The extension to the case where the liquid is bounded externally by a fixed ellipsoid [lambda] = [lambda]1 is made in a similar manner, by putting [phi] = xy([chi] + M), (10) and the ratio of the effective angular inertia in (9) is changed to a1² - b1² abc (B0 - A0) - (B1 - A1) + --------- ------ a1² + b1² a1b1c1 --------------------------------------------------. (11) a² - b² a1² - b1² abc ------- - --------- ------ - (B0 - A0) + (B1 - A1) a² + b² a1² + b1² a1b1c1 Make c = [oo] for confocal elliptic cylinders; and then _ /[oo] ab ab / /b² + [lambda] \ A[lambda] = | ----------------------------------------------------- = ------- ( 1 - / ------------- ), (12) _/[lambda] (a² + [lambda])[root]([4·a² + [lambda]·b² + [lambda]) a² - b² \ \/ a² + [lambda] / ab / /a² + [lambda] \ B[lambda] = ------- ( / ------------- - 1 ), C[lambda]= 0; a² - b² \ \/ b² + [lambda] / and then as above in § 31, with a = c ch [alpha], b = c sh [alpha], a1 = [root](a² + [lambda]) = c ch [alpha]1, b1 = c sh [alpha]1 (13) the ratio in (11) agrees with § 31 (6). As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and Oy. A torsion of the ellipsoidal surface will give rise to a velocity function of the form [phi] = xyz[Omega], where [Omega] can be expressed by the elliptic integrals A_[lambda], B_[lambda], C_[lambda], in a similar manner, since _ / [oo] [Omega] = L | d[lambda]/P³ _/ [lambda] 48. The determination of the [phi]'s and [chi]'s is a kinematical problem, solved as yet only for a few cases, such as those discussed above. But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R. The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds. Conversely, if the kinetic energy T is expressed as a quadratic function of x1, x2, x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.