ON THE STABILITY OF THE MOTION OF SATUKN's RINGS. 321 



So that for every term of p of the form 



p = A cos(ms + nt + a) (34), 



there is a corresponding term in cr, 



2&m + - RM 

 a- = -^ A sin (ms + nt + a) (35). 



10. Let us now fix our attention on the motion of a single satellite, 

 and determine its motion by tracing the changes of p and cr while t varies 

 and s is constant, and equal to the value of s corresponding to the satellite 

 in question. 



We must recollect that p and cr are measured outwards and forwards from 

 an imaginary point revolving at distance 1 and velocity , so that the motions 

 we consider are not the absolute motions of the satellite, but its motions 

 relative to a point fixed in a revolving plane. This being understood, we may 

 describe the motion as elliptic, the major axis being in the tangential direc- 

 tion, and the ratio of the axes being nearly 2 - , which is nearly 2 for n^ and 4 



71 



and is very large for n., and n t . 



The time of revolution is , or if we take a revolution of the ring as 



ft 



the unit of time, the time of a revolution of the satellite about its mean 



position is - . 

 n 



The direction of revolution of the satellite about its mean position is in 

 every case opposite to that of the motion of the ring. 



11. The absolute motion of a satellite may be found from its motion 

 relative to the ring by writing 



r=l+p = l+Acos (ms + nt + a), 





2 - A sin (ms + nt + a). 

 n 



VOL. I. 41 



