ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 317 



For every pure impossible root of the form / In' there are terms of 

 the forms 



At*"' 1 cos (ms + a). 



Although the negative exponential coefficient indicates a continually diminishing 

 displacement which is consistent with stability, the positive value which neces- 

 sarily accompanies it indicates a continually increasing disturbance, which would 

 completely derange the system in course of time. 



For every mixed root of the form \/ In' + n, there are terms of the form 



Af* cos (ms + nt + a). 

 If we take the positive exponential, we have a series of m waves travelling 



71 / 



with velocity and increasing in amplitude with the coefficient e +n>t . The 



negative exponential gives us a series of m waves gradually dying away, but 

 the negative exponential cannot exist without the possibility of the positive one 

 having a finite coefficient, so that it is necessary for the stability of the motion 

 that the four values of n be all real, and none of them either impossible 

 quantities or the sums of possible and impossible quantities. 



We have therefore to determine the relations among the quantities K t L, 

 M, N, R, S, that the equation 



n _ {S+ 1 R (3K+ L - N)} n 3 



-4 (a ~RMn + {SS+ - R (K-L}} - RN- I R*M* = U= 



r 1 /* r" P 1 



may have four real roots. 



7. In the first place, U is positive, when n is a large enough quantity, 

 whether positive or negative. 



It is also positive when n = 0, provided S be large, as it must be, com- 



pared with - RL, - RM and - RN. 

 P* /* /* 



If we can now find a positive and a negative value of n for which U 

 is negative, there must be four real values of n for which U=0, and the four 

 roots will be real. 



