468 ME. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
3 f n f~ n sin 3 6 'cos" 'XflQd'X, 
111 U Jo l« 2 
n j o {« 2 + pr + 2pa cos 0 } ;; 
sin 3 Odd 
= — jXTTiV’V 
r 
J o 
[a~ + p“ + 2 pa cos #} ,J 
The integral of which is 
~ 0 3 [(/> 2 + a3 ){(/ ) + a )"” (P' a )} —pa{p+ci -f-(p-a)}] 
op 
Writing v and cr for /x, p for doublets outside the sphere A, we obtain 
, , 47 rcdvv 
— f7r/aa and — 
whence 
OCT’ 
2 T =- M 1 v 2^ 3 +- 3 
1 I a° cr 3 
Now any v at the distance cr produces an image in A consisting of a doublet v[- 
at a distance together with a line sink stretching from this to the centre, whose 
line magnitude is — — X distance from the centre. Hence the whole amount of the 
co- 
image is 
Now every /x except /x 0 forms part of an image of some v, and of that v only. 
Hence 
and 
2 — = 22 ~ — 2 ^ 7 * 
cr., a° ar 
2T=-^^o+32 / x} 
=iM 1 a 3 |l + 32 
. . (7) 
The 2 extending to the whole mass of images inside A. 
8 . If A has also a motion along B A, together with one perpendicular to it, T has 
no term depending on u, v; for it is clear that if the sign of v is changed, then the 
kinetic energy must be the same as before. 
If B moves also perpendicularly to B A, T will have additional terms in % 3 , and 
r O x , v. 2 . The coefficient of v.d will be analogous to that for v x , whilst that for v v v 2 , as 
