MR. W. M. HICKS OK THE MOTION OF TWO SPHERES IN A FLUID. 457 
curves and the sections by planes perpendicular to the axis being small circles of 
constant radii, and he arrives at the result that their action on one another may be 
represented by supposing electric currents to flow round them; and I have recently 
solved the problem of the motion of two cylinders in any manner with their axes 
always parallel. The velocity potentials for the motion of the two cylinders are 
found in general as definite integrals, which, when the cylinders move as a rigid 
body, are expressed in a simple finite form as elliptic functions of bipolar coordinates. 
The functions involved in the coefficients of the velocities in the expression for the 
energy have a close analogy with those for spheres arrived at in the following 
investigation. 
1. Our first aim will be to find the velocity potential for the motion of the fluid in 
which a sphere is fixed and in which a source of fluid exists. By the image of the 
source in general is meant that collocation of sources or sinks within the sphere which 
produces outside of it a fluid motion which in conjunction with the original source has 
no normal motion across the sphere : in other words, that “ mass ” of positive or 
negative sources which produces across the surface of the sphere a normal flow equal 
and opposite to that of the outside source. When this “ image ” is found, the way is 
theoretically clear to finding the velocity potential when two spheres are fixed in the 
fluid, and thence, by distributing over the surface of the spheres sources proportional 
to the normal motion of the surface at that point, to determine the velocity potential 
when the two spheres are moving in any manner. In the case of an electrical point 
the image is, as is well known, a negative point at the inverse point of the other. In 
the case of fluid motion the image is, as will be shown, a positive source at the inverse 
point, together with a negative line sink stretching from this point to the centre of 
the sphere. 
Fig. 1. 
2. Take O the centre of the sphere for origin and let the axis of z pass through the 
source S. Let the radius of the sphere be a, and the distance of S from the centre 
be b. Then the velocity potential will clearly be symmetrical about 0 S. The 
velocity potential for the unit source at S can be expanded in the series 
_ 1 _1_1 ^oo r n 
R v /-r 2 -26r co*W+¥~ b ^ l n+l n 
