464 MR. W. M. HICKS OX THE MOTION OF TWO SPHERES IN A FLUID. 
Put, and choose x so as to make the constant term vanish. To find x 
we have 
o 4- c~ — b~ 0 
.x-- x-j- a~ = 0 
Fig. 4. 
Now let C, C 2 he the inverse points of the spheres, and O the middle point of 
Cj Co. Put C] Co=2X. Then 
OA=\/k’+a 2 = f ^——= r \ say.(2) 
X=\/r^—a~ .(3) 
c=3* 1 +r 2 
Further, P being any point on the sphere A, denote the constant ratio 
and let q . 2 be the similar constant for the sphere B. Then 
ar 
CfP 
by !\ 
_A+r 1 —a A + q_ a 
^ 1 A — ?\ + a a r-, — A | 
F.( 4 ) 
A—5’ 2 + b _ b _— A | 
■ A T r 3 — b A +b 
The equation to determine x now becomes 
x 2 — 2 r l x-\- a 2 = 0 
The roots of which are r 1= p\. 
Choosing the positive sign, the equation of differences becomes 
U„Un-l — [X,— ~ \X l — = 0 
Now a?=x l x o whence writing 7 for u n we get 
<r -:( c A) 
•hO— a) 
r, 
•‘]( c *2) 
