1902.] Figure of Equilibrium of a Rotating Mass of Liquid. 179 



treated as a statical problem, if the mass be subjected to a rotation 

 potential. The energy lost in the concentration of such a system from 

 a condition of infinite dispersion consists of two parts. The first of 

 these, say W, is the lost energy of the system at rest ; the second is 

 equal to the kinetic energy, say T, of the system in motion. The 

 whole lost energy, say E, is equal to W + T, and the condition for a 

 figure of equilibrium is that E shall be stationary for all variations, 

 subject to constant angular velocity. 



It might appear at first sight that the condition for secular stability 

 is that E shall be a maximum. But M. Poincare has shown that this 

 condition is insufficient, and that it is necessary for stability that the 

 whole energy, say U, which is equal to - W + T, shall be a minimum 

 for all variations, subject to the condition of constancy of angular 

 momentum. 



He has, however, adduced another consideration, which enables us to 

 determine the stability from the variations of E, without a direct con- 

 sideration of the function U. He has shown, in fact, that if for given 

 angular momentum slightly less than that of the critical Jacobian 

 ellipsoid, from which the pear-shaped figures bifurcate, there is only 

 one possible figure, namely, the Jacobian; and if for slightly greater 

 angular momentum there are two figures, namely, the Jacobian and 

 the pear,* then exchange of stability between the two series must 

 occur at the bifurcation. If, on the other hand, the smaller momentum 

 corresponds with the two figures and the larger with only one, one of 

 the two (namely, the Jacobian) must be stable, and the other (namely,, 

 the pear) unstable. 



The question is then completely answered by the value of the 

 momentum of the pear; if it is greater than that of the critical 

 Jacobian, the pear is stable, and if less, unstable. It suffices then to 

 determine the pear from the variations of E with constant angular 

 velocity, and afterwards to evaluate the angular momentum. 



In the first approximation the pear-shaped figure is represented by 

 the third zonal harmonic inequality with reference to the longest axis 

 of the critical Jacobian ellipsoid. In proceeding to the higher 

 approximation I suppose that its amplitude is measured by a para- 

 meter e, which is to be regarded as a quantity of the first order. We 

 must now also suppose the ellipsoid to be deformed by every other 

 harmonic, but with amplitudes of order e 2 . In the first approximation 

 W was proportional to e 2 , but it now becomes necessary to go as far 

 as the order e 4 . A change in the sign of e means that the figure is 

 rotated in azimuth through 180°. As this rotation cannot affect the 

 energy, the odd powers of e must be absent from the expression for W T 

 We have further to find the moment of inertia, as far as the terms 



* For the sake of simplicity, I speak of one pear instead of two in azimuths- 

 differing by 180°. 



