and [psi] = -[mu], a constant, over the surface of the sphere, so that there is no flow across. When the source S is inside the sphere and H outside, the line sink must extend from H to infinity in the image system; to realize physically the condition of zero flow across the sphere, an equal sink must be introduced at some other internal point S´. When S and S´ lie on the same radius, taken along Ox, the Stokes' function can be written down; and when S and S´ coalesce a doublet is produced, with a doublet image at H. For a doublet at S, of moment m, the Stokes' function is 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. The corresponding expression for two orthogonal cylinders will be / a1² a2² a² \ [psi]´ = Uy ( 1 - --- - --- + -- ). (9) \ r1² r2² r² / With a2 = [oo], these reduce to / a^5 \ x / a^4 \ x [psi]´ = ½Uy² ( 1 - --- ) ---, or Uy ( 1 - --- ) ---, (10) \ r^5 / a \ r^4 / a for a sphere or cylinder, and a diametral plane.