Expressed as a differential relation, with the value of U from (11), _ _ d | d[psi] | d[alpha] --------- | [alpha][psi] + 2(a² + [lambda])[alpha]--------- | - [psi]--------- = 0, (14) d[lambda] |_ d[lambda] _| d[lambda] d[psi] d / d[psi] \ 3[alpha]--------- + 2(a² + [lambda])-------- ( [alpha]-------- ) = 0, (15) d[lambda] d[lambda] \ d[lambda]/ and integrating d[psi] (a² + [lambda])^3/2 [alpha]-------- = a constant, (16) d[lambda] so that we may put _ / Md[lambda] [psi] = | ----------------, (17) _/ (a² + [lambda])P P² = 4(a² + [lambda])(b² + [lambda])(c² + [lambda]), (18) where M denotes a constant; so that [psi] is an elliptic integral of the second kind. The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which d[psi] 2(a² + [lambda])--------- + [psi] = 0, (19) d[lambda] and this is the infinite boundary ellipsoid if we make the upper limit [lambda]1 = [oo]. The velocity of the ellipsoid defined by [lambda] = 0 is then d[psi]0 U = -2a²--------- - [psi]0 d[lambda] _ M / [oo] Md[lambda] = --- - | ---------------- abc _/ 0 (a² + [lambda])P M = --- (1 - A0), (20) abc with the notation _ / [oo] abc d[lambda] A or A_[lambda] = | ---------------- _/[lambda] (a² + [lambda])P _ d / [oo] d[lambda] = -2abc--- | ---------, (21) da²_/ [lambda] P so that in (4) M UxA xA_[lambda] [phi] = ---xA = ------, [phi]1 = -----------, (22) abc 1 - A0 1 - A0 in (1) for an ellipsoid. The impulse required to set up the motion in liquid of density [rho] is the resultant of an impulsive pressure [rho][phi] over the surface S of the ellipsoid, and is therefore _ _ _ _ / / / / | | [rho][phi]l dS = [rho][psi]0 | | xl dS _/_/ _/_/ = [rho][psi]0 (volume of the ellipsoid) = [psi]0 W´, (23) where W´ denotes the weight of liquid displaced. Denoting the effective inertia of the liquid parallel to Ox by [alpha]W´. the momentum [alpha]W´U = [psi]0W´ (24) [psi]0 A0 [alpha] = ------ = ------; (25) U 1 - A0