GRAVITATIONAL STABILITY OF THE EARTH. 193 



examine the two coses in which v = and v = |. We have now to discuss the 

 conditions of gravitational instability in respect of the values 0, 1, 2, ... of the number 

 n which specifies the type of vibration. 



Instability in Reaped of Radial Displacements. 



19. The case in which n = is the case of a sphere vibrating radially. This case 

 is not very easily included in the foregoing analysis, and it is very easy to investigate 

 it independently. Let U denote the radial displacement. Then U is a function of r, 

 and we have 



XTT VTT TT 2 



u = - U , v = 2. U, w = - U, A = -- + -- . 



r r r fir r 



2U 



, = -- 

 fir 



We go back to the equations (15) of the type 



where W, the additional potential, is a function of r. This equation is 



d /rfU . 2U\ . \ld* 2 eZ\/U\ O x d/U 

 r _- + _ )+n\x -7-5 + - -^- -1 + 2- -3- ( 

 r\dr r I [ \dr r dr/ \r ] rdr\r 



Now 



2 dW 



., . . (74) 



r r dr 



fd\J A 2U\ 



, , --- j = 47J-yp ( -= h- ), 

 or r dr \dr r J 



and therefore we may write 



dW 

 ^.-iryipfcU+B, 



where 



j- + - R = 0, or Rr* = const. 

 r r 



Since dW/dr is finite at r = 0, we must have R = 0, and thus equation (74) becomes 

 after division by (X + 2/i) a;/r 



2 + + w + 3s>U + h>U = 0,. . . . (75) 



r dr r* \ dr / 



where .f* = j7rypo S /(X + 2/x) and h* = p a p t /(\ + 2p.). This equation can be solved by 

 means of a series, which is convergent for all finite values of r, in the form 



U - A r - 



235 2.4.3.5.7 



I i ( . 



VOL. CCVII. A. 2 C 



