ON THE STABILITY OF THE MOTION OF SATURN'S EINGS. 301 



- [ZRrjan + (R + S) M} {ZRrjm} {RJfn 1 - SN} 

 + {Rtf - Rrf -(R + S) L} {RIM} {(R + S)N] 



+ (SM) (flr.V) (R + S) M- (SM) (2Rr t a>n) (R + S)N\=0 ...... (17). 



+ {2Rrjun + (R + S)M} {Rtfri} {(R + S)M} 



- {Rn* - Rw 1 -(R + S)} {/2r.V} {RtW - SN} 



By multiplying up, and arranging by powers of n and dividing by Rn', 



this equation becomes 



An t + Bn*+C=0 ................................ (18), 



where 



...... (19). 



C=R{(R+S)lt- 3Sr '} <S + (R + S) {(R + S)V + Sr*} (LN- 



Here we have a biquadratic equation in n which may be treated as a 

 quadratic in ?i 2 , it being remembered that n stands for the operation -5- . 



PROB. IV. To determine whether the motion of the ring is stable or 

 unstable, by means of the relations of the coefficients A, B, C. 



The equations to determine the forms of r lt lt and fa are all of the form 



and if n be one of the four roots of equation (18), then 



u = De nt 



will be one of the four terms of the solution, and the values of r lt lt and 

 fa will differ only in the values of the coefficient D. 



Let us inquire into the nature of the solution in different cases. 



(1) If n be positive, this term would indicate a displacement which 

 must increase indefinitely, so as to destroy the arrangement of the system. 



(2) If n be negative, the disturbance which it belongs to would gradually 

 die away. 



(3) If n be a pure impossible quantity, of the form a/ 1, then there 

 will be a term in the solution of the form D cos (at + a), and this would indi- 



cate a periodic variation, whose amplitude is D, and period - - . 



a 



