -v(a-x) ^ f m 

^ sm v sm 



aoi)f 



where x is supposed to lie between and a, y between and 6, 
and y between and y . This formula is however true so long 

as x lies between and 2a, and y between y and y . 

rb rb ra 

If we compare the two expressions for I I udy dy dx 

Jo Jo Jo 

obtained from (101), taking 2 for the sign of summation corre 
sponding to odd values of n from 1 to oo, putting a = rb, and 
replacing S 1/n 2 by its value 7r 2 /8, we shall get the formula 



which is true for all positive values of r, and likewise for all 
negative values, since the left-hand side of (102) is not changed 
when - r is put for r. In integrating the second side of (101), 
supposing that we integrate for y before integrating for y , we 
must integrate separately from # = to y = y, and from y = y 



300 ON THE CRITICAL VALUES OF 

to y = b, since the algebraical expression of the quantity to be 
integrated changes when y passes the value y. 

It would be useless to go on with these transformations, which 
may be multiplied to any extent, and which cease to be useful 
when they are separated from physical problems to which they 
relate, and of which we wish to obtain solutions. 

It may be observed that instead of supposing, in the case of 
the parallelepiped, the value of u known for all points of the 
surface, we might have supposed the value of the flux known, 
subject of course to the condition that the total flux shall be 
zero. This would correspond to the following problem in fluid 
motion, u taking the place of the quantity usually denoted by </>, 
"To determine the initial motion at any point of a homogeneous 
incompressible fluid contained in a closed vessel of the form of 
a rectangular parallelepiped, which it completely fills, supposing 
the several points of the surface of the vessel suddenly moved 
in any manner consistent with the condition that the volume be 
not changed." In this case we should expand u in a series of 
cosines instead of sines, and employ the formula (D) instead of (B). 
We might, again, suppose the value of u known for the faces 
perpendicular to one or two of the axes, and the value of the 
flux known for the remaining faces. In this case we should 
employ sines involving the co-ordinates perpendicular to the first 
set of faces, and cosines involving the others. 

The formulae would also be modified by supposing some one or 
more of the faces to move otf to an infinite distance. In this 
case some of the series would be replaced by integrals. Thus, in 
the case in which the value of u at the surface is known, if we 
supposed a to become infinite we should employ the integral (50) 
instead of the series (3), as far as relates to the variable x, and 
the formula (b) instead of (B). If we were considering a rect 
angular bar infinitely extended both ways we should employ the 
integral (65). Of course, if we had already obtained the result for 
the case of the parallelepiped, the shortest way would be thence 
to deduce the result for the case of the bar infinite in one or in 
both directions, but if we began with considering the bar it would 
be best to start with the integrals (50) or (65). 

50. To give one example of transformations of this kind, 
let us suppose b to become infinite in (101). Ol serving that 



THE SUMS OF PERIODIC SEMES. (SECT. IV. APPLICATIONS.) 301 

v = n7r/b, Av = 7T/6, we get on passing to the limit 

2 



= -2 (e^-e-^e-^ sin/^ ..... (103). 

Multiply both sides of this equation by dx dy, and integrate from 
x = to # = a, and from y = to y = oo . With respect to the 
integration of the second side, it is only necessary to remark that 
when y becomes greater than y r , y and y must be made to change 
places in the expression written^ down in (103). As to the in 
tegration of the first side, if we first integrate from y = to y = Y, 
we get, putting f(v t x) for the fraction involving x, 



I f(v, x) sin vy (1 cos v Y) . 
J o v 



Now let Y become infinite; then the term involving cos^Fmay 
be omitted, not because cosz^F vanishes when F becomes infinite, 
which is not true, but because, as may be rigorously proved, the 
integral in which it occurs vanishes when F becomes infinite. 
If we write 1 for a, as we may without loss of generality, we 
get finally 

"?=;![ 2,4(1-6*) (104). 



51. Hitherto in satisfying the general equation v^ = 0, to 
gether with the particular conditions at the surface, the value 
of u has been expanded in a double series involving two of the 
variables, and the functions of the third variable which enter as 
coefficients into the double series have been determined by an 
ordinary differential equation such as (98). We might however 
expand u in a triple series, and thus satisfy at the same time 
the equation v^ = and the conditions at the surface, without 
using an ordinary differential equation at all, but simply by means 
of the terms introduced into the series by differentiation, which 
are given by the formulae at the beginning of this Section ; and 
then by summing the triple series once, which may be done in 
any one of three ways, we should arrive at the same results as 
if we had employed in succession three double series, involving 
circular functions of x and y, y and z, z and x respectively, and 
the corresponding ordinary differential equations. I am indebted 



302 ON THE CRITICAL VALUES OF 

for this method to my friend Prof. William Thomson, to whom 
I shewed the method given in Art. 48. 

Let us take the case of the permanent state of temperature 
in a rectangular parallelepiped, supposing the temperature at the 
several points of the surface known. For more simplicity suppose 
the temperature zero at the surface, except infinitely close to 
the point (x, y ) in the face for which = 0, so that all the 
functions f, &c. are zero, except f a (x, y), and / 3 (x, y) itself zero 
except for values of x, y infinitely close to x t y respectively ; and 
let fff a (x, y) docdy = 1, provided the limits of integration include 
the values x = x, y = y . Let u be expanded in the triple series 



sin ixz .............. (105), 

where yu,, v, is mean the same as in Art. 48. Then 



a? = ^ p \ 



sn ^ sn 

/ 3 0, y) sin 



Now the expansion of f s (x, y) in a double series is 
4/a6 . 22 sin fix sin vy sin px sin vy, 

that is to say with this understanding, that the result is to be 
substituted in (106); for it would be absurd to speak, except 
by way of abbreviation, of a quantity which is zero except for 
particular values of x and y y for which it is infinite. The values 
of tfuldx* and d^u/dy* will be obtained by direct differentiation. 
We have therefore for the direct developement of TJU in a triple 
series 



O 2 + i? + *r 2 ) A mnp 

SOT , ,} 

+ j- sin IJLX sin vy \ sin px sin vy sin isz. 

But \7w being equal to zero, each coefficient will be equal to 
zero, from whence we get A mnp , and then 

u j 222 -3 - - i sin fj,x f sin vy sin px sin vy sin ixz ..... (107). 



One of the three summations, whichever we please, may be 
performed by means of the known formulae 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 303 
c e k ^-^ -e k ^~^ . 



t ^v t c 
M + S 1P1V = 2~-6- >^^>^>Q ...... (109), 

which may be obtained by developing the second members be 
tween the limits z and z = c, y / = and y t b by the formulae 
(2), (22), and observing that the expansions hold good within 
the limits written after the formulae, since 6* (c ~* } e~* (c ~*> has 
the same magnitude and opposite signs for values of z equidistant 
from c, and e*^-J +e-*(&-tU has the same magnitude and sign 
for values of y t equidistant from 6. If in equation (107) we 
perform the summation with respect to p, by means of the formula 
(108), we get the equation (99) : if we perform the summation 
with respect to n, by means of the formula (109), we get the 
equation (100). 

52. The following problem will illustrate some of the ideas 
contained in this paper, although, in the mode of solution which 
will be adopted, the formulas given at the beginning of this Section 
will not be required. 

A hollow conducting rectangular parallelepiped is in com 
munication with the ground : required to express the potential, 
at any point in the interior, due to a given interior electrical 
point and to the electricity induced on the surface. 

Let the axes be taken as in Art. 48. Let of, y, z be the 
co-ordinates of the electrical point, m the electrical mass, v the 
required potential. Then v is determined Jirst by satisfying the 
equation y v = 0, secondly by being equal to zero at the surface, 
thirdly by being equal to m/r infinitely close to the electrical 
point, r being the distance of the points (x, y, z}, (x, y , z \ and 
"by being finite and continuous at all other points within the 
parallelepiped. 

Let v = m/r + v l , so that v^ is the potential due to the elec 
tricity induced on the surface. Then v^ is finite and continuous 
within the parallelepiped, and is determined by satisfying the 
general equation yt^ = 0, and by being equal to m/r at the 
surface. Consequently v l can be determined precisely as u in 
Art. 48 or 51. This separation however of v into two parts 
seems to introduce a degree of complexity not inherent in the 



304 ON THE CRITICAL VALUES OF 

problem ; for v itself vanishes at the surface ; and it is when 
the function expanded vanishes at the limits that the application 
of the series (2) involves least complexity. On the other hand 
we cannot immediately expand v in a triple series of the form 
(105), on account of its becoming infinite at the point (x , y\ z }. 

Suppose, therefore, for the present that the electricity is 
diffused over a finite space ; then it is evident that we may 
suppose the electrical density, p, to change so gradually, and pass 
so gradually into zero, that the derivatives of v, of as many orders 
as we please, shall be continuous functions. We may now suppose 
v expanded in a triple series, so that 



v = -nMy, sn fuff sn vy sn vrz : 
and we shall have 

yy = SSS (yu, 2 + v* + or 2 ) A mnp sin fix sin vy sin iaz. 
But we have also, by a well-known theorem, y v = 4firp ; and 

p = S2S-Knnp sin fix sin vy sin wz, 



where 



g ra rb re 

~r~ P sm I* 31 ? sin v y s i n **% dx dy dz, 

abcJo Jo Jo 



p being the same function of # , y\ z that p is of x, y, z. W 
get therefore by comparing the two expansions of yv 



whence the value of v is known. We may now,,if we like, suppose 
the electricity condensed into a point, which gives 

8m 
Rp = -7- sm px sm vy sin TSZ , 



sin fix sin vy sin vrz sin fix sin vy sin r^. . . (110). 

One of the summations may be performed just as before. We 
thus get, by summing with respect to p, 



8-Trm 



ab q e* c -6-<i c 

sin/Aic sin i/y sin fix sin z/y ....... (HI); 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 305 

where # 2 = //, 2 + v z , and z is supposed to be the smaller of the 
two z t z. If z be greater than z, we have only to make z and z 
change places in (111). 

53. The equation (110) shows that the potential at the 
point (x, y, z) due to a unit of electricity at the point (sc t y\ z) 
and to the electricity induced on the surface of the parallelepiped 
is equal to the potential at the point (x t y, z) due to a unit of 
electricity at the point (x, y, z) and to the electricity induced 
on the surface. This however is only a particular case of a general 
theorem proved by Green*. 

Of course the parallelepiped includes as particular cases two 
parallel infinite planes, two parallel infinite planes cut at right 
angles by a third infinite plane, &c. The value of v being known 
the density of the induced electricity at any point of the surface 
is at once obtained, by means of a known theorem. 

If we suppose a ball-pendulum to oscillate within a rectangular 
case, the value of < belonging to the motion of the fluid which 
is due to the direct motion of the ball and to the motion reflected 
from the case can be found in nearly the same manner. The 
expression reflected motion is here used in the sense explained 
in Art. 6 of my paper, "On some Cases of Fluid Motionf." In 
the present instance we should expand in a triple series of 
cosines. 

54. Let a hollow cylinder, containing one or more plane 
partitions reaching from the axis to the curved surface, be filled 
with homogeneous incompressible fluid, and made to oscillate 
about its axis, both ends being closed : required to determine 
the effect of the inertia of the fluid on the motion of the cylinder. 

If there be more than one partition, it will evidently be suffi 
cient to consider one of the sectors into which the cylinder is 
divided, since the solution obtained may be applied to the others. 
In the present case the motion is such that udx + vdy + wdz (ac 
cording to the usual notation) is an exact differential d(f>. The 
motion considered is in two dimensions, taking place in planes 
perpendicular to the axis of the cylinder. Let the fluid be referred 
to polar co-ordinates r y 6 in a plane perpendicular to the axis, r 
being measured from the axis, and 6 from one of the boundin^ 



* Essay on Electricity, p. 19. 

t See p. Ill of the present Volume. [Ante, p. 28.] 



20 



306 ON THE CKITICAL VALUES OF 

partitions of the sector considered, being reckoned positive when 
measured inwards. Let the radius of the cylinder be taken for the 
unit of length, and let a. be the angle of the sector, and o> the 
angular velocity of the cylinder at the instant considered. It will 
be observed that a = 2?r corresponds to the case of a single partition. 
Then to determine </> we have the general equation 



dr* r dr r 2 d6 2 
with the conditions 

i g= cor, when = or = a (113), 

. T^=0, whenr = l (114), 

and, that <> shall not become infinite when r vanishes. 

Let r = e~ A , and take 0, X for the independent variables ; then 
(112), (113), (114) become 

(115), 



= -, when ^0 or a ....... ..... (116), 

du 

^ = 0, whenX=0 ........................ (117). 

aA- 

Let $ be expanded between the limits 6 = and 9 = a in a series 
of cosines, so that 

(118), 



A , A rt being functions of X. Then we have by the formula 
(D) and the condition (116) applied to the general equation (115) 



dK* 



whence 



A. = 4.6 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 307 

Since <> is not to be infinite when r vanishes, that is when X 
becomes infinite, we have in the first place A Q = 0, A n 0. We 
have then by the condition (117) 



H7T 



when n is odd, and B n = Q when n is even. If then we omit 
which is useless, and put for X its value, we get 



C S nv 



The series multiplied by r 2 may be summed. For if we 
expand sin 2(6 Ja) between the limits 6 = 0, 6 a in a series of 
cosines, we get 



. . Q ^ 82 cos a 

sin (20 - a) = - S cos 



whence 

W7r/a cos mr#a co 



2 
mr (nV 4a 2 ) 2 cos a 



. /C . A . ,-t^\ 
r 2 sm(20-a)...(120). 



In determining the motion of the cylinder, the only quantity 
we care to know is the moment of the fluid pressures about the 
axis. Now if the motion be so small that we may omit the square 
of the velocity we shall have, putting (f> = u>f(r t 6), 



where p is the pressure, >|r () a function of the time t, whose 
value is not required, and where the density is supposed to be 1, 
and the pressure due to gravity is omitted, since it may be taken 
account of separately. The moment of the pressure on the curved 
surface is zero, since the direction of the pressure at any point 
passes through the axis. The expression (119) or (120) shows 
that the moments on the plane faces of the sector are equal, and 
act in the same direction ; so that it will be sufficient to find the 
moment on one of these faces and double the result. If we con 
sider a portion of the face for which 6 = whose length in the 
direction of the axis is unity, we shall have for the pressure on an 
element dr of the surface dw/dt . f (r, 0) dr ; and if we denote the 
whole moment of the pressures by G dco/dt, reckoned positive 

202 



308 ON THE CRITICAL VALUES OF 

when it tends to make the cylinder move in the direction of 
positive, we shall have 

(7 = 2 P/(n 0)r<Zr. 

.0 

Taking now the value of /(/ , 0) from (120), and performing the 
integration, we shall have 

(7=jtana-16a 3 2 0/ - N 1 , - ^~ 2 ......... (121). 

4 (mr 2y) mr (mr + 2a/ 

The mass of the portion of fluid considered is Ja; and if we 
put 

C = i*fc", 

and write S7T/2 for a, so that s may have any value from to 4, we 
shall have 



k 9 - tan - 2 

K> - Uclll o ^W 



o Q . , 

STT 2 TT (n s)n(n + s) 

When s is an odd integer, the expression for 7/ 2 takes the 
form oo oo , and we shall easily find 

2 



where all odd values of n except s are to be taken. 

The quantity k may be called the radius of gyration of the 
fluid about the axis. It would be easy to prove from general 
dynamical principles, without calculation, that if k be the corre 
sponding quantity for a parallel axis passing through the centre of 
gravity of the fluid, h the distance of the axes, 

k" 2 = k*+h* ....................... (124), 

in fact, in considering the motion of the cylinder, which is sup 
posed to take place in two dimensions, the fluid may be replaced 
by a solid having the same mass and centre of gravity as the 
fluid, but a moment of inertia about an axis passing through the 
centre of gravity and parallel to the axis of the cylinder different 
from the moment of inertia of the fluid supposed to be solidified. 
If K , K be the radii of gyration of the solidified fluid about the 
axis of the cylinder and a parallel axis passing through the centre 
of gravity respectively, we shall have 

.__, T ^ 4 sin -J a 8 . STT /-,-K\ 

J fiT 2 = i = A 2 + A 2 , h = - ^- = sm-r- .... (125). 
3 a 3.97T 4 N 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 309 

If we had restricted the application of the series and the 
integrals involving cosines to those cases in which the derivative 
of the expanded function vanishes at the limits, we should have 

.00 

expanded <f> in the definite integral I (0, 0) cos ft\d/3, and 

Jo 

the equation (115) would have given 



f, ^ denoting arbitrary functions, which must be determined by the 
conditions (116). We should have obtained in this manner 



* 



_ 6 



cos 



B log d/3... (126), 



= s2 r i - 6-p* 

Try. Jo l + e-0* 



(127). 



It will be seen at once that & 2 is expressed in a much better form 
for numerical computation by the series in (122) than by the 
integral in (127). Although the nature of the problem restricts 
a. to be at most equal to 2?r, it will be observed that there is no 
such restriction in the analytical proof of the equivalence of the 
two expressions for $, or for k 2 . 

In the following table the first column gives the angle of the 
cylindrical sector, the second the square of the radius of gyration 
of the fluid about the axis of the cylinder, the radius of the 
cylinder being taken for the unit of length, the third the square of 
the radius of gyration of the fluid about a parallel axis passing 
through the centre of gravity, the fourth and fifth the ratios of 
the quantities in the second and third to the corresponding quanti 
ties for the solidified fluid. 



a 


2 


k 2 


fc 2 
K* 


fc 2 
K* 





50000 


05556 


1-0000 


1-0000 


45 


45385 


03179 


9077 


4079 


90 


39518 


03492 


7904 


2499 


135 


34775 


07442 


6955 


3283 


180 


31057 


13044 


6211 


4078 


225 


28101 


18261 


5620 


4547 


270 


25703 


21700 


5141 


4718 


315 


23720 


22858 


4744 


4652 


360 


22051 


22051 


4410 


4410 



310 ON THE CRITICAL VALUES OF 

55. When a. is greater than TT, it will be observed that the 
expression for the velocity which is obtained from (119) becomes 
infinite when r vanishes. Of course the velocity cannot really 
become infinite, but the expression (119) fails for points very near 
the axis. In fact, in obtaining this expression it has been assumed 
that the motion of the fluid is continuous, and that a fluid particle 
at the axis may be considered to belong to either of the plane 
faces indifferently, so that its velocity in a direction normal to 
either of the faces is zero. The velocity obtained from (119) 
satisfies this latter condition so long as a is not greater than TT. 
For when a < TT the velocity vanishes with r, and when a. = TT the 
velocity is finite when r vanishes, and is directed along the single 
plane face which is made up of the two plane faces before con 
sidered. 

But when a is greater than TT the motion which takes place 
appears to be as follows. Let OABC be a section of the cylindrical 
sector made by a plane perpendicular to the axis, and cutting it in 
0. Suppose the cylinder to be turning round in the direction 
indicated by the arrow at B. Then the fluid in contact with OA 
and near will be flowing, relatively to OA, towards 0, as indi 




cated by the arrow a. When it gets to it will shoot past the 
face OC ; so that there will be formed a surface of discontinuity 
Oe extending some way into the fluid, the fluid to the left of Oe 
and near flowing in the direction AO, while the fluid to the 



THE SUMS OF PEEIODIC SERIES. (SECT. IV. APPLICATIONS.) 311 

right is nearly at rest. Of course, in the case of fluids such as 
they exist in nature, friction would prevent the velocity in a direc 
tion tangential to Oe from altering abruptly as we pass from a 
particle on one side of Oe to a particle on the other ; but I have 
all along been going on the supposition that the fluid is perfectly 
smooth, as is usually supposed in Hydrodynamics*. The extent of 

[* It may be said that the motion of a perfect fluid which is at first at rest, 
and is then set in motion by the action of solid bodies of finite curvature in contact 
with it, is unique and continuous, so that no surface of discontinuity can be 
formed ; and that that being always true will be true in the limit, when we suppose 
the curvature at a certain point or along a certain line to become infinite, in such 
a manner as to pass in the limit to a salient conical point or edge intruding into 
the fluid, and therefore even in this case no surface of discontinuity can be formed. 

This may be perfectly true in one sense and yet not in another. A perfect fluid 
is an ideal abstraction, representing something which does not exist in nature. All 
actual fluids are more or less viscous, and we arrive at the conception of a perfect 
fluid by starting with fluids such as we find them, and then in imagination making 
abstraction of the viscosity. Similarly any edge that we can mechanically form is 
more or less rounded off ; but we have no difficulty in conceiving of an edge 
perfectly sharp. The motion that belongs to a perfect fluid and perfectly sharp 
edge may be regarded as the limit, if unique limit there be, of the motion which 
belongs to a slightly viscous fluid interrupted by a solid presenting a salient, 
slightly rounded edge, when both the viscosity of the fluid and the radius of cur 
vature of the edge are supposed to vanish. For the sake of clear ideas we may 
suppose that the mass of fluid we are dealing with is contained in a vessel differing 
from that mentioned in the text in being bounded on the side towards the centre by 
^cylindrical surface of very small radius a, coaxial with the outer cylinder. Then 
we may represent the motion, in a sense which the reader will readily apprehend, 
by f(fj,, a), where fj, denotes the coefficient or index of internal friction. We pro 
posed to contemplate the limit of /(,u, a) when /* and a vanish. But for anything 
that appears to the contrary there may be no such unique limit, but the limit 
lim.a-o lim.p=o/(Ai, a) may be one thing, and the limit lim.^o lim. a =o /(/*, a) a 
totally different thing ; and I am strongly disposed to believe that such is actually 
the case. When a is finite and yu = 0, we pass to a case of motion of a perfect fluid 
similar to that in the text, and capable of being attacked by a similar analysis, but 
in which the motion nowhere becomes infinite ; and the limit to which this tends as 
a vanishes does not present a surface of discontinuity, but the velocity near the 
centre increases indefinitely as a decreases indefinitely. But when on the other 
hand fjt. though small remains finite, and a diminishes without limit, the motion 
which would be investigated by an analysis resembling that in the text would be 
such that near the centre there would be an enormous gliding, which would call 
into play a great tangential force, in which work would be consumed, that is, 
converted into heat. This I believe would be an unstable condition, and what 
would actually take place would be that so large a local consumption of work would 
be avoided by the fluid rushing past the corner, somewhat as represented in the 
figure, carrying with it by adhesion a narrow stratum in which there would be very 
great molecular rotation, inasmuch as the fluid of which the stratum consists had 
previously been pent up between the radial wall of the vessel on one side of it, 



312 ON THE CRITICAL VALUES OF THE SUMS OF PERIODIC SERIES. 

the surface of discontinuity Oe will be the less the smaller be the 
motion of the cylinder; and although the expression (119) fails 
for points very near 0, that does not prevent it from being sensibly 
correct for the remainder of the fluid, so that we may calculate k * 
from (122) without committing a sensible error. In fact, if 7 be 
the angle through which the cylinder oscillates, since the extent of 
the surface of discontinuity depends upon the first power of 7, the 
error we should commit would depend upon 7*. I expect, there 
fore, that the moment of inertia of the fluid which would be 
determined by experiment would agree with theory nearly, if not 
quite, as well when a > TT as when a < TT, care being taken that 
the oscillations of the cylinder be very small. 

As an instance of the employment of analytical expressions 
which give infinite values for physical quantities, I may allude to 
the distribution of electricity on the surfaces of conducting bodies 
which have sharp edges. 

56. The preceding examples will be sufficient to show the 
utility of the methods contained in this paper. It may be observed 
that in all cases in which an arbitrary function is expanded 
between certain limits in a series of quantities whose form is 
determined by certain conditions to be satisfied at the limits, the 
expansion can be performed whether the conditions at the limits 
be satisfied or not, since the expanded function is supposed per 
fectly arbitrary. Analogy would lead us to conclude that the 
derivatives of the expanded functions could not be found by direct 
differentiation, but would have to be obtained from formulae 
answering to those at the beginning of this Section. If such 
expansions should be found useful, the requisite formula? would 
probably be obtained without difficulty by integration by parts. 
This is in fact the case with the only expansion of the kind which 
I have tried, which is that employed in Art. 45. By means of 
this expansion and the corresponding formulae we might determine 
in a double series the permanent temperature in a homogeneous 
rectangular parallelepiped which radiates into a medium whose 

which had no radial motion and but little in a perpendicular direction, and the 
rapidly rushing fluid on the other side. The smaller /* is made, the narrower will 
this stratum be, but not, so far as I can see, the shorter; and a very narrow 
stratum in which there is intense molecular rotation passes, or may pass, in the 
limit to a surface of discontinuity. 

The above is what was referred to by anticipation in the footnote at p. 99.] 



THE SUMS OF PERIODIC SERIES. (SECT. IV. APPLICATIONS.) 313 

temperature varies in any given manner from point to point ; or 
we might determine in a triple series the variable temperature in 
such a solid, supposing the temperature of the medium to vary in 
a given manner with the time as well as with the co-ordinates, 
and supposing the initial temperature of the parallelepiped given 
as a function of the co-ordinates. This problem, made a little 
more general by supposing the exterior conductivity different for 
the six faces, has been solved in another manner by M. Duhamel 
in the Fourteenth Volume of the Journal de lEcole Poly technique. 
Of course such a problem is interesting only as an exercise of 
analysis. 



ADDITIONAL NOTE. 

If the series by which r 2 is multiplied in (119) had been left 
without summation, the series which would have been obtained for 
A/ 2 would have been rather simpler in form than the series in (122), 
although more slowly convergent. One of these series may of 
course be obtained from the other by means of the development 
of tan x in a harmonic series. When s is an integer, k 2 can be 
expressed in finite terms. The result is 

& 2 = 8s- 1 7T 2 log e 2 + 8s 1 7T- 2 {2- 1 + 4" 1 . . . 

+ (s - I) 1 } + 47T- 2 {2- 2 + 4- 2 . . . + (s - I)" 2 ) - J, (s odd,) 

F = 8s- 1 7T- 2 {r 1 + 3- 1 . . . + (s - 1) 1 } + 47T- 2 {r 2 + 3~ 2 . . . 

+ (s - 1)" 2 ) - J. (s even.) 

Moreover when 2s is an odd integer, or when a = 45, or = 135, &c., 
k * can be expressed in finite terms if the sum of the series 
1 2 + 5~* -f 9~ 2 + ... be calculated, and then be regarded as a known 
transcendental quantity. 



[Not before published. (See page 229.)] 

SUPPLEMENT TO A PAPER ON THE THEORY or OSCILLATORY 

WAVES. 

THE labour of the approximation in proceeding to a high order, 
when conducted according to the method of the former paper 
whether we employ the function </> or ty, depends in great measure 
upon the circumstance that the two equations which have to be 
satisfied simultaneously at the free surface are both composed in a 
rather complicated manner of the independent variables, and in the 
elimination of y the length of the process is still further increased 
by the necessity of expanding the exponentials in y according to 
series of powers, giving for each exponential a whole set of terms. 
This depends upon the circumstance that of the limits of y belong 
ing to the boundaries of the fluid, one instead of being a constant 
is a function of x, and that too a function which is only known 
from the solution of the problem. 

If we convert the wave motion into steady motion, and refer 
the fluid to two independent variables of which one is the para 
meter of the stream lines or a function of the parameter, and the 
other is x or a quantity which extends with x from -co to + oo , 
we shall ensure constancy of each independent variable at both its 
limits, but in general the equations obtained will be of great com 
plexity. It occurred to me however that if from among the infinite 
number of systems of independent variables possessing the above 
character we were to take the functions </>, i/r, where 

<f> =f(ud% -f vdy), ty = $(udy vdx), 

simplicity might be gained in consequence of the immediate rela 
tion of these functions to the problem. 



SUPPLEMENT TO A PAPER ON OSCILLATORY WAVES. 315 

We know that <, ^ are conjugate solutions of the equation 



satisfying the equations 

dcf> d^r d(f> _ dty , . 

dx dy dy dx " 

so that if the form of either be assigned, satisfying of course the 
equation (1), the other may be deemed known, since it can be 
obtained by the integration of a perfect differential. If now we 
take (/>, -fy for the independent variables, of which x and y are re 
garded as functions, we get by changing the independent variables 
in differentiation 

d<}> _ 1 dy d</> _ 1 dx d^r _ 1 dy d^r _ldx , , 
dx^Sd^ d^ = ~Sd$ ~dx~~Sd^ ~dj~" 

a dx dy dx dy 

where ^ TT TT ~~ TT j > 

d^jr d(f> 



whence from (2) 

dx _ dy dx _ _ dy ... 

~d$~d^ ~d^~~d$" 

so that x, y are conjugate solutions of the equation 

tfx d*x . 



We have also from (4) 



We get from (3), (4) and (6) 

\ 2 (dty\* 1 (fda;\* f dx 

+(-/) =^\ ,, + 
j \dxj 8 [\d<t>J \ 

whence 

(8), 



- 

where C is an arbitrary constant. 

The mode of proceeding is the same in principle whether the 
depth of the fluid be finite or infinite ; but as the formulae are 
simpler in the latter case, it may be well to consider it separately 
in the first instance. 



316 SUPPLEMENT TO A PAPER ON THE 

If be the velocity of propagation, c will be the horizontal 
velocity at a great depth when the wave motion is converted into 
steady motion. The difference between </> and ex will be a 
periodic function of x or of (/>. We may therefore assume in ac 
cordance with equation (5) 

x = -^ + ^(ApWI* + B i e-Wl*)*\nim<l>lc ......... (9). 

c 

No cosines are inserted in this equation because if we take, as we 
may, the origins of x and of <f> at a trough or a crest (suppose a 
trough), x will be an odd function of </>, in accordance with what 
has already been shown at page 212. Corresponding to the above 
value of x we have 



y = - + (A.e im ^ c - Re- im ^ c ) cos im^/c ..... (10), 
c 

the arbitrary constant being omitted, as may be done provided we 
leave open the origin of y. 

The origin of ty being arbitrary, we may take, as it will be 
convenient to do, ^ = at the free surface. We see from (10) that 
ty increases negatively downwards ; and therefore of the two ex 
ponentials that with im-jr/c for index is the one which must be 
omitted, as expressing a disturbance that increases indefinitely in 
descending. 

We may without loss of generality shorten the formulas during 
a rather long approximation by writing 1 for any two of the con 
stants which depend differently on the units of space and time. 
These constants can easily be reintroduced in the end by rendering 
the equations homogeneous. We may accordingly put m = 1 and 
c = 1. The expressions for x and y as thus shortened become, on 
retaining only the exponential which decreases downwards, 

l> .................. (11), 

l> .................. (12). 

At the free surface i/r = 0, and we must therefore have for ty 



which gives 

(C + 2 A. cos fy) (1 - 22t4 4 cos ty + 2*% f + VtijAA> cos [(t - ; 



THEORY OF OSCILLATORY WAVES. 317 

where in the last term within parentheses each different combina 
tion of unequal integers i, j is to be taken once. 

On account of the complicated form of this equation, we can 
proceed further only by adopting some system of approximation. 
The most obvious is that adopted in the former paper, namely to 
proceed according to powers of the coefficient of the term of the 
first order. If we multiply out in equation (13), and replace pro 
ducts of cosines by cosines of sums and differences, we may arrange 
the equation in the form 

jB + S l cos $ + 2 cos 2< + . . . = 0, 

where the several It s are series of terms involving the coefficients 
A. And as the equation has to be satisfied independently of <, 
we must have separately 

= 0, ^=0, 2 = 0, &c. 

A slight examination of the process will show that A. is of the 
order ", and that consequently the product of any number of the 
A s is of the order marked by the sum of the suffixes, and that B. 
is of the order i. In proceeding therefore to any desired order we 
can see at once what terms need not be written down, as being of 
a superior order. 

Thus in proceeding to the fifth order we must take the six 
equations B Q = 0, B l = 0, . . . B 6 = 0, which when written at length 
are 

C (1 + A* + 4^1 2 2 ) - A* + 2A*A 2 - 2A* - J g* = 0, 



+ 3^4,- 544, = 0, 

C (- 4J 2 + 644) + A 2 - A? + 34 2 J 2 - 444 = 0, 
(-64+ 844) + J 3 - 344 + 44M a + 2A l A?- 5^J 4 = 0, 



c (- 

These equations may be looked on as giving, the first, the 
arbitrary constant C, the second, the velocity of propagation, and 
the succeeding ones taken in order the values of the constants 
A 2 , A 3 , A 4 , A 5 , respectively. I say " may be looked on as giving", 
for it is only when we restrict ourselves to the terms of the lowest 
order in each equation that those quantities are actually given in 
succession ; the equations contain also terms of higher orders ; and 



318 SUPPLEMENT TO A PAPER ON THE 

to get the complete values of the quantities true to the order to 
which we are working, we must use the method of successive sub 
stitutions. As to the second equation, if we take the terms of 
lowest order in the first two we get C = J^" 1 , and then by substi 
tution in the second equation 1 = g, the constant A^ dividing out. 
The equation 1 =g becomes on generalizing the units of space and 
time c 2 = g/m, and accordingly gives the velocity of propagation to 
the lowest order of approximation. 

On eliminating the arbitrary constant in the above equations, 
and writing b for A i9 the results become 



(14), 
x = - + 6e*sin - (b 2 + PV / sin 20 + (f b 5 + if 6 5 )^sin 30 

............ (15), 



-f 6V* cos 40 + 1^ 6V* cos 50 ............ (16). 

The equation (14) gives to the fifth order the square of the 
velocity of propagation in the wave motion; and (15), (16) give 
the point where the parameters 0, >|r have given values, and also, 
by the aid of the formulae previously given, the components of the 
velocity, and the pressure, in the steady motion. These same 
equations (15), (16), if we suppose ^jr constant give implicitly the 
equation of the corresponding stream line, or if we suppose 
constant the equation of one of the orthogonal trajectories. 

To find implicitly the equation of the surface, we have only to 
put i|r= in (15), (16), which gives 

as = - + b sin - (V + |- 6 4 ) sin 20 - (f tf + if b 5 ) sin 30 

-f & 4 sin 40+ ^& B sin5 ............... (17), 

y = b cos - (6 2 + i 6 4 ) cos 20 - (f b* + if I 5 } cos 30 

-|6 4 cos40-}- % 5 6 5 cos50 ............... (18). 

It is not necessary to form the explicit equation, but we can do so 
if we please, most conveniently by the aid of Lagrange s theorem. 
The result, carried to the fourth order only, which will suffice for 
the object more immediately in view, is 

6 3 ) cos x- (i& 2 + V & 4 ) cos 2# 

+ f b 3 cos 3a? - J6 4 cos 4^... (19). 



THEORY OF OSCILLATORY WAVES. 319 

If we put b + b 3 = a, we have to the fourth order 

b = a - | a 3 , 

and substituting in (19) we get 
2/4-1 a 2 i a 4 = a cos x (J a 2 + -|J a 4 ) cos 2% + f a 3 cos 3# 

- J a 4 cos4# (20). 

The expression (14) for the square of the velocity of propagation, 
and the equation of the surface (20), agree with the results pre 
viously obtained by the former method (see p. 221) to the degree 
of approximation to which the latter were carried, as will be seen 
when we remember that the origins of y are not the same in the 
two cases ; but it would have been much more laborious to obtain 
the approximation true to the fifth order by the old method. 

It has already been remarked (p. 211) that the equation of the 
profile in deep water agrees with a trochoid to the third order, 
which js-as. far as the approximation there proceeded. This is no 
longer, true when we proceed to the fourth order. On shifting 
the origin of y so as to get rid of the constant term, the equation 
(20) of the profile becomes 

y a cos x ( J a 2 + JJ a 4 ) cos Zx + f a 3 cos 3# J a 4 cos 4#. . . (21). 

On the other hand, the equation of a trochoid is given impli 
citly by the pair of equations 

x = a6 + j3 sin 0, y = j3cos0 + y. 

In order that x may have the same period in the trochoid as in 
the profile of the wave, we must have a = 1. We get then by 
development to the fourth order, choosing 7 so as to make the 
constant term vanish, 

y = (p- f 3 ) cos x - (4 /3- J 4 ) cos 2^ + /3 3 cos 3x - J/3 4 cos 4a?, 

and putting 

/3-t/3 ! = , 

we get to the fourth order 

y = aco$x- (i a 2 + V a*) cos Zx + f a 3 cos 3# - J a 4 cos 4^.. .(22). 

Hence if y w , y t denote the ordinates for the wave and trochoid 
respectively, we have to the fourth order 



Hence the wave lies a little above the trochoid at the trough and I/ 
crest, and a little bejow it in the shoulders. 



320 SUPPLEMENT TO A PAPER ON THE 

This result agrees well witli what might have been expected. 
It has been shown (p. 227) that the limiting form for a series of 
uniformly propagated irrotational waves is one presenting edges of 
120, and that the inclination in this limiting form is in all proba 
bility restricted to 30, whereas in the trochoidal waves investigated 
by Gerstner and Kankine the limiting form is the cycloid, presenting 
accordingly cusps, and an inclination increasing to 90. Hence the 
limiting form must be reached with a much smaller value of the 
parameter a in the former case than in the latter. Hence when 
a is just large enough to make the difference of form of the irrota 
tional and trochoidal waves begin to tell, since the limiting form 
is more nearly approached in the former case than in the latter, we 
should expect the curvature at the summit to be greater, while at 
the same time as the general inclination is probably rather less, 
and the inclination begins by increasing more rapidly as we recede 
from the summit, the troughs must be shallower and flatter for an 
equal mean height of wave. 

Let us proceed now to the case of a finite depth. As before 
we may choose the units of space and time so that c and m 
shall each be 1, and we may choose for the value of the para 
meter ^/r at the surface. Let & be its value at the bottom. 
Then since dfyjdy = at the bottom we have from (3) and (4) 
dy/d<p = when ty + k = 0, and consequently 



whence writing A.e ik for A. we have 

o * i 

iDMty ........... (23), 



Putting for shortness 

e * + e-* = S t , e ik -e- ik =D it 

we have by the condition at the free surface 

(C + ^,A t D t cos ty) {1 - 2&A t iS t cos i</> + $iAj3 t cos /</>/ 



As the expressions are longer than in the case of an infinite 
depth, and the problem itself of rather less interest, I shall content 



THEORY OF OSCILLATORY WAVES. 321 

myself with proceeding to the third order. We have to this 
order from (25), on taking account of the relations 



(C + AJ)^ cos + A 2 D 2 cos 2</> + A 3 D 3 cos 30) 

x | 1 - 2A& cos cf) - 4A& cos 20 - 6A 3 S 3 cos 30 1 
{+ A*S Z + 4AtAj8 9 cos -f 2^ 2 cos 20 + ^^cos 30j 

-** 

]\Iultiplying out, retaining terms up to the third order only, 
arranging the terms according to cosines of multiples of 0, and 
equating to zero the coefficients of the cosines of the same 
multiple, we get the four equations 



j 

3 



C (- 4,A& + 2A*) - AS t D t + A,D, = 0, 
C (- QA S 



A slight examination of the process of approximation will 
show that, whatever be the order to which we proceed, C, and 
the coefficients A z , A^ ... with even suffixes, will contain only 
even powers, and the coefficients A 3 , A 5) ... with odd suffixes 
only odd powers, of the first coefficient A lt Writing b for A lt we 
may therefore assume, in proceeding to the third order only, 



Substituting in the last three equations of the preceding group, 
which after the substitution may be divided by b, b 2 , b* respect 
ively, arranging, and equating coefficients of like powers of 6, 
we get 

- 2^a -f D, = 0, 

) 7 - 2^/3 + S& 4- A = 0, 



(D, - 66 8=0. 

21 



322 SUPPLEMENT TO A PAPER ON THE 

The substitution for C and the coefficients A 2 , A a , ... of series 
according to even or odd powers of b with indeterminate coeffi 
cients was hardly worth making in proceeding to the third order 
only, but seems advantageous when we want to proceed to a 
rather high order. In proceeding to the n ih order it is to be noted 
that the coefficients of C in the group of n + 1 equations got by 
equating to zero the coefficients of cosines of multiples of </> 
(including the zero multiple, or constant term), are of the orders 
0, 1, 2, ... n in b, so that C being determined only to b n ~ l in the 
equations after the first, the terms of the order n in the first 
equation (which could only occur when n is even) are not required, 
but this first equation need only be carried as far as to n 1. 
In fact, in proceeding to the orders 1, 2, 3, 4, 5, 6, ..., the velocity 
of propagation is given to an order not higher than 0, 1, 2, 3, 
4, 5, ... in b, and therefore actually to 0, 0, 2, 2, 4, 4, ... since 
it involves only even powers of b. 

The last equations give in succession 



(26), 



(28), 
(29), 



and then by substituting in the first equation of the group on 
the middle of p. 321, we get 



We get now from (23), (24), after rendering the equations ho 
mogeneous, 

x = - 
c 



...... (31), 

cos m$/c + &c ...... (32), 



THEORY OF OSCILLATORY WAVES. 323 

the expression for y after the first term differing from that for x 
only in having a minus sign before the second exponential in each 
term, and cosines in place of sines. We have also from (30) 

= + A (S t +aS, + 12) V ............. (33), 

9 t ^1-^1 

which gives the velocity of propagation according to one of its 
possible definitions (see Art. 3, p. 202). In these expressions 
it is to be observed that 

imk/c J) e imk/c _ e - 



We might of course in the numerators of the coefficients have 
used expressions proceeding according to powers of 8 t instead of 
according to the functions 8 l) S 2 , S 3 ... 

Let h be the value of y at the bottom, which is a stream line 
for which ty= k, then we have from (24) generalized as to 
units 

k = ch ............................... (34), 

so that it remains only to specify the origin of y and the meaning 
of c. To the first order of small quantities we have 



C 

and at the surface 



/c ......... (35), 

< ) cos m^/c ......... (36), 



lc ................................. (37), 

c 

y = bD t cos ra</>/c ......................................... (38). 

Since y in (38) is a small quantity of the first order, we may 
replace <f>/c in its expression by #, in accordance with (37), which 
gives for the equation of the surface 

y = bD t cos mx, 

so that to this order of approximation the origin is in the plane 
of mean level, and therefore h denotes the mean depth of the fluid. 
Also since u = dfy/dx we have to the first order from (3), (4), (6) 

u = Fjjj] = j- i + ^ ( e (*+A:)/c + e -m(*+*)/c) cos m^ 
= -c-c.mb (e h ~ri + e - m(h -^) cos mx, 



324 SUPPLEMENT TO A PAPER ON THE 

and consists therefore of two parts, one representing a uniform 
flow in the negative direction with a velocity c, and the other 
a motion of periodic oscillation. To this order therefore there 
can be no question that c should be the horizontal velocity in 
a positive direction which we must superpose on the whole mass 
of fluid in order to pass to pure wave motion without current. 
In passing to the higher orders it will be convenient still to regard 
this constant as the velocity of propagation, and accordingly as 
representing the velocity which we must superpose, in the positive 
direction, on the steady motion in order to arrive at the wave 
motion ; but what, in accordance with this definition, may be 
the mean horizontal velocity of the whole mass of fluid in the 
residual wave motion, or what may be the mean horizontal velocity 
at the bottom, &c., or again what is the distance of the origin from 
the plane of mean level, are questions which we could only answer 
by working out the approximation, and which it would be of 
very little interest to answer, as we may just as well suppose 
the constant h defined by (34) given as suppose the mean depth 
given, and similarly as regards the flow. 

Putting \/r = in (31) and (32), we have implicitly for the 
equation of the surface the pair of equations 

x - - + SJ) sin m^/c --- ^ ($ 2 + 1) S z mb* sin 2m<j)/c 
c U 






4) SX&" sin 



y = Dp cos infyjc - - ($ 8 + l) D 2 mb 2 cos 2mc/>/c 



+ Q n 4 (3$ 4 + 4$ 2 + 4) D 3 m 2 & 3 cos 

The ratios of the coefficients of the successive cosines in y or 
sines in x to what they would have been for an infinite depth, 
supposing that of cos md>/c the same in the two cases, are 

i i 

multiplied respectively by 

# 



THEORY OF OSCILLATORY WAVES. 325 



for the cosines in y, and by 



8. 8, 8, 

n> n , . n a 



1 "! 

for the sines in #. Expressed in terms of D t , the first three ratios 
become 



1, l + W~\ I 

and increase therefore as the depth diminishes, and consequently 
D 1 diminishes. The same is the case with the multipliers 
D 2 /A 2 > DJD?, SJD 1} &c, and on both accounts therefore the 
series converge more slowly as the depth diminishes. Thus for 
1)^ = 3 the first three ratios are 1, 2, 3$. 1^ = 3 corresponds to 
A/X = 125, nearly, so that the average depth is about the one- 
eighth of the length of a wave. 

The disadvantage of the approximation for the case of a finite 
as compared with that of an infinite depth is not however quite so 
great as might at first sight appear. There can be little doubt 
that in both cases alike the series cease to be convergent when 
the limiting wave, presenting an edge of 120, is reached. In the 
case of an infinite depth, the limit is reached for some determinate 
ratio of the height of a wave to the length, but clearly the same 
proportion could not be preserved when the depth is much 
diminished. In fact, high oscillatory waves in shallow water tend 
to assume the character of a series of disconnected solitary waves, 
and the greatest possible height depends mainly on the depth of 
the fluid, being but little influenced by the length of the waves, 
that is, the distance from crest to crest. To make the comparison 
fair therefore between the convergency of the series in the cases of 
a finite and of an infinite depth, we must not suppose the co 
efficient of cos m<p/c the same in the two cases for the same length 
of wave, but take it decidedly smaller in the case of the finite 
depth, such for example as to bear the same proportion to the 
greatest possible value in the two cases. 

But with all due allowance to this consideration, it must be 
confessed that the approximation is slower in the case of a finite 
depth. That it must be so is seen by considering the character of 
the developments, in the two cases, of the ordinate of the profile 
in a harmonic series in terms of the abscissa, or of a quantity 
having the same period and the same mean value as the abscissa. 
The flowing outline of the profile in deep water lends itself readily 



326 THEOKY OF OSCILLATORY WAVES. 

to expansion in such a series. But the approximately isolated 
and widely separated elevations that represent the profile in very 
shallow water would require a comparatively large number of 
terms in their expression in harmonic series in order that the 
form should be represented with sufficient accuracy. In extreme 
cases the fact of the waves being in series at all has little to 
do with the character of the motion in the neighbourhood of the 
elevations, where alone the motion is considerable, and it is not 
therefore to be wondered at if an analysis essentially involving the 
length of a wave should prove inconvenient. 



INDEX TO VOL. I. 



Aberration of light, 134, 153; Fresnel s 

theory respecting, 141 
Airy, Sir G. B., on tides and waves, 163, 

165, 169, 171 

Angular velocities of a fluid, 81, 112 
Axes of extension, 82 

Babinet s result as to non-influence of 
earth s motion on interference ex 
plained, 142 

Ball pendulum, resistance to, 180, 186; 
resistance to a, within a concentric 
spherical case, 41, 181; in presence of 
a distant plane, 43 ; within rectangu 
lar box, 111, 305 

Box, motion of fluid within a closed, of 
the form of a parallelepiped, 60, 66, 
194, 288 ; equilateral triangular prism, 
65 ; elliptic cylinder, 65 ; sector of 
cylinder, 305 

Cauchy s proof of a fundamental pro 
position in hydrodynamics, 107, 160 

Challis, Prof., aberration, 138; hydro- 
dynamical theorem, 160; ball pendu 
lum, 180 

Convergency, essential and accidental, 
241 ; infinitely slow, 281 

Current, superficial, in water an accom 
paniment of waves, 208 

Cylinder, motion of a piston and of the 
air within a, 69 

Cylindrical surfaces, (circular) motion 
of perfect fluid between, 30 ; (elliptic) 
approximate motion within or outside, 
54 ; (circular) motion of viscous fluid 
between, 102 



Determinateness of problems in fluid 
motion, 21 

Discharge of air through small orifices, 
paradox relating to, 176 

Discontinuity, determination of, in a 
function expressed by series or inte 
grals of periodic functions, 239, 271 ; 
propositions respecting, in the sums of 
infinite series, &c., 279 ; of motion in 
a fluid, 310 

Doubly refracting crystals, formula for 
determining the principal indices of, 
148 

Earnshaw, S., on solitary waves, 169 
Eddies, production of resistance by, 53, 

99 ; production of, 311 
Elastic solids, isotropic, equations of 
equilibrium &c. of, 113; necessity of 
two arbitrary constants in the equa 
tions, 120 

Fresnel s theory of non-influence of 
earth s motion on the reflection and 
refraction of light, 141 

Friction, internal, of fluids, theory of, 75, 
182; production of eddies by, 99, 311 

Gerstner s investigation of a special case 
of possible waves, 219 

Green, notice of his papers on waves, 
162 ; on sound, 178 ; on the motion of 
fluid about an ellipsoid, 54, 179 

Hydrodynamics, report on, 157 
Impulsive motion of fluids, 23 



328 



INDEX TO VOL. I. 



Instability of motion, 53, 311 
Integrating factor of homogeneous dif 
ferential equations, 130 

Kelland, Prof., long waves in canal of 
any form, 163 ; oscillatory waves in tri 
angular canal, 165 

Lee- way of a ship, effect of waves on, 

208 

Lines of motion (see Stream lines) 
Luminiferous ether, equations of mo 
tion of, 124; constitution of, 153 

Motion of fluids, some cases of, 17; sup 
plement, 188 

Newton s solution of velocity in a vortex, 
correction in, 103 

Parallelepiped, rectangular, motion of 
fluid within, 60, 66, 288; experiments 
as to the motion, 194; different ex 
pressions for permanent temperature 
in, 295, 302; expression for the poten 
tial in a hollow conducting, due to an 
interior electric point, 303 

Periodic series, critical values of the 
sums of, 236 

Pipe, linear motion of fluid in a, 105; 
production of eddies in a, 99 

Poisson s theory of elastic solids, 116; 
of viscous fluids, 118, 182; reduction 
of his two arbitrary constants in the 
latter case to one, 119, 184 

Poisson s solution of the problem of a 
ball pendulum, correction in, 42, 49 

Eankine s investigation of a special case 
of possible rotational waves, 219; of 
the limiting form of irrotational waves, 
225 

Eectangle, different expressions for the 
permanent temperature in, 290 



Eeflection, principle of, as applied to 

the motion of liquids, 28 
Eesistance referable to instability of 

motion and eddies, 52, 99 

Saint- Venant and Wantzel, discharge of 
air through small orifices, 176 

Saint-Venant, equations of motion of a 
viscous fluid, 183 

Sound, intensity and velocity of, theo 
retical effect of viscosity of air on, 100 

Spheres, motion of fluid between two 
concentric, 38; non-concentric, 230 

Steady motion of incompressible fluids, 1 

Stream lines, determination whether a 
given family of curves can be a set of, 
in two dimensions, 5, 9; for motion 
symmetrical about an axis, 15 

Thomson, F. D. , demonstration of a the 
orem due to him, 7 

Thomson, Sir W., hydrodynamical ap 
plication of his method of images, 230; 
expression suggested by, for perma 
nent temperature in rectangular pa 
rallelepiped, 301 

Tides (see Waves) 

Triangular prism, (equilateral) motion of 
fluid within, 8, 65 

udx + vdy+wdz an exact differential, 
proposition relating to, 1, 20, 106, 
158 

Uniqueness of xpf ession for p, q, or 0, 
23 

Waves and tides, report respecting, 161 
Waves, theory of oscillatory, of finite 
height, 197, 314 ; of small, at the com 
mon surface of two liquids, 212; 
greatest height of, at the surface of a 
liquid, 225 ; Gerstner and Eankine s 
investigation of a special possible case 
of, 219; solitary, 168, 325 



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AN ELEMENTARY TREATISE ON QUATERNIONS, 

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COUNTERPOINT. 

A Practical Course of Study, by Professor G. A. MACFARREN, M.A., 
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A CATALOGUE OF AUSTRALIAN FOSSILS 

(including Tasmania and the Island of Timor), Stratigraphically and 
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TOMY, VERTEBRATE AND INVERTEBRATE, 

for the Use of Students in the Museum of Zoology and Comparative 
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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 Quarto, Plates, *, u. 

A CATALOGUE OF THE COLLECTION OF 
CAMBRIAN AND SILURIAN FOSSILS 

contained in the Geological Museum of the University of Cambridge, 
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CATALOGUE OF OSTEOLOGICAL SPECIMENS 

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THE MATHEMATICAL WORKS OF 

ISAAC BARROW, D.D. ,- 
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ASTRONOMICAL OBSERVATIONS 

made at the Observatory of Cambridge by the Rev. JAMES CHALLIS, 
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ASTRONOMICAL OBSERVATIONS 

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THE CAMBRIDGE UNIVERSITY PRESS. 15 



LAW. 

AN ANALYSIS OF CRIMINAL LIABILITY. 

By E. C. CLARK, LL.D., Regius Professor of Civil Law in the 
University of Cambridge, also of Lincoln s Inn, Barrister at Law. 
Crown 8vo. cloth, 75. 6d. 

A SELECTION OF THE STATE TRIALS. 
By J. W. WiLLiS-BUND, M.A., LL.B., Barrister-at-Law, Professor of 
Constitutional Law and History, University College, London. Vol. I. 
Trials for Treason (13271660). Crown 8vo. cloth, iSs. 

" A great and good service has been done volumes of the State Trials." Contemporary 

to all students of history, and especially to Review. 

those of them who look to it in a legal aspect, " This work is a very useful contribution 
by Prof J. W. Willis-Bund in the publica- to that important branch of the constitutional 
tion of a Selection of Cases from the State history of England which is concerned with 
Trials . . . Professor Willis- Bund has been the growth and development of the law of 
very careful to give such selections from the treason, as it may be gathered from trials be- 
State Trials as will best illustrate those fore the ordinary courts. The author has 
points in what may be called the growth of very wisely distinguished these cases from 
the Law of Treason which he wishes to those of impeachment for treason before Par- 
bring clearly under the notice of the student, liament, which he proposes to treat in a future 
and the result is, that there is not a page in volume under the general head Proceedings 

the book which has not its own lesson in Parliament. "/"^ Academy. 

In all respects, so far as we have been able "This is a work of such obvious utility 

to test it this book is admirably done." that the only wonder is that no one should 

Scotsman. have undertaken it before. ... In many 

,. , . ~ , respects therefore, although the trials are 

"Mr Willis-Bund has edited A Selection more or ]ess abridgedj this is for the ordinary 

of Cases from the State Trials which is studen t s purpose not only a more handy, 

likely to form a very valuable addition to bu( . a more useful work than Rowell s." 

the standard literature. . . ihere can Saturday Review. 

be no doubt, therefore, of the interest that Within the boards of this useful and 

can be found in the State trials. But they handy book the student will find everything 

are large and unwieldy, and it is impossible he can desire {n the way of lists of cases 

for the general reader to come across them . n at i ength or re ferred to, and the 

Mr WUlUBund has therefore done good statutes bearing on the text arranged chro- 

service in making a selection that is m the no l O gically. The work of selecting from 

first volume reduced to a commodious torm. Howell s bulky series of volumes has been 

The Examiner. done with much judgment, merely curious 

"Every one engaged, either in teaching cases being excluded, and all included so 

or in historical inquiry, must have felt the treated as to illustrate some important point 

want of such a book, taken from the unwieldy of constitutional law." Glasgow Herald. 

Vol. II. In the Press. 

THE FRAGMENTS OF THE PERPETUAL 
EDICT OF SALVIUS JULIANUS, 

collected, arranged, and annotated by BRYAN WALKER, M.A. LL.D., 
Law Lecturer of St John s College, and late Fellow of Corpus Christi 
College, Cambridge. Crown 8vo., Cloth, Price 6s. 

"This is one of the latest, we believe mentaries and the Institutes . . . Hitherto 

quite the latest, of the contributions made to the Edict has been almost inaccessible to 

legal scholarship by that revived study of the ordinary English student and such a 

the Roman Law at Cambridge which is now student will be interested as well as perhaps 

so marked a feature in the industrial life surprised to find how abundantly the extant 

of the University. ... In the present book fragments illustrate and clear up points which 

we have the fruits of the same kind of have attracted his attention in the Commen- 

thorough and well-ordered study which was taries, or the Institutes, or the Digest. - 

brought to bear upon the notes to the Com- Law Times. 

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1 6 PUBLICATIONS OF 



THE COMMENTARIES OF GAIUS AND RULES 
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With a Translation and Notes, by J. T. ABDY, LL.D., Judge of County 
Courts, late Regius Professor of Laws in the University of Cambridge, 
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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 
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to practised scholars, whose knowledge of almost indispensable." Spectator. 
classical models does not always avail them "The notes are learned and carefully corn- 
in dealing with the technicalities of legal piled, and this edition will be found useful 
phraseology. Nor can the ordinary diction- to students." Law Times, 
aries be expected to furnish all the help that " Dr Abdy and Dr Walker have produced 
is wanted. This translation will then be of a book which is both elegant and useful." 
great use. To the ordinary student, whose Athenceum. 

SELECTED TITLES FROM THE DIGEST, 

annotated by B. WALKER, M.A., LL.D. Part I. Mandati vel 
Contra. Digest XVII. i. Crown 8vo., Cloth, 5^. 

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Part II. De Adquirendo rerum dominio and De Adquirenda vel amit- 
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GROTIUS DE JURE BELLI ET PACIS, 

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HISTOKY. 

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 de 
partment, we might perhaps look forward to 
a time when our historians would raise the 
cry of protection for native industry. Of 
the unquestionably 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 competi 
tion with German works and disowns the 
presumption of teaching us Germans our own 
history, yet casts into the shade by its bril 
liant superiority all that we have ourselves 
hitherto written about Stein.... In five long 
chapters Seeley expounds the legislative and 
administrative reforms, the emancipation of 
the person and the soil, the beginnings of 
free administration and free trade, in short 
the foundation of modern Prussia, with more 
exhaustive thoroughness, with more pene 
trating insight, than any one had done be 
fore." Deutsche Rundschau. 

"Dr Busch s volume has made people 
think and talk even more than usual of Prince 
Bismarck, and Professor Seeley s very learned 
work on Stein will turn attention to an earlier 
and an almost equally eminent German states 
man It is soothing to the national 

self-respect to find a few Englishmen, such 
as the late Mr Lewes and Professor Seeley, 



doing for German as well as English readers 
what many German scholars have done for 
us. " Times. 

" 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 under 
stand the Germany of to-day one must study 
the Germany 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." Athe* 
neeum. 

"The book before us fills an important 
gap in English nay, European historical 
literature, 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 proper historic light We con 
gratulate Cambridge and her Professor of 
History on the appearance of such a note 
worthy production. And we may add that it 
is something upon which we may congratulate 
England that on the especial field of the Ger 
mans, history, on the history of their own 
country, by the use of their own literary 
weapons, an Englishman has produced a his 
tory of Germany in the Napoleonic 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. cloth (734 pp.), \is. 

the University during the troublous times of 
the Reformation and the Civil War." Athe- 



"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 

continue his labours, and give us a history of 



nceum. 

" Mr Mullinger s work is one of great 
learning and research, which can hardly fail 
to become a standard book of reference on 
the subject. . . . We can most strongly recom 
mend this book to our readers." Spectator. 



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iS 



PUBLICATIONS OF 



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 
of almost inexhaustible amusement, by his 
torians it will be found a work of considerable 
service on questions respecting our social 
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thoroughness with which Mr Mayor has dis 
charged his editorial functions are creditable 
to his learning and industry." Athenceum. 

" The work displays very wide reading, 
and it will be of great use to members of the 
college and of the university, and, perhaps, 
of still greater use to students of English 
history, ecclesiastical, political, social, literary 



and academical, who have hitherto had to be 
content with Dyer. " Academy. 

" It may be thought that the history of a 
college cannot beparticularlyattractive. The 
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thing more than a mere special interest for 
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HISTORY OF NEPAL, 

translated by MUNSHI SHEW SHUNKER SINGH and PANDIT SHRI 
GUNANAND ; edited with an Introductory Sketch of the Country and 
People by Dr D. WRIGHT, late Residency Surgeon at Kathmandu, 
and with facsimiles of native drawings, and portraits of Sir JUNG 
BAHADUR, the KING OF NEPAL, &c. Super-royal 8vo. Price 2is. 



" The Cambridge University Press have 
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torian but also to the ethnologist; Dr 

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graphic plates are interesting." Nature. 

"The history has appeared at a very op 
portune moment... The volume... is beautifully 
printed, and supplied with portraits of Sir 
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lent coloured sketches illustrating Nepaulese 
architecture and religion." Examiner. 



"Von nicht geringem Werthe dagegen sind 
die Beigaben, welche Wright als Appendix* 
hinter der history folgen lasst, Aufzah- 
lungen namlich der in Nepal iiblichen Musik- 
Instrumente, Ackergerathe, Miinzen, Ge- 
wichte, Zeittheilung, sodann ein kurzes 
Vocabular in Parbatiya und Newari, einige 
Newari songs mit Interlinear-Uebersetzung, 
eine Konigsliste, und, last not least, ein 
Verzeichniss der von ihm mitgebrachten 
Sanskrit-Mss., welche jetzt in der Universi- 
tats-Bibliothek in Cambridge deponirt sind." 
A. WEBER, Literaturzeituug, Jahrgang 
1877, Nr. 26. 



THE ARCHITECTURAL HISTORY OF THE 
UNIVERSITY AND COLLEGES OF CAMBRIDGE, 

By the late Professor WILLIS, M.A. With numerous Maps, Plans, 

and Illustrations. Continued to the present time, and edited 

by JOHN WILLIS CLARK, M.A., formerly Fellow 

of Trinity College, Cambridge. [In the Press. 



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SCHOLAR ACADEMICAE: 

Some Account of the Studies at the English Universities in the 
Eighteenth Century. By CHRISTOPHER WORDSWORTH, M.A., 
Fellow of Peterhouse ; Author of " Social Life at the English 
Universities in the Eighteenth Century." Demy octavo, cloth, 15^. 



"The general object of Mr Wordsworth s 
book is sufficiently apparent from its title. 
He has collected a great quantity of minute 
and curious information about the working 
of Cambridge institutions in the last century, 
with an occasional comparison of the corre 
sponding state of things at Oxford. It is of 
course impossible that a book of this kind 
should be altogether entertaining as litera 
ture. To a great extent it is purely a book 
of reference, and as such it will be of per 
manent value for the historical knowledge of 
English education and learning. "Saturday 
Review. 

" In the work before us, which is strictly what 
it professes to be, an account of university stu 
dies, we obtain authentic information upon the 
course and changes of philosophical thought 
in this country, upon the general estimation 
of letters, upon the relations of doctrine and 
science, upon the range and thoroughness of 
education, and we may add, upon the cat 
like tenacity of life of ancient forms.... The 
particulars Mr Wordsworth gives us in his 
excellent arrangement are most varied, in 



teresting, and instructive. Among the mat 
ters touched upon are Libraries, Lectures, 
the Tripos, the Trivium, the Senate House, 
the Schools, text-books, subjects of study, 
foreign opinions, interior life. We learn 
even of the various University periodicals 
that have had their day. And last, but not 
least, we are given in an appendix a highly 
interesting series of private letters from a 
Cambridge student to John Strype, giving 
a vivid idea of life as an undergraduate and 
afterwards, as the writer became a graduate 
and a fellow." University Magazine. 

"Only those who have engaged in like la 
bours will be able fully to appreciate the 
sustained industry and conscientious accuracy 
discernible in every page. . . . Of the whole 
volume it may be said that it is a genuine 
service rendered to the study of University 
history, and that the habits of thought of any 
writer educated at either seat of learning in 
the la--t century will, in many cases, be far 
better understood after a consideration of the 
materials here collected." Academy. 



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CATALOGUE OF THE HEBREW MANUSCRIPTS 

preserved in the University Library, Cambridge. By Dr S. M. 
SCHiLLER-SziNESSY. Volume I. containing Section I. The Holy 
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A CATALOGUE OF THE MANUSCRIPTS 

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INDEX TO THE CATALOGUE. Demy Odavo. 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 
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Hockington, Cambridgeshire. Demy Oclavo. js. 6d. 

A CHRONOLOGICAL LIST OF THE GRACES, 

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CATALOGUS BIBLIOTHEOE BURCKHARD- 
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THE CAMBRIDGE UNIVERSITY PRESS. 21 



Cambrfoge 35tble for 



GENERAL EDITOR : J. J. S. PEROWNE, D.D., DEAN OF 
PETERBOROUGH. 



THE want of an Annotated Edition of the BIBLE, in handy portions, 
suitable for School use, has long been felt. 

In order to provide Text-books for School and Examination pur 
poses, 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 Peter 
borough, has undertaken the general editorial supervision of the work, 
and will be assisted by a staff of eminent coadjutors. Some of the 
books have already been undertaken by the following gentlemen : 

Rev. A. CARR, M.A., Assistant Master at Wellington College. 

Rev. T. K. CHEYNE, Fellow of Balliol College, Oxford. 

Rev. S. Cox, Nottingham. 

Rev. A. B. DAVIDSON, D.D., Professor of Hebrew, Edinburgh. 

Rev. F. W. FARRAR, D.D., Canon of Westminster. 

Rev. A. E. HUMPHREYS, M.A., Fellow of Trinity College, Cambridge. 

Rev. A. F. KIRKPATRICK, M.A., Fellow of Trinity College. 

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., WardenofSt Augustine s Coll., Canterbury. 

Rev. H. C. G. MOTJLE, M. A., Fellow of Trinity College. 

Rev. W. F. MOULTON, D.D., Head Master of the Leys School, Cambridge. 

Rev. E. H. PEROWNE, D.D., Master of Corpus Christi College, Cam 
bridge, Examining Chaplain to the Bishop of St Asaph. 

The Ven. T. T. PEROWNE, M.A., Archdeacon of Norwich. 

Rev. A. PLUMMER, M.A., Master of University College, Durham. 

Rev. E. H. PLUMPTRE, D.D., Professor of Biblical Exegesis, King s 
College, London. 

Rev. W. SANDAY, M.A., Principal of Bishop Hatjield Hall, Durham. 

Rev. W. SiMCOX, M.A., Rector of Wcyhill, Hants. 

Rev. ROBERTSON SMITH, M.A., Professor of Hebrnv, Aberdeen. 

Rev. A. W. STREANE, M.A., Fellow of Corpus Christi Coll., Cambridge. 

The Ven. H. W. WATKINS, M.A., Archdeacon of Northumberland. 

Rev. G. H. WHITAKER, M.A., Fellow of St John s College, Cambridge. 

Rev. C. WORDSWORTH, M.A., Rector of Glaston, Rutland. 



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22 PUBLICATIONS OF 



THE CAMBRIDGE BIBLE FOR SCHOOLS. Continued. 

Now Ready. Cloth, Extra Fcap. 8vo. 

THE BOOK OF JOSHUA. Edited by Rev. G. F. 
MACLEAR, D.D. With i Maps. is. 6d. 

THE BOOK OF JONAH. By Archdn. PEROWNE. is. 6d. 

THE GOSPEL ACCORDING TO ST MATTHEW. 

Edited by the Rev. A. CARR, M.A. With i Maps. is. 6d. 

THE GOSPEL ACCORDING TO ST MARK. Edited 
by the Rev. G. F. MACLEAR, D.D. (with 2 Maps), is. 6d. 

THE GOSPEL ACCORDING TO ST LUKE. By 

the Rev. F. W. FARRAR, D.D. (With 4 Maps.) 4-r. 6d. 

THE ACTS OF THE APOSTLES. By the Rev. 

Professor LUMBY, D.D. Part I. Chaps. I XIV. With i Maps. 
is. 6d. 

PART II. Preparing. 

THE EPISTLE TO THE ROMANS. By the Rev. 

H. C. G. MOULE, M.A. S.T. 6d. 

THE FIRST EPISTLE TO THE CORINTHIANS. 

By the Rev. Professor LIAS, M.A. With a Map and Plan. is. 

THE SECOND EPISTLE TO THE CORINTHIANS. 

By the Rev. Professor LIAS, M.A. is. 

THE GENERAL EPISTLE OF ST JAMES. By the 
Rev. Professor PLUMPTRE, D.D. is. 6d. 

THE EPISTLES OF ST PETER AND ST JUDE. 
By the Rev. Professor PLUMPTRE, D.D. is. 6d. 



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THE CAMBRIDGE UNIVERSTY PRESS. 23 

THE CAMBRIDGE BIBLE FOR SCHOOLS.-^***/.) 
Preparing. 

THE FIRST BOOK OF SAMUEL. By the Rev. 

A. F. KIRKPATRICK, M.A. {Immediately. 

te. 

THE BOOK OF JEREMIAH. By the Rev. A. W. 
STREANE, M.A. {Nearly ready. 

THE BOOKS OF HAGGAI AND ZECHARIAH. By 

Archdeacon P BROWNE. 

THE BOOK OF ECCLESIASTES. By the Rev. 

Professor PLUMPTRE. 

THE GOSPEL ACCORDING TO ST JOHN. By 
the Rev. \V. SANDAY and the Rev. A. PLUMMER, M.A. 

{Nearly ready. 



In Preparation. 
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. 

THE GOSPEL ACCORDING TO ST MATTHEW. By the 
Rev. A. CARR, M.A. {In the Press. 

The books will be published separately, as in the "Cambridge Bible 
for Schools." 



London : Cambridge Warehouse, \ 7 Paternoster Row. 



24 PUBLICATIONS OF 



THE PITT PRESS SERIES. 



I. GREEK. 

THE ANABASIS OF XENOPHON, BOOK VI. With 

a Map and English Notes by ALFRED PRETOR, M.A., Fellow of 
St Catharine s College, Cambridge ; Editor of Persius and Cicero ad Atticum 
Book I. Price is. bd. 

" 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, and geography are briefly but very effectually elucidated." Tkt 
Examiner. 

BOOKS I. III. IV. & V. By the same Editor. 2s. each. 
BOOK II. By the same Editor. Price 2s. 6d. 

"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. 

"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 

" Had we to introduce a young Greek scholar to Xenophon, we should esteem ourselves for 
tunate in having Pretor s text-book as our chart and guide." Contemporary Review. 

AGESILAUS OF XENOPHON. The Text revised 

with Critical and Explanatory Notes, Introduction, Analysis, and Indices. 
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LUCIANI SOMNIUM CHARON PISCATOR ET DE 

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HISTOIRE DU SIECLE DE LOUIS XIV. PAR 

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