1892.] Maclam^ins Liquid Spheroid. 35 



Substituting from (19), (21) and (22) in (15), the condition of 

 stability is that 



\ p^C% (p,q, - p,^q„) 2^ + ^3 (1-5 pcf (1 + 7^)'^ A' 



(23), 



should be positive. 



Comparing this result with that given by Mr Bryan*, which 

 is Poincare's so-called condition of secular stability, and having 

 regard to the difference of notation, we see that the condition of 

 stability (except for the simultaneous values of 7W = 0, n = 2) is 

 that 



x-i^i n + ml -^" ^" 



which agrees. 



12. We shall now discuss this result in the special case in 

 which m = 0, so that the displacement is symmetrical with respect 

 to the axis of revolution, in which case provided n is not equal to 

 2, the condition becomes 



Pi<li-Pn(ln>^ (25). 



If e be the excentricity, e = (1 + 7^)"% and consequently when 

 6 = 0, 7 = 00 , and when e = l, 7 = 0. Now Poincare and Mr Bryan 

 have both shown that (24) is essentially positive provided n — m 

 is odd, and consequently the spheroid is stable when m = 0, and 

 n is odd, that is for displacements represented by harmonics of 

 odd degree. But, when m — and n is even, the first term of (25) 

 vanishes when 7 = 0, whilst the second term remains finite, and 

 consequently the spheroid becomes unstable for such a displace- 

 ment, provided its excentricity is sufficiently great. The values 

 of the excentricity for which instability commences are given by 

 the equation 



'Pi<ii-P2n%n = ^ (26), 



where w = 2, 3 



From these results we conclude that Maclaurin's spheroid is 

 unstable for a zonal displacement of any even order which is 

 greater than 2, provided the excentricity is sufficiently great ; and 

 the inference to be drawn from these results is that the least 

 value of e which makes the left-hand side of (26) vanish and 

 change sign, is a spheroid of bifurcation, which forms the limit 

 between the spheroidal system and another system of surfaces of 



* Proc. Boy. Soc, vol. xlvii. p. 369, equation (7). 



3—2 



