322 ON THE STABILITY OP THE MOTION OF SATURN'S RINGS. 



When n is nearly equal to <a, the motion of each satellite in space is 

 nearly elliptic. The eccentricity is A, the longitude at epoch s, and the longi- 

 tude when at the greatest distance from Saturn is for the negative value n, 



- R (2 K+ L - iM - 4N) t + (m + 1 ) * + a, 

 and for the positive value n 4 



-R(2K+L + 4M-tN)t~(m+l)s-a. 



fid) 



We must recollect that in all cases the quantity within brackets is negative, 

 so that the major axis of the ellipse travels forwards in both cases. The chief 

 difference between the two cases lies in the arrangement of the major axes of 

 the ellipses of the different satellites. In the first case as we pass from one 

 satellite to the next in front the axes of the two ellipses lie in the same 

 order. In the second case the particle in front has its major axis behind that 

 of the other. In the cases in which n is small the radius vector of each 

 satellite increases and diminishes during a periodic time of several revolutions. 

 This gives rise to an inequality, in which the tangential displacement far exceeds 

 the radial, as in the case of the annual equation of the Moon. 



12. Let us next examine the condition of the ring of satellites at a given 

 instant. We must therefore fix on a particular value of t and trace the changes 

 of p and o- for different values of s. 



From the expression for p we learn that the satellites form a wavy line, 

 which is furthest from the centre when (ms + nt + a) is a multiple of 2,ir, and 

 nearest to the centre for intermediate values. 



From the expression for cr we learn that the satellites are sometimes in 

 advance and sometimes in the rear of their mean position, so that there are 

 places where the satellites are crowded together, and others where they are 

 drawn asunder. When n is positive, B is of the opposite sign to A, and the 

 crowding of the satellites takes place when they are furthest from the centre. 

 When n is negative, the satellites are separated most when furthest from the 

 centre, and crowded together when they approach it. 



The form of the ring at any instant is therefore that of a string of beads 

 forming a re-entering curve, nearly circular, but with a small variation of distance 



