THE SUMS OF PERIODIC SERIES. 575 



By expanding u in the double series SSFsin fia: sin ■zzrar we should get 



M = — 22 — ;t 3-j sin fxv' sin fix sin Tirs; (100), 



ac s e" — e «" 



where s^ = ^u^ + sr^, and ^' is the greater of the two y, y'. The third expansion would be derived 

 from the second by interchanging the requisite quantities. In these formal* a: may have any 

 positive value less than 2 c. 



We should get in a similar manner in the case of two variables ai, y 



" = T 2; —^ — sin „y' sin vy = - ^ -, — ^ sin nx, ... (101), 



b e ~ e a 6^—6 



where .i' 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 . 



If we compare the two expressions for / / / udydy'dx obtained from (101), taking 2o for 



•'o •'o •'o 

 the sign of summation corresponding to odd values of n from 1 to co , putting a = rb, and 



1 , TT^ 



replacing 2o ^ by its value — , we shall get the formula 



"IT 



1 1 1 - e-"" 1 1 _ e~ .;r^ 



-r^»^TTP^^ + '-^V3 ZE = T6^ ('o^)' 



1 + e ' 



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), sup- 

 posing that we integrate for y before integrating for y , we must integrate separately from y = 

 to y = y , and from y = y' io y = h, 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 ])hysical problems to which tliey 

 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 follow- 

 ing problem in fluid motion, u taking the place of the quantity usually denoted by <p, " 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 tlie 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 !< in a series of cosines 

 instead of .sines, and employ the fomula (Z)) 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 perpendi- 

 cular to the first set of faces, and cosines involving the others. 



The formulfc would also be modified by supposing some one or more of the faces to move off 

 to ,111 infinite distance. In this case some of the series woulil be replaced by integrals. Thus, in 

 the case in wliich tlie value of ii at the surface is known, if we supposed a to become mfiiiile we 

 should employ the integral (50) instead of the series (.')), as far as relates to the variable ,/•, and the- 

 formula (fc) instead of {Ii). If we were considering a rectangular I)ar inluiitely extended both 

 ways wc should employ the integral (fiS). Of course, if we had already obtained the result for the 



4 E 2 



