GRAVITATIONAL STABILITY OF THE EARTH. 191 



17. We may express A. and B, in more convenient forms by the use of the 



identities 



nP.-(n+l)aU = 0.- 



M*-a 

 da ' aj 



_, - - 



a da 



a da 

 We find 



f 1 

 B " = ~ i 



'a J Q',,}. . . . (68) 



We may also express C. and D,, in simpler forms by the use of the equations 

 connecting the t/ functions. We find 



<70) 



The Frequency Equation and the Condition of Gravitational Instability. 



18. Exactly as in the problem of the vibrating sphere which is free from gravitation, 

 it follows from the equations of type (62) that we must have at once 



A..+ 0.^ = 0, and B.w.+D^,, = ...... (71) 



and the frequency equation is of the form 



A.D.-BLC. - o .......... (72) 



The forms of all the functions which enter into the expressions of A,, B., C,, D. have 

 been determined. 



To investigate gravitational instability, we have to determine the conditions which 



