484 MR. W. M. HICKS OK THE MOTION OF TWO SPHERES IN A FLUID. 
_ A 3 — B 
u~~ A^B 
When the spheres are equal ii. : =~u l and the motion is the same as that of a single 
sphere in a fluid bounded by a plane, and moving perpendicularly to the plane. 
For this particular case 
7 
,2 
4T 
A + B 
or if u denote the velocity relative to the fixed plane r=2u, and 
rr== 
A + B 
(A + B) 0 3 
A + B 0 
where (A+B),^ u 0 are the values of A+B, and u at the point of projection. If the 
sphere is projected from contact with the plane 
(A+B) 0 =m+^m + f m (J-S 3 — l+iS 3 ) 
‘3030853 in 
= m+‘803085w' 
and at an infinite distance 
A + B 
Hence the ratio of the limiting velocity to the initial velocity is 
A /{l + -60617°7^_} 
where p is the density of the sphere. 
For densities 0, 1, 10, the values of this ratio are respectively 1*2661, 1‘0963, 
1*0143. The greatest value is when the density of the sphere is zero, and the least 
is when in' — 0 (no fluid) or m = co , the ratio then being, as it ought to be, unity. 
In the case where the spheres are unequal and projected with no momentum from 
contact their initial velocities must be opposite and in the ratio of the quantities 
ma+lm^-fm^flD 3 log, F(1 +y) + S 3 j 
and 
m l + \m \—f {ID 3 log, r( 1 + x) + S 3 } 
x and y denoting the quantities 
b a 
a + b’ ct + b 
