so that for the combination
| ψ = my2 ( | a3 | 1 | − | 1 | ) = m | y2 | ( | a3 | − | ƒ3 | ), | |
| ƒ3 | PH3 | PS3 | ƒ3 | PH3 | PS3 |
(7)
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 ψ′ to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle π/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
| ψ′ = ½ Uy2 ( 1 − | a13 | − | a23 | + | a3 | ), |
| r13 | r23 | r3 |
(8)