222 PROFESSOR A. E. H. LOVE ON THE 



while the body is rotating, as if rigid, about the axis of z with angular velocity a> ; 

 and we shall seek a strained state in which the body could exist without the 

 application of any external forces, this state being such that, in the absence of 

 rotation, the distribution of density would be hemispherical. In the notation of 47 

 the equations of steady motion of the body are 



av ap av ap av ap M 



- P <J X = P ---, - p(a y = p--, "P^-fc. . (HO) 



The initial state is determined by the same equations with p , V , > substituted for 

 p, V, P. Now the initial figure is an oblate spheroid, and the initial form of V is 



V = const. - {A' 

 where A' and C' are constants ; also the initial form of P is 



p = const. +/> |V 



= const, -ifl, {(A'-o, 2 



When we write, as in 47, 



and neglect terms which cancel on account of the values of p and V , and 

 also neglect terms which are of the second order in the small quantity the 

 equations (110) become 



(111) 



3x 82: ' 

 8W_ X 8| 



8W_ X 



Now we have the equations 



2A' + C' = 4 Try/ao, V 2 W = 4 777/3,, 

 and therefore we can eliminate W and obtain the equation 



where s" is written for f 7ryp 2 /A. If w were zero, A' and C' would both be equal to 

 sTryp,,, and we therefore put 



A", C' = 



