d y²
m-- cos PSx = -m---; (5)
df PS³
and for its image at H the Stokes' function is
d a³ y²
m-- cos PHx = -m-- ---; (6)
df f³ PH³
so that for the combination
/a³ 1 1 \ y² / a³ f³\
[psi] = my² ( -- --- - --- ) = m-- ( --- - --- ), (7)
\f³ PH³ PS³/ f³ \PH³ PS³/
and this vanishes over the surface of the sphere.
There is ao Stokes' function when the axis of the doublet at S does
not pass through O; the image system will consist of an inclined
doublet at H, making an equal angle with OS as the doublet S, and of a
parallel negative line doublet, extending from H to O, of moment
varying as the distance from O.
A distribution of sources and doublets over a moving surface will
enable an expression to be obtained for the velocity function of a
body moving in the presence of a fixed sphere, or inside it.
The method of electrical images will enable the stream function [psi]´
to be inferred from a distribution of doublets, finite in number when
the surface is composed of two spheres intersecting at an angle
[pi]/m, where m is an integer (R. A. Herman, _Quart. Jour. of Math._
xxii.).
Thus for m = 2, the spheres are orthogonal, and it can be verified
that
/ a1³ a2³ a³ \
[psi]´ = ½Uy² ( 1 - --- - --- + -- ), (8)
\ r1³ r2³ r³ /
where a1, a2, a = a1a2/[root](a1² + a2²) is the radius of the spheres
and their circle of intersection, and r1, r2, r the distances of a
point from their centres.