318-320] ORDINARY AND SECULAR STABILITY. 595 



oscillation in the two types (when e < '9529) have been calculated 

 by Love*. 



The theory of the stability and the small oscillations of 

 Maclaurin's ellipsoid, when the disturbance is unrestricted, has 

 been very fully worked out by Bryanf, by a method due to 

 Poincare'. It appears that when e<'9529 the equilibrium is 

 thoroughly stable. For sufficiently great values of e there is 

 of course instability for other types, in addition to the one above 

 referred to. 



320. In the investigations here cited dissipative forces are 

 ignored, and the results leave undetermined the more important 

 question of 'secular' stability. This is discussed, with great 

 command of mathematical resources, by Poincare. 



If we consider, for a moment, the case of a fluid covering a 

 rigid nucleus, and subject to dissipative forces affecting all relative 

 motions, there are two forms of the problem. It was shewn in 

 Art. 197 that if the nucleus be constrained to rotate with constant 

 angular velocity (n) about a fixed axis, or (what comes to the same 

 thing) if it be of preponderant inertia, the condition of secular 

 stability is that the equilibrium value of V T should be station- 

 ary, V denoting the potential energy, and T the kinetic energy of 

 the system when rotating as a whole, with the prescribed angular 

 velocity, in any given configuration. If, on the other hand, the 

 nucleus be free, the case comes under the general theory of 

 'gyrostatic' systems, the ignored coordinates being the six co- 

 ordinates which determine the position of the nucleus in space. 

 The condition then is (Art. 235) that the equilibrium value of 

 V+K should be a minimum, where K is the kinetic energy of 

 the system moving, as rigid, in any given configuration, with the 



* " On the Oscillations of a Rotating Liquid Spheroid, and the Genesis of the 

 Moon," Phil Mag., March, 1889. 



t " The Waves on a Rotating Liquid Spheroid of Finite Ellipticity," Phil. Trans., 

 1889 ; " On the Stability of a Rotating Spheroid of Perfect Liquid," Proc. Roy. Soc., 

 March 27, 1890. The case of a rotating elliptic cylinder has been discussed by 

 Love, Quart. Journ. Math., t. xxiii. (1888). 



The stability of a rotating liquid annulus, of relatively small cross-section, 

 has been examined by Dyson, I.e. ante p. 166. The equilibrium is shewn to 

 be unstable for disturbances of a "beaded" character (in which there is a periodic 

 variation of the cross-section as we travel along the ring) whose wave-length exceeds 

 a certain limit. 



