Professor Baker, On the stability of rotating liquid ellipsoids 191 

 Then if f , rj be arbitrary we have 



■^ J y 



substituting the forms of P^^, etc., and utilising the two identities 

 remarked at starting which here take the forms 



°^ xdx _ 3 f^ 

 y^ 

 xHx /'^ „ 



'o y^ .'0 



we have 



cc^ 



3x2 + 2a;(c+n+| + | 



7^5 ' 



'^^("+i) + 2Kf + I) + ''' 



dx 



where 



in which, if we put 



t 



Q=Ux^ + 2Vx + W, 



ty^^ (IX 



y5 



ph 



, u = ch, f 1 = cf , 7^1 = hr}, 



the values of C/, V, W are given by 

 4/? 4 



^ C^C2 = (f,2 _ ^^^^ + ^^2) + 1|^2 (10.^ + 1) + ^^2,, (3t-U~ 1), 

 ^Vc = i (^1^ - ^,rj, + 77^2) + i|^2 (3,^ + i + 4) + r^i^l, (2f _ 1), 



~W= {3t - 1) ii,^ - liT^i + 7^,2) + ^^2 (2f + 1). 



Hence as c, ^, ^ are positive, if ^, 17 be any real quantities, each 

 of U, V, W is necessarily positive provided 2^ > 1. For this, being 



2 11 



->~+^, orc< 2ab/{a + b), 

 cab 



involves, because of 4:d^b^/{a + b)^ < ab, also c^ < ab, and hence 



t> u, 3t-u-l = t-u + 2t-l>0. 



Thus for all real values of |, rj, the quadratic form in ^, 17 

 denoted by Q is necessarily positive, and the form 



4>ni' + ^Kiv + ^22V' 

 necessarily negative, the sign of y being taken positive as stated. 



