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



These four quantities may have for each satellite any four arbitrary values, 

 as the position and motion of each satellite are independent of the rest, at the 

 beginning of the motion. 



Each of these quantities is therefore a perfectly arbitrary function of s, the 

 mean angular position of the satellite in the ring. 



But any function of s from s = to s = 2ir, however arbitrary or discontinuous, 

 can be expanded in a series of terms of the form A cos (.? + a) + A' cos (2s + a) + &c. 

 See 3. 



Let each of the four quantities p, - , <r, --j- be expressed in terms of such 

 a series, and let the terms in each involving tns be 



(37), 



(38), 

 o- = G cos (ms +g) .............................. (39), 



(40). 



These are the parts of the values of each of the four quantities which are 

 capable of being expressed in the form of periodic functions of ms. It is 

 evident that the eight quantities E, F, G, H, e, f, g, h, are all independent and 

 arbitrary. 



The next operation is to find the values of L, M, N, belonging to disturb- 

 ances in the ring whose index is in [see equation (8)], to introduce these 

 values into equation (28), and to determine the four values of n, (,, n,, n,, 4 ). 



This being done, the expression for p is that given in equation (32), which 

 contains eight arbitrary quantities (A t , A t , A t , A u a,, a,, a,, a 4 ). 



Giving t its original value in this expression, and equating it to Ecos(ms + e), 

 we get an equation which is equivalent to two. For, putting ms = 0, we have 



A l cos <LI + A t cos a, + A, cos a, + A t cos a 4 = E cos e ............ (41). 



And putting ms = , we have another equation 



i 



A l sin a, + A* sin o, + -4, sin a, + A 4 sin a 4 = E sin e 



