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



Dividing each term by 1f, we get 



= ............ (32). 



The first condition gives /less than '8279. 



The second condition gives f greater than "815865. 



Let us assume as a particular case between these limits /='82, which 

 makes the ratio of the mass of the particle to that of the ring as 82 to 18, 

 then the equation becomes 



l-82n 4 +-8114nV+-9696<a 4 = ...................... (33), 



which gives J~^ln = '5916co or '3076w. 



These values of n indicate variations of r,, lt and $,, which are com- 

 pounded of two simple periodic inequalities, the period of the one being 1*69 

 revolutions, and that of the other 3 '2 51 revolutions of the ring. The relations 

 between the phases and amplitudes of these inequalities must be deduced from 

 equations (14), (15), (16), in order that the character of the motion may be 

 completely determined. 



Equations (14), (15), (16) may be written as follows: 



................ (34), 



(35), 

 }^ = ....... (36). 



By eliminating one of the variables between any two of these equations, 

 we may determine the relation between the two remaining variables. Assuming 

 one of these to be a periodic function of t of the form A cos vt, and remem- 



bering that n stands for the operation -j- , we may find the form of the other. 

 Thus, eliminating Q l between the first and second equations, 



{(3-g)a-l s hn}<t> l = Q ............ (37). 



