stability of rotating masses of liquid 



201 



wherein, by what has appeared, the last term is the same as 



hc-P 



I 



^h 9, 

 b, f, m 

 f, c, n 



The solution, above omitted, arising by taking x = 0, requires 

 by + fz — k'jn = 0, fy + cz — k'n =0, y' ^ 0, 

 so that, if we introduce a, given hy a = hy + gz ~ k'l, we obtain 



I 



m 

 n 

 and the quadratic form arising in considering the stability is 



ae + ibrj^+M + H' + ^^b'rj'^ 

 Assuming then that the ellipsoids up to the form of bifurcation, 

 that is for k' <0, are stable (as is well known), we can infer that 

 b, be — p, b' and a are all positive, and hence that 



h, g, I 



b, f, m 



f, c, n 

 is positive. 



Hence, returning to the quadratic form above wherein x is not 

 zero, we infer (1) that the pear-shaped figure, so far as the increment 

 hH is appropriately represented by the form above, is stable if k' 

 is positive. By Darwin and Jeans this conclusion is made to depend 

 on general reasoning, due in the first place to (Liapounofi and) 

 Schwarzschild, writing in correction of Poincare {Inaugural Dis- 

 sertation, Miinchen, 1896, or Neue Annalen d. Sternwarte Munchen, 

 III, 1898, 275); and (2) that a necessary and sufficient condition 

 for this is that the determinant 



S {h'^/b'), 

 h, 



9, 



h, 9 

 b, f 



should be positive. It is however easy to verify that this deter- 

 minant is, in the notation of Sir George Darwin, equal to 



-ihc-f^)N+^ r + ^C -^(J>j' 



where 



N = 2 



A. ■ <^^'^ 



(2)^2 



+ 



a 





+ 2 



aw 



