39. As an application of moving axes, consider the motion of liquid filling the ellipsoidal case x² y² z² --- + --- + --- = 1; (1) a² b² c² and first suppose the liquid to be frozen, and the ellipsoid to be rotating about the centre with components of angular velocity [xi], [eta], [zeta]; then u = - y[zeta] + z[eta], v = - z[xi] + x[zeta], w = - x[eta] + y[xi]. (2) Now suppose the liquid to be melted, and additional components of angular velocity [Omega]1, [Omega]2, [Omega]3 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function b² - c² c² - a² a² - b² [phi] = - [Omega]1 ------- yz - [Omega]2 -------zx - [Omega]3 -------xy, (3) b² + c² c² + a² a² + b² as may be verified by considering one term at a time. If u´, v´, w´ denote the components of the velocity of the liquid relative to the axes, 2a² 2a² u´ = u + yR - zQ = ------- [Omega]3 y - ------- [Omega]2 z, (4) a² + b² c² + a² 2b² 2b² v´ = v + zP - xR = ------- [Omega]1 z - ------- [Omega]3 x, (5) b² + c² a² + b² 2c² 2c² w´ = w + xQ - yP = ------- [Omega]2 x - ------- [Omega]1 y, (6) c² + a² b² + c² P = [Omega]1 + [xi], Q = [Omega]2 + [eta], R = [Omega]3 + [zeta]. (7) Thus x y z u´ --- + v´ --- + w´ --- = 0, (8) a2 b2 c2 so that a liquid particle remains always on a similar ellipsoid. The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, become 1 dp du du du du ----- -- + 4[pi][rho]Ax + -- - vR + wQ + u´-- + v´-- + w´-- = 0, ... , ... , (9) [rho] dx dt dx dy dz where _ / [oo] abcd[lambda] A, B, C = | ---------------------------------------------- _/ 0 (a² + [lambda], b² + [lambda], c² + [lambda])P P² = 4(a² + [lambda]) (b² + [lambda]) (c² + [lambda]). (10) With the values above of u, v, w, u´, v´, w´, the equations become of the form 1 dp ----- -- + 4[pi][rho]Ax + [alpha]x + hy + gz = 0, (11) [rho] dx 1 dp ----- -- + 4[pi][rho]By + hx + [beta]y + fz = 0, (12) [rho] dy 1 dp ----- -- + 4[pi][rho]Cz + gx + fy + [gamma]z = 0, (13) [rho] dz and integrating p[rho]^(-1) + 2[pi][rho](Ax² + By² + Cz²) + ½([alpha]x² + [beta]y² + [gamma]z² + 2fyz + 2gzx + 2hxy) = const., (14)