190 Professor Baker, On the stability of rotating liquid ellipsoids 



On the stability of rotating liquid ellipsoids. By Professor 

 H. F. Baker. 



[Read 9 February 1920.] 



The present note deals only with the paedagogic problem of 

 reducing the algebraic treatment of the stabiUty of rotating elUp- 

 soids and spheroids, for elhpsoidal or spheroidal displacements 

 only, to the simplest possible terms. It is a modification of what 

 is given on pp. 69-71 of Mr Hargreaves paper on the Domains of 

 steady motion for a liquid elhpsoid, and the oscillations of the 

 Jacobian figure, Camb. Phil. Trans., xxii, 1914. Mr Hargreaves 

 refers to the paper of C. 0. Meyer, Crelle, xxiv, 1842, for one of 

 the identities he uses, but does not do himself justice in that he 

 refrains from pointing out that Meyer's work is vitiated in an 

 important point by a mistake of algebra. 



§ 1. Supposing a, b, c to be real positive quantities in descending 

 order of magnitude, put 



fix) ^{x + a){x + b) {X + c), fix) = ^^ y' =f{x); 



then we can verify without difficulty that 



^^3r ;ry>)^^ p^^ ^- x^i?>f{x)-xf'{x)] ^^ 



'o y^ 4J0 y'' ' Jo y^ Jo y^ 



Next put, y being throughout taken positive, 

 dx , 1 



cb = i — , a + 6=y, ahc = p, 



^ Jo y h 



so that 



/ (x) = {x + c) (x2 + I + I j = x3 + x2 (c + ^) + X (I + I) + ;, ; 



in this regard ^3 as a constant, so that /(a;) is regarded as depending 

 on the two variable quantities h and c; so considered, denote it 

 by P, and write 



r, _dP p„8P p_9'? , 



dh' ^ ac ' " dh 



and correspondingly write 





