MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 491 
19. If instead of supposing tlie sphere (b) free to move we suppose it held fast, and 
require to find the force necessary to do so, we get a different result from the fore¬ 
going. Suppose the sphere (a) moving in any manner, the sphere (b) being for the 
moment at rest, and suppose a constant force F acting on b. 
The equation of motion for B is 
Ai - B.r,+K( 2«i ■- i%) : ^(Ai - 2B) = F 
Suppose now that F is of such a magnitude that u 2 being zero it makes x. 2 also zero. 
Then F is the force required to keep (b) at rest at the moment when the motion of (a) 
is given by u x , x v Hence 
F=-B^+K s |(A 1 -2B) 
Let X\ = L sin Kf, L being small. Then neglecting cubes of small quantities 
F= (B+^ drj LK 3 sin K t+\ ~ (A 1 -2B)L s K a cos 5 K t 
and 
dr=x 1 = L sin K t 
.-. F=BLK 2 sin Kt+AJSK 2 jfr+i|;(A 1 -2B)| +\ cos 2K t 
This is the force at the time t necessary to keep (6) at rest. Hence the mean force 
clA 
is a force = towards (a), which is equal and opposite to the force of (a) on 
dA 
(b). Since is negative, the action is an attractive one 
= -iD>K*fs 
_ _ ^2 dA x 
dr 
Taking for A 1 only the first term of Q, which is equivalent to neglecting twelfth and 
higher inverse powers of r 
ab \ 8 ' 
and the force 
= 9to / 1 it, 
Q 
V 
VI 
Ai—-j 1 + 3(^ 2 
a?Wr 
-5 3 
(P 2 -6 2 ) 4 
= 9— ■ ( -- X weight of fluid displaced by 
(25) 
For example, for equal spheres at a distance 4<x (distance between their surfaces 
= 2 a), the mean square of velocity of (a) being the same as for oxygen at a tempe- 
