ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 331 



own body, but can likewise exercise an influence over their relations to each 

 other, so as to overrule their natural tendency to crowd together, and distribute 

 and preserve them in the form of a ring. 



"We have traced the motion of each satellite, the general shape of the 

 disturbed ring, and the motion of the various waves of disturbance round the 

 ring, and determined the laws both of the natural or free waves of the ring, 

 and of the forced waves, due to extraneous disturbing forces. 



We have now to consider the cases in which such a permanent motion of 

 the ring is impossible, and to determine the mode in which a ring, originally 

 regular, will break up, in the different cases of instability. 



The equation from which we deduce the conditions of stability is 



U = n* - jo>= + - R (2K+ L - N)\ ri 1 - 4o> - RMn 

 I V- P 



+ { 3o>' - - R (2K+ L)\ - RN - - 2 KM* = 0. 

 II* J V" P" 



The quantity, which, in the critical cases, determines the nature of the 

 roots of this equation, is JV. The quantity M in the third term is always 

 small compared with L and N when ra is large, that is, in the case of the 

 dangerous short waves. We may therefore begin our study of the critical cases 

 by leaving out the third term. The equation then becomes a quadratic in if, 

 and in order that all the values of n may be real, both values of n a must be 

 real and positive. 



The condition of the values of w* being real is 



>0 (61), 



/* ' P-' 



which shews that CD* must either be about 14 times at least smaller, or about 14 

 times at least greater, than quantities like - RN. 



That both values of n* may be positive, we must have 



.1. J-\ / _ fr . f Tl T\ /\ 



3^-~R(2K + L)\-RN>0 



..(62). 

 422 



