Mathematical and Physical Papers, Volume 1 (Cambridge Library Collection - Mathematics)

Cambridge Library CoLLeCtion Books of enduring scholarly value

Mathematical Sciences From its pre-historic roots in simple counting to the algorithms powering modern desktop computers, from the genius of Archimedes to the genius of Einstein, advances in mathematical understanding and numerical techniques have been directly responsible for creating the modern world as we know it. This series will provide a library of the most influential publications and writers on mathematics in its broadest sense. As such, it will show not only the deep roots from which modern science and technology have grown, but also the astonishing breadth of application of mathematical techniques in the humanities and social sciences, and in everyday life.

Mathematical and Physical Papers vol.1. Sir George Stokes (1819-1903) established the science of hydrodynamics with his law of viscosity describing the velocity of a small sphere through a viscous fluid. He published no books, but was a prolific lecturer and writer of papers for the Royal Society, the British Association for the Advancement of Science, the Victoria Institute and other mathematical and scientific institutions. These collected papers (issued between 1880 and 1905) are therefore the only readily available record of the work of an outstanding and influential mathematician, who was Lucasian Professor of Mathematics in Cambridge for over fifty years, Master of Pembroke College, President of the Royal Society (1885-90), Associate Secretary of the Royal Commission on the University of Cambridge and a Member of Parliament for the University.

Cambridge University Press has long been a pioneer in the reissuing of out-of-print titles from its own backlist, producing digital reprints of books that are still sought after by scholars and students but could not be reprinted economically using traditional technology. The Cambridge Library Collection extends this activity to a wider range of books which are still of importance to researchers and professionals, either for the source material they contain, or as landmarks in the history of their academic discipline. Drawing from the world-renowned collections in the Cambridge University Library, and guided by the advice of experts in each subject area, Cambridge University Press is using state-of-the-art scanning machines in its own Printing House to capture the content of each book selected for inclusion. The files are processed to give a consistently clear, crisp image, and the books finished to the high quality standard for which the Press is recognised around the world. The latest print-on-demand technology ensures that the books will remain available indefinitely, and that orders for single or multiple copies can quickly be supplied. The Cambridge Library Collection will bring back to life books of enduring scholarly value across a wide range of disciplines in the humanities and social sciences and in science and technology.

Mathematical and Physical Papers vol.1. Volume 1 George Gabriel Stokes

C A M B R I D G E U n I V E R SI t y P R E S S Cambridge new york Melbourne Madrid Cape town Singapore São Paolo Delhi Published in the United States of America by Cambridge University Press, new york www.cambridge.org Information on this title: www.cambridge.org/9781108002622 © in this compilation Cambridge University Press 2009 This edition first published 1880 This digitally printed version 2009 ISBn 978-1-108-00262-2 This book reproduces the text of the original edition. The content and language reflect the beliefs, practices and terminology of their time, and have not been updated.

MATHEMATICAL AND

PHYSICAL PAPEES.

SonOon: CAMBRIDGE WAREHOUSE, 17, PATERNOSTER ROW.

: DE1GHTON, BELL, AND CO. %tif}iS: V. A. BROCKHAUS.

MATHEMATICAL AND

PHYSICAL PAPEES

GEORGE GABRIEL STOKES, M.A., D.C.L., LL.D., F.R.S., FELLOW OF PEMBKOKE 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.

Cambrftrge: AT THE UNIVERSITY PRESS. 1880 [The rights of translation and reproduction are reserved.]

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

PKEFACE. 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. However, 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 consequence 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

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 distinguish 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 innovation 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 Metropolitana, though the proposal seems never to have been taken up. Mathematicians 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 usuallyprinted in the middle of the page. The difficulty may be avoided by using, in lieu of the bar between the numerator and denominator, 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 appropriated to mean a colon. The symbol " -=-" is certainly distinctive, but it is inconveniently long, and dy -f- dx for a differential coefficient would hardly be tolerated. Now simple fractions are frequently 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 dy/dx. 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 -=- a), and not (sin mrx) -f- a. This welding action may be arrested when necessary by a stop: thus sin n6. /r" means (sin nO) -i-r" and not sin (n0 -=- r"). 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 xnla or by xnla. I do not think the objection 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 xnl-a, where the " ( " would serve to recall the parentheses in the expression written at length. G. G. STOKES. CAMBRIDGE,

August 16, 1880.

CONTENTS. PAGE

On the Steady Motion of Incompressible Fluids On some cases of F l u i d Motion

.

.

.

.

.

.

.

.

On the Motion of a Piston a n d of t h e Air in a Cj'linder

. .

.

.

. .

. .

.

1

.

17

.

.

69

On the Theories of t h e I n t e r n a l Friction of F l u i d s in Motion, and of t h e Equilibrium a n d Motion of Elastic Solids SECTION I . — E x p l a n a t i o n of t h e Theory of F l u i d Motion proposed. F o r m ation of t h e Differential E q u a t i o n s . Application of these E q u a t i o n s to a few simple eases . . . . . . . . . . SUCTION I I . — Objections to Lagrange's proof of the theorem t h a t if udx+vdy + wdz is a n exact differential at a n y one i n s t a n t it is always so, t h e pressure being supposed equal in all directions. Principles of M. Cauchy's proof. A new proof of the theorem. A physical interpretation of t h e circumstance of t h e above expression being a n exact differential SECTION III.—Application of a m e t h o d analogous to t h a t of Seetion I. to t h e determination of t h e equations of equilibrium and motion of elastic solids . . . . . . . . . . . SECTION IV.—Principles of Poisson's theory of elastic solids, a n d of t h e oblique pressures existing in fluids in motion. Objections to one of his hypotheses. Reflections on t h e constitution, a n d equations of motion of t h e luminiferous ether in v a c u u m . . . . . On the Proof of t h e Proposition t h a t (Mx + Ny)'1

75

78

106

113

116

is an I n t e g r a t i n g Factor of

the Homogeneous Differential E q u a t i o n M+N

dyjdx = 0

.

.

.

130

On the Aberration of L i g h t

134

On Fresnel's Theory of t h e Aberration of Light

141

On a F o r m u l a for determining the Optical Constants of Doubly Refracting Crystals

148

On the Constitution of t h e Luminiferous E t h e r , viewed with reference to the Aberration of Light

153

X

CONTENTS. PAOR

Report on Eecent Researches onHydrodynamics . . . . . . I. G e n e r a l t h e o r e m s c o n n e c t e d w i t h t h e o r d i n a r y e q u a t i o n s of F l u i d Motion I I . T h e o r y of w a v e s , i n c l u d i n g t i d e s . . . . . . . I I I . T h e d i s c h a r g e o f g a s e s t h r o u g h s m a l l orifices I V . T h e o r y of s o u n d V . S i m u l t a n e o u s o s c i l l a t i o n s of fluids a n d s o l i d s . . . . . V I . F o r m a t i o n of t h e e q u a t i o n s of m o t i o n w h e n t h e p r e s s u r e i s n o t s u p posed equal i n all directions S u p p l e m e n t t o a M e m o i r o n s o m e c a s e s of F l u i d M o t i o n . . . . . O n t h e T h e o r y of O s c i l l a t o r y W a v e s O n t h e R e s i s t a n c e of a F l u i d t o t w o O s c i l l a t i n g S p h e r e s . . . . . O n t h e C r i t i c a l V a l u e s of t h e S u m s of P e r i o d i c S e r i e s . . . . .

157 158 161 176 178 179 182 188 197 230 237

S E C T I O N L — M o d e of a s c e r t a i n i n g t h e n a t u r e of t h e d i s c o n t i n u i t y of a f u n c t i o n w h i c h i s e x p a n d e d i n a s e r i e s of s i n e s o r c o s i n e s , a n d of o b t a i n i n g t h e d e v e l o p m e n t s of t h e d e r i v e d f u n c t i o n s . . . . 239 S E C T I O N I I . — M o d e o f a s c e r t a i n i n g t h e n a t u r e of t h e d i s c o n t i n u i t y of the integrals w h i c h are analogous t o t h e series considered i n Section I., a n d of o b t a i n i n g t h e d e v e l o p m e n t s of t h e d e r i v a t i v e s of t h e expanded functions 271 S E C T I O N I I I . — O n t h e d i s c o n t i n u i t y of t h e s u m s o f i n f i n i t e s e r i e s , a n d of t h e v a l u e s of i n t e g r a l s t a k e n b e t w e e n i n f i n i t e l i m i t s . . . . 279 S E C T I O N I V . — E x a m p l e s o f t h e a p p l i c a t i o n of t h e f o r m u l a s p r o v e d i n t h e preceding sections . . . . . . . . . . 286 S u p p l e m e n t t o a p a p e r o n t h e T h e o r y of O s c i l l a t o r y W a v e s . . . . 314 Index 327

ERRATA. P . 1 0 3 , 1. 14, for t h e i r read t h e r e . P . 1 9 3 , 1. 3 , for $>=* r e a d pv=h.

MATHEMATICAL AND PHYSICAL PAPERS.

[From the Transactions of the Cambridge Philosophical Society, Vol. VII. p. 439.]

ON THE STEADY MOTION OF INCOMPKESSIBLE FLUIDS. [Bead April 25, 1842.]

IK 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

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

2

ON THE STEADY 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 parameter of a line of motion. I shall first shew that in the case considered G is absolutely constant, and then that whenever it is, udx + vdy + wdz is an exact differential *. To determine the value of G 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 P

(2),

l

px 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 pv and the velocity will be indefinitely small. Consequently C is more nearly equal to G^ than any assignable quantity : therefore G 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 distance 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 p dx

„

du dx

dv dx

dw dx'

. but

1 dp „ du —~- = X — u^j p dx dx

du v--. dy

du w -~r ; dz

, whence

fdv v[~7 \dx

du\ (dw 7- + w 1 dyj \dx

du\ . 7- = ". dzj

[* See note, page 3.]

ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS.

Similarly,

whence*

dv\ dx) ' •nh fdu dw\ fdv div\ u I -. dx)'* \dz dy)~ ' dv du dw dv du dw dx dy ' dy ~dz' dz dx' fdw \dy

W

dv\

fdu \dy

U

and therefore udx + vdy + wdz is an exact differential. When udx + vdy + wdz is an exact differential, equation (1) may be deduced in another way -f*, 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

and p is given by the equation P The differential equation to a line of motion is 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 dimensions, 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 restricted to the two cases above mentioned.] t See Poisson, Traite de Mecanique.

1—2

4

ON THE STEADY MOTION OF INCOMPEESSIBLE 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 =C will be the equation to the system of lines of motion, C being the parameter. U may have any value which allows d TJ]dy and — d TJ/dx 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

^ l y) +f(x - V^l y). Now it will be easily seen that U must be wholly real for all values of x and y, at least within certain limits. But F(a) may be put under the form F1 (a) + V— 1 F2 (a), where Fx (a) and Fi (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(x + V^l y)+F(x - V^I y) 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,

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 a2 + /32 = 0. Consequently a and /3 -must be, one real, the other imaginary, or both partly real and partly imaginary. Putting then a = a1 + *J— l a 2 , /3 = fi1 + V— 1 /32, introducing the condition that a2 + /32 = 0, and replacing imaginary exponentials by sines and cosines, we find that the most general value of V is of the form U = X^e'l(c0ST-a:-sinY-2'+a). cos n (sin 7 . x + cos 7 . y + 6), where J., n, 7, a and h 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", in F (x) or f(x), and then sum the series by the formula

in a series,

coswi•0 + V - 1 sinM0 = cos"