194 Professor Baker, On the stability of rotating liquid ellipsoids 

 while, along th.e series considered, he = — ^f^^hlcfi^^', so that 



IS [(a - 6)2] = 



^22 4_ 2i? , " 

 7^3- + c2 9'12 



8A 



</>22 



We proved however (§ 1) that 



wherein P, Q, R are positive, and |, 77 are arbitrary. This gives 



- {4>22n + </'i2^) = ^ (- icAl + h^-q) + Rri; 

 hence, replacing ^, -q respectively by 2^0/0^ and Ijh^, we infer 



, . 1 vJi T ah 



wherein j — ^— , ^ a + 



hi ' ¥ 



h c ' a + 6 ' 



is necessarily positive. ' 



Thus, as (^22 is negative, the equation 



SA = - 1 c^22 (t? + ^ ^12)"' 8 [(« - W] 



shows that SA has a sign opposite to that of 8 [(a — 6)^]. 



We have thus proved that as the axes 2 -\/a, 2 -\/6 become more 

 unequal, the energy H, the moment of inertia ^Mjh, and the angular 

 momentum /x, all constantly increase, while the angular velocity to 

 constantly diminishes. 



In Mr Hargreaves' notation {Camb. Phil. Trans., xxii, 1914, 61), 

 if momentarily m be used for the whole mass, instead of M, 



^ 7.2JL ^ 2JL ^ 7.JL 

 = _ A^c^ii, -— - = - c2</.22, ^—7^ = Ch(pj^2- 



3w/2 ^^^' 3m/2 ^^2. 3^/2 



He obtains L > 2N, M > N. The preceding work estabhshes 



M>2N. 



§ 5. For the series of spheroids of rotational equiHbrium, the 

 variables h, c are not appropriate, for a reason which will appear. 

 Writing a + x, b + x respectively for a, b and A + ^, c + 77 for h, c, 

 we have at once, x, y being small, 



^= -h^(^x + y) + h^{x + yf + etc., 

 \ (x^ -- -2. 



_c(^^y\ , j^" , ^y , y 



\a 

 and hence 



- a| + (/.il + </.2^ + I (</.n^^ + Hi2^r^ + 9^22^') = T^ + Y2 + etc.. 



