Two equal spheres, intersecting at 120°, will require _ _ | x a³ a^4(a - 2x) a³ a^4(a + 2x) | [psi]´ = ½Uy² | --- - ---- + ----------- + ---- - ----------- |, (11) |_ a 2r1³ 2r1^5 2r2³ 2r2^5 _| with a similar expression for cylinders; so that the plane x = 0 may be introduced as a boundary, cutting the surface at 60°. The motion of these cylinders across the line of centres is the equivalent of a line doublet along each axis. 47. The extension of Green's solution to a rotation of the ellipsoid was made by A. Clebsch, by taking a velocity function [phi] = xy[chi] (1) for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface [lambda] may be in rotation about Oz without disturbing the motion if / 1 1 \ dx ( ------------ + ------------ ) [chi] + 2--------- \ a² + [lambda] b² + [lambda] / d[lambda] R = - -----------------------------------------------------, (2) 1/(b² + [lambda] - 1/(a² = [lambda]) and that the continuity of the liquid is secured if d[chi] (a² + [lambda])^3/2 (b² + [lambda])^3/2 (c² + [lambda]) ½--------- = constant, (3) d[lambda] _ / [oo] Nd[lambda] N B_[lambda] - A_[lambda] [chi] = | ------------------------------- = --- . -----------------------; (4) _/[lambda] (a² + [lambda])(b² + [lambda])P abc a² - b² and at the surface [lambda] = 0, / 1 1\ N B0 - A0 N 1 ( -- + -- ) --- ------- - --- ---- \a² b²/ abc a² - b² abc a²b² R = - ----------------------------------, (5) 1/b² - 1/a² N 1/b² - 1/a² --- = R --------------------------, (6) abc 1 / 1 1\ B0 - A0 ---- - ( -- + -- ) ------- a²b² \a² b²/ a² - b² (a² - b²)²/(a² + b²) = R -------------------------------. (a² - b²)/(a² + b²) - (B0 - A0) The velocity function of the liquid inside the ellipsoid [lambda] = 0 due to the same angular velocity will be [phi]1 = Rxy(a² - b²)/(a² + b²), (7) and on the surface outside N B0 - A0 [phi]0 = xy[chi]0 = xy--- -------, (8) abc a² - b² so that the ratio of the exterior and interior value of [phi] at the surface is [phi]0 B0 - A0 ------ = -------------------------------, (9) [phi]1 (a² - b²)/(a² + b²) - (B0 - A0) and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid [lambda] = 0.