x² y² z² ------------- + ------------- + ------------- = 1 (3) a² + [lambda] b² + [lambda] c² + [lambda] and a velocity function of the form [phi] = x[psi], (4) where [psi] is a function of [lambda] only, so that [psi] is constant over an ellipsoid; and we seek to determine the motion set up, and the form of [psi] which will satisfy the equation of continuity. Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px py pz l = -------------, m = -------------, n = ------------- (5) a² + [lambda] b² + [lambda] c² + [lambda] p²x² p²y² p²z² 1 = ---------------- + ---------------- + ----------------, (6) (a² + [lambda])² (b² + [lambda])² (c² + [lambda])² p² = (a² + [lambda])l² + (b² + [lambda])m² + (c² + [lambda])n², (7) = a²l² + b²m² + c²n² + [lambda], dp d[lambda] 2p-- = ---------; (8) ds ds Thence d[phi] dx d[psi] ------ = --[psi] + x------ ds ds ds dx d[psi] dp = --[psi] + 2(a² + [lambda])--------- l--, (9) ds d[lambda] ds so that the velocity of the liquid may be resolved into a component -[psi] parallel to Ox, and -2(a² + [lambda])l d[psi]/d[lambda] along the normal of the ellipsoid; and the liquid flows over an ellipsoid along a line of slope with respect to Ox, treated as the vertical. Along the normal itself d[phi] / d[psi] \ ----- = ( [psi] + 2(a² + [lambda])-------- )l, (10) ds \ d[lambda] / so that over the surface of an ellipsoid where [lambda] and [psi] are constant, the normal velocity is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to Ox d[psi] U = -[psi] - 2(a² + [lambda])---------, (11) d[lambda] and so the boundary condition is satisfied; moreover, any ellipsoidal surface [lambda] may be supposed moving as if rigid with the velocity in (11), without disturbing the liquid motion for the moment. The continuity is secured if the liquid between two ellipsoids [lambda] and [lambda]1, moving with the velocity U and U1 of equation (11), is squeezed out or sucked in across the plane x = 0 at a rate equal to the integral flow of the velocity [psi] across the annular area [alpha]1 - [alpha] of the two ellipsoids made by x = 0; or if _ / [lambda]1 d[alpha] [alpha]U - [alpha]1U1 = | [psi]-------- d[lambda], (12) _/ [lambda] d[lambda] [alpha] = [pi][root](b² + [lambda]·c² + [lambda]). (13)