180 PROFESSOR A. E. H. LOVE ON THE 



of the systems of equations (25) and (26). It appears on trial that ,, v lt w^ can 

 have the forms 



(28) 



where P M (r) and Q a (r) are certain functions of r. Also it is clear that u a , v a , w a can 

 have the forms 



n \ 3s 2 v 8F M /oftX 



* *- (29) 



Further, the forms of u 3 , v 3 , w 3 are known from the analysis of the problem of the 

 vibrating sphere which is free from gravitation. We have 



I, (30) 



\ G 5 Qtl / CtJu ii T"~ .L O*t/ \i / _ 



where 



T\ ' \ 1.,. Jl.,.l \ /. /' \ * 



X B and (f) n are spherical solid harmonics of degree n, and the expressions for v a , u' 3 are 

 to be obtained from the expression for u 3 by cyclical interchanges of the letters x, y, z. 

 It appears that to a single term f n o> n in the expression for A there corresponds a 

 definite term F n in the expression for E. Further, when we form the boundary 

 conditions, it appears that the terms of u a , v a , w a which contain % represent a free 

 vibration, and the frequency of this vibration is determined by the same equation as 

 if the sphere were free from gravitation. Also it appears that to any term f n <a n in 

 the expression for A there corresponds a definite function < in the expression for 

 Us, v 3 , w s . The solution expressed by a single term /& of A and the corresponding 

 terms of (MI, v,, w^, (u a , v a , w a ) and (u 3 , t* 3 , 3 ) determines a normal mode of vibration. 

 We shall therefore omit x>., and reduce all the summations to typical terms. 

 8. If in the equation 



(V a + h") A + G 3 A + V?- = . ... (19 bis) 



or 



we put/ B (r) . to, for A, we find that^,(r) must satisfy the equation 



= 



dr r dr \ dr 

 or 



}/ n = ..... (32) 



