192 Professor Baker, On the stability of rotating liquid ellipsoids 



§ 2. Now consider a mass of homogeneous (incompressible) 

 fluid, of mass M, whose surface has the equation 



3,2 y2 2,^ 



— + ^ + -=1, 

 a c 



rotating with angular velocity co about the axis of z. Let /, /x, — W 

 be respectively the moment of inertia, moment of momentum and 

 potential energy of gravitation and 



we have 



MM 3 



I = '^-{a + b), fi^-'^-{a + b)co, W = ^M^, 



and if we put 



7. 10 „ 25 2 , 1 



3M2' ' " 3M3^' a + b' 



we have K = ah — cJ3. 



We are concerned to make H a minimum when n. and M, and there- 

 fore also p, = abc, are retained constant, and hence to make K 

 a minimum regarded as a function of h and c, retaining a as con- 

 stant. 



The necessary conditions of rotational equilibrium 



dh ' dc ^' 

 give </>! = c, ^2 = 0, 



and the condition of stability is that 



^ui^ + ^4>i2iv + 9^22''?^ 



should be negative for all real values oi ^, tj. 

 We have 



, 1 f^xdx/ 1 p 



^^ = -2], ^(" + ^-3 



and, in the famihar way, this vanishes for a proper value of jJ, 

 which is such that 



As this involves also p > ^c^/h, or, in a previous notation 

 2t > 1, we see from § 1 that i/ie condition of stability is satisfied for 

 the ellipsoidal forms of rotational equilibrium when ellipsoidal dis- 

 placements are considered. 



