460 MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
We have to find the limit when b= co 
y 
W 
k of 
** 1 Vr"-Ihr cos e + V v 7 r 3 + 2&r cos $ + b~ 
+2 ( 1 1 'l 
5 \ \/V 3 — 2Ar cos 0-+■ A 3 + 2Ar cos 0 + A~/ 
1 ^ A —r cos 0 + \/r~ — 2Ar cos 0 + A 2 1 + cos 6 1 
ci ^ A T - 'F cos 6 T" -h 2 A?’ cos 0 -P A~ 1 — cos 0 j 
When b is large and X small this is easily shown to be 
d>= —f; i 2 r cos d+ 
L 
i 3 cos 0 . A 
Hence the limit is 
cf) — — h (2s’-(- .j 
If the velocity of the fluid at an 
origin, then 
infinite distance parallel to x is u towards the 
2k=u 
Also impressing* on the whole system a velocity u, the sphere moves with velocity u in 
an infinite fluid, and the potential function is 
a?ux _ ahi cos 0 
W r= ~ 2r 3 
The well-known form of <f> in this case. 
4. If now two spheres A, B are present in the fluid, and we consider the series of 
images resulting from the first image in A, we see that they very rapidly become 
extremely complicated, e.g., the first image is a source and line sink; the image of 
this in B consists of (1) a source and line sink, (2) the image of the first line sink or a 
line sink (segment of a circle), and an area source bounded by this last line sink and 
two straight lines from the centre. It is, therefore, hopeless in this way to find first 
the velocity potential for a source in the presence of the two spheres, and thence the 
potential for any motion of the spheres. But now suppose A fixed and B moving in 
any direction. If A were not present the velocity potential of B would be that due 
to a doublet at its centre, whose axis lies in the direction of the motion of B. The 
effect of the introduction of A will be to produce a series of images of this doublet, 
lying inside A and B. This method dispenses with the necessity of integrating over 
the spheres when we have found the velocity potential for the doublet. In the special 
