reducing, when the liquid extends to infinity and B1 = 0, to A B [phi] = ½Ux --, [psi] = - ½Uy² --; (5) B0 B0 so that in the relative motion past the body, as when fixed in the current U parallel to xO, / A \ / B \ [phi]´ = ½Ux ( 1 + -- ), [psi]´ = ½Uy² ( 1 - -- ). (6) \ B0 / \ B0 / Changing the origin from the centre to the focus of a prolate spheroid, then putting b² = pa, [lambda] = [lambda]´a, and proceeding to the limit where a = [oo], we find for a paraboloid of revolution p B p B = ½ -------------, -- = -------------, (7) p + [lambda]´ B0 p + [lambda]´ y² ------------- = p + [lambda]´ - 2x, (8) p + [lambda]´ with [lambda]´ = 0 over the surface of the paraboloid; and then [psi]´ = ½U [y² - p[root](x² + y²) + px]; (9) [psi] = -½Up [[root](x² + y²) - x]; (10) [phi] = -½Up log [ [root](x² + y²) + x]. (11) The relative path of a liquid particle is along a stream line [psi]´ = ½Uc², a constant, (12) p²y² - (y² - c²)² p²y² - (y² - c²)² x = -----------------, [root](x² + y²) = ----------------- (13) 2p(y² - c²) 2p(y² - c²) a C4; while the absolute path of a particle in space will be given by dy r - x y² - c² -- = - ----- = -------, (14) dx y 2py y² - c² = a²e^(-x/p). (15) 46. Between two concentric spheres, with a² + [lambda] = r², a² + [lambda]1 = a1², (1) A = B = C = a³/3r³, a³ a³ a³ a³ -- + 2 --- -- + 2 --- r³ a1³ r³ a1³ [phi] = ½Ux -----------, [psi] = ½Uy² ----------; (2) 1 - a^4/a1² 1 - a³/a1³ and the effective inertia of the liquid in the interspace is A0 + 2A1 a1³ + 2a³ --------- W´ = ½ --------- W´. (3) 2A0 - 2A1 a1³ - a³ When the spheres are not concentric, an expression for the effective inertia can be found by the method of images (W. M. Hicks, _Phil. Trans._, 1880). The image of a source of strength [mu] at S outside a sphere of radius a is a source of strength [mu]a/[f] at H, where OS = [f], OH = a²/f, and a line sink reaching from the image H to the centre O of line strength - [mu]/a; this combination will be found to produce no flow across the surface of the sphere. Taking Ox along OS, the Stokes' function at P for the source S is [mu] cos PSx, and of the source H and line sink OH is [mu](a/[f]) cos PHx and -([mu]/a)(PO - PH); so that / a PO - PH \ [psi] = [mu] (cos PSx + --- cos PHx - ------- ), (4) \ [f] a /