320 



ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 



If we put n= 0), 



U= -- 



Therefore the corrected values of n are 



n= { (a+ _L R (2K + L-4N)} + RM. 

 2p.oi u.(a 



(29). 



The small values of n are nea.rly/3-RN: correcting them in the same 

 way, we find the approximate values 



(30). 



The four values of n are therefore 



n 1= -o- 



.__ 



,= + 3-RN- 



V ft /ACD 



RM 



(31), 



and the complete expression for p, so far as it depends on terms containing ms, 

 is therefore f = -A.\ cos (ms + nj, + a,) + A t cos (ms + nji + a,) 



+ A, cos (ms + njt, + a,) + A t cos (ms + n t t + a t ) ............... (32), 



and there will be other systems, of four terms each, for every value of m in 

 the expansion of the original disturbance. 



We are now able to determine the value of <r from equations (12), (20), by 

 putting /8 = a, and 



2<un + - RM 



B= 



(33). 



