MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 459 
to ( - of the source at S, together with a line sink extending from O to the distance 
the line density of the sink being - X source at S. 
Performing the integration for V, we find finally that the whole velocity potential 
for a unit source at S is 
_ _ — 1 -... 
SP - — 2 hr cos 6,+ b 2 b y/r* — 2A ,r cos 6 + A, 3 
where \= 
b 
+b°g 
\—r cos 0 + \/r° — 2 \r cos 0 + A. 3 
r(I — cos O') 
It is easy to verify this value for by direct differentiation. 
If we apply the same method to find the velocity potential for the motion of fluid 
inside a sphere under the influence of a source inside, the integral becomes infinite 
unless the source is zero. The case is of course physically impossible since if fluid is 
generated within the sphere it must pass through the boundary. But if we also place 
an equal sink at any point within, the motion is then possible, and the expression 
becomes finite. S being the source let S' be its inverse point with reference to the 
sphere, and S" any point on the line S S' produced to infinity. Then the “image” of 
S is a source at S', and a line distribution of sinks of line density - from S' to 
infinity. Let S T be an equal sink, then its image and that of S will produce potentials 
with finite derivatives. In fact, the potential at P will be 
1 a 1 a 1 1 
S^wfs'P - L’shP - a ° g 
OS '—r cos 0 + S'P 1 — cos 0 \I 
OSfi —r cos 6 / + S\P 1 — cos 0 } 
where 0, 9 / are the angles O P makes respectively with O S, O S,. 
3. The expression found for the motion when there is a single source outside the 
sphere enables us to deduce the velocity potential for a single sphere moving through 
an infinite fluid. Taking the direction of motion as the axis of x, from which we will 
suppose 6 measured, we may arrange a surface distribution of sources proportional to 
cos 0d$ and integrate over the surface of the sphere, or we may employ the simpler 
method used in a paper in the ‘ Quarterly Journal of Mathematics’ for March, p. 128. 
The first gives us the velocity potential when the sphere moves by an integration 
which would be laborious. The other gives directly the potential, when the sphere is 
fixed and the fluid moves past it, by means of an easy differentiation. Putting a 
source at x—b and an equal sink at x— — b, let these move off to infinity, increasing 
indefinitely as they do so, yet so that the motion at a finite distance from the origin 
is finite. In the limit we clearly get the case of fluid flowing past the sphere. 
