36 Mr Basset, On the Stability of [Nov. 28, 



revolution which are probably stable, at any rate if they do not 

 differ very greatly from the spheroidal form. When the angular 

 velocity is sufficiently great, it seems certain that a ring-shaped 

 figure exists ; and consequently if the angular velocity be con- 

 ceived to diminish, the ring would close up until it assumed the 

 form shown in the figure ; and if the angular velocity still further 



diminished, the depressions at the poles would disappear, and the 

 liquid would finally assume a spheroidal form. That there must be 

 a spheroidal surface, which forms the limit between the spheroidal 

 form and a series of surfaces which ultimately become annular, 

 appears to me so probable as almost to amount to a certainty. 



13. We must now consider the case in which ni= 0, and u = 2. 

 Mr Bryan has noticed that in this case the left-hand side of the 

 inequality (25) vanishes and changes sign for a certain value of 

 the excentricity, and he has endeavoured to account for this by 

 stating that* "it does not indicate that the spheroid in question is 

 secularly unstable for this particular type of displacement. Its 

 meaning is that the spheroid is more oblate than that form for 

 which the angular velocity is a maximum." This explanation is 

 incorrect ; for, as we shall now show, the displacement represented 

 by w = 2 is a spheroidal displacement for which the figure is stable, 

 and the error into which he has fallen has doubtless arisen from 

 his not having noticed that Poincare has omitted the last term 

 in (15). 



Since h = /^w, 



^o = fk^PcV(l+y)^ (27), 



it follows from (22) that 



// 307r ' 



also since m^ = 2-npr^ {(1 -\- 87^) cot~^ 7 — 37), 



the condition of stability when ?i = 2 becomes 



7?i (7) - P, (7) ^2 (7) + i {(1 + H) cot-^ 7 - 37} > 0. 

 * Proc. Roy. Soc, vol. xlvii. p. 371. 



