578 Mr. stokes, ON THE CRITICAL VALUES OF 



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 = S22 J„ sin fix sin vy sin stz ; 

 and we shall have 



V« = - 222 (;a" + v' + w')A,„„p sin ijljc sin vy sin ■ars;. 

 But we have also, by a well-known theorem, yu = — iTvp; and 



p = 2225„„;, sin /mx sin vy sin ■jjr«, 



where R„„ = —-- / / p' sin fix' sin u y' sin stx' dot' dy dx', 



o,ocJ„ Jg J^ 



p being the same function of x, y', z that p is of x, y, z. We get therefore by comparing the two 



expansions of y« 



^^y = 47r (m' + 1-' + sr-)-' R,„„^, 



whence the value of v is known. We may now, if we like, suppose the electricity condensed 

 into a point, which gives 



„ Sm . , . , . 



aoc 



V = — ; — 222 Oi^ + v~ + OT^)~' sin fix' sin vy sin ■zsrs' sin/x.r sin i;y sin tstz ('10). 



aoc 



One of the summations may be performed just as before. We thus get, by summing witli 

 respect to p, 



S-^m ^^ 1 (e'- - e-"') je"''""' - e-^"' -'') . , . , . . . , . 



V = — r— 2,2 ■ ■ sin fix sin vy sm fix sin vy (Hi), 



ab q e'l" — e ''° ' 



where 9° = ni' + i-'', and z is supposed to be the smaller of the two z, z . If z be greater than z\ 

 we have only to make x and z change places in (ill). 



53. The equation (llO) shows that the potential at the point (x, y, z) due to a unit of electri- 

 city at the point (»', y , z) and to the electricity induced on the surface of the parallelepiped is equal 

 to the potential at the point {x, 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, 8ec. 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 (h 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 Motion^." 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 



• Essay on Electricity, p. 19. -f See p. UI of the present Volutne. 



