so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; and 4c²(c² - a²) / c² - a² \² [alpha] = ------------[Omega]2² - ( -------[Omega]2 - [eta] ) (c² + a²)² \ c² + a² / 4b²(a² - b²) / a² - b² \² - ------------[Omega]3² - ( -------[Omega]3 - [zeta] ), (15) (a² + b²)² \ a² + b² / with similar equations for [beta] and [gamma]. If we can make (4[pi][rho]A + [alpha])x² = (4[pi][rho]B + [beta])b² = (4[pi][rho]C + [gamma])c², (16) the surfaces of equal pressure are similar to the external case, which can then be removed without affecting the motion, provided [alpha], [beta], [gamma] remain constant. This is so when the axis of revolution is a principal axis, say Oz; when [Omega]1 = 0, [Omega]2 = 0, [xi] = 0, [eta] = 0. (17) If [Omega]3 = 0 or [theta]3 = [zeta] in addition, we obtain the solution of Jacobi's ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting a = b, Maclaurin's solution is obtained of the rotating spheroid. In the general motion again of the liquid filling a case, when a = b, [Omega]3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to d[xi] 2c² d[eta] 2a² ----- = - -------[Omega]2 [zeta], ------ = -------[Omega]1 [zeta], dt a² + c² dt a² + c² d[zeta] 2c² ------- = -------([Omega]2 [xi] - [Omega]2 [eta]) (18) dt a² + c² d[Omega]1 a² + c² --------- = [Omega]2 [zeta] + -------[eta][zeta], dt a² - c² d[Omega]2 a² + c² --------- = [Omega]1 [zeta] + -------[xi][zeta]; (19) dt a² - c² of which three integrals are a² [xi]² + [eta]² = L - --[zeta]², (20) c² (a² + c²)² [Omega]1² + [Omega]2² = M + ------------ [zeta]², (21) 2c²(a² - c²) a² + c² [Omega]1 [xi] + [Omega]2 [eta]N = + ------- [zeta]²; (22) 4c² and then / d[zeta]\² 4c^4 ( ------- ) = --------- ([Omega]2[xi] - [Omega]1²[eta])² \ dt / (a² + c²)