476 MR. J. H. JEANS ON GRAVITATIONAL INSTABILIT 



/ replacing c. This is found to be 



v 4TC (frr)' / 'an 

 s ~ ' 2 



r 



The condition now to he satisfied is clearly that 



at all points on the boundary r = a + 6S n . This requires that V l V 2 V 3 shall vanish, 

 to within a constant, at all points outside this boundary, and therefore, in particular, 

 at r = c. It will be readily seen that 



V V V (V } 



v 1 3 - v m V ' m/n = 



while V 3 is exactly the value of the terms in V m that involve a, when a is put equal 

 to zero. The conditions sought are, therefore, simply that all the terms in a which 

 occur in V m shall vanish at every point of the sphere. 



19. We may now equate the coefficients of the separate harmonics, and obtain 



O n T I i \ Afl T / \ ^7fA 



fr-U (71) 



On account of the simplifications made possible by these eqxiations, equation (68) 

 may be put in the form 



<- = ac-'^v, ( KC> /3)- 



n > ..... (72 ) 



The elimination of AO and C from (59), (60), and (70) gives 



- ^\ Js / 2 ( K ' a ' -ct ) - / _ ^ 

 " 



K' J v 7(?aT a) " 2W IT J v , (*a) ' 



while similarly the elimination of A n and C n from (61), (62), and (71) gives 



i 4. J,,^/.,(^, -$)3*,.( K 'a, -a)\ __ i _ uf\ L J, + ./. (K) J./, (*)! 

 J. +Vi ( K 'a, ft) J,, (,'a, a) I ' 2W I J. + . fc M J-/ 3 M/ ' 



For brevity we introduce a function u n (x, 9), a generalisation of the u n of 11, the 



