486 ME. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
(A r B)(A r E)+ B (A 2 —B) 3 —& 2 A 2 
* (A a —B) 2 —ft 2 A ^(Aj-B) 2 -^” 
This enables us to find within what limits k must lie, for £ must have real roots, 
and therefore 
or 
{(A 1 -B)(A 2 -B)+FB} 2 -{(A 1 -B) 3 -FA 1 }{(A 3 -B) 3 -FA 2 }>0 
k~ (A x A 2 — B~) (p—k 2 ) > 0 
Hence k~ may be any positive quantity less than p. The greatest possible value of 
this is when the spheres are infinitely distant, and then 
p = rn Y + m 2 +^(m / 1 +m' 2 ) 
To each value of ^ will correspond two states of motion, the initial velocities in each 
case being opposite. For example, if ^ is positive, i.e., both velocities in the same 
direction, the two states will be when (a) is the foremost, and when (b) is the foremost; 
if £ be negative, the two states will be, one in which the balls begin to move towards 
each other, the other in which they begin to move from each other. Thus for every 
given value of k there are four possible states of motion. 
If ever tq=0 then £=0, and the spheres must be at such a distance that 
(A 2 -B) 2 -FA 2 =0 
Now, supposing k given, this can only happen if k~ lies between the greatest and 
least values of — —■ — -. The least value is when the spheres are in contact, the 
greatest when they are at an infinite distance, the value then being m. 2 + bn 
If %=0, then k- must lie between the greatest and least values of 
( A,—B ) 2 
a 2 
Now 
(A 2 -B)N(A 1 -B ) 2 
A 2 < A, 
as 
(A 1 A 2 -B 2 )(A 2 »A 1 )>0 
as 
a 2 >a, 
If we suppose a > b then A, > A 2 , and calling kp, k.p the least values of the above 
limits k } < k. 
Hence if 
