MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 475 
If x is of the form 
2m+1 
2 5 
Q=(2m+ir{s' 3 -v(^l) 3 } 
= (2m+1) 3 1 -0517998 - 1 — 
Also a finite expression, and in this case 
In the particular cases 
2m —1, 
--to 
2m +1 
a=%b Q= -39859 
a=jb Q=-84535 
The expressions for Q directly in terms of r, a, b are the same functions of 
a?-\-b 2 —r 3 as the corresponding expressions for external spheres are of r 3 — d 2 —W. 
13. The series for Q' may, as in the case of Q, be shown to be convergent. 
When the spheres are in contact 
Also the general term in r, a, b is given by 
( 2 ab) n 
2 n ~ l a n b n 
u„=- 
2 rt 
2p-\-l n—2 p—1 
x 
■ (15) 
It is easily seen that both Q, Q' for external spheres diminish as x — i.e., r — 
increases. 
Hence for external spheres —7 are both negative. 
When one sphere is inside the other, Q decreases as x increases— i.e., as r diminishes. 
Hence in this case ^ is positive. 
The values of the first three terms of Q, Q' are 
for Q 
and for Q' 
ah 
aW 
r~—b~\ 5 [(r 3 -J 2 )2-«VJ ’ [x 6 + « 2 x 4 -2aW 
a?W 
a &\ 3 / a 3 Z > 3 \ 3 
r ] ’ \ ra 2 ] L r(ad —a 2 5 2 ) 
aW 
y 
■ ■ (16) 
