310 ON THE STABILITY OF THE MOTION OF 8ATURN*S RINGS. 



the original values r,, 0,, ^,, and also their first differential coefficients with 

 respect to t, are arbitrary, it would appear that six arbitrary constants ought 

 to enter into the equation. The reason why they do not is that we assume 

 r, and 6, as the mean values of r and 6 in the actual motion. These quantities 

 therefore depend on the original circumstances, and the two additional arbitrary 

 constants enter into the values of r e and . In the analytical treatment of the 

 problem the differential equation in n was originally of the sixth degree with a 

 solution 7i* = 0, which implies the possibility of terms in the solution of the 

 form Ct + D. 



The existence of such terms depends on the previous equations, and we find 

 that a term of this form may enter into the value of 0, and that r, may contain 

 a constant term, but that in both cases these additions will be absorbed into 

 the values of and r . 



PART II. 



ON THE MOTION OF A RING, THE PARTS OF WHICH ARE NOT RIGIDLY CONNECTED. 



1. IN the case of the Ring of invariable form, we took advantage of the 

 principle that the mutual actions of the parts of any system form at all times 

 u system of forces in equilibrium, and we took no account of the attraction 

 between one part of the ring and any other part, since no motion could result 

 from this kind of action. But when we regard the different parts of the ring 

 as capable of independent motion, we must take account of the attraction on 

 each portion of the ring as affected by the irregularities of the other parts, and 

 therefore we must begin by investigating the statical part of the problem in 

 order to determine the forces that act on any portion of the ring, as depending 

 on the instantaneous condition of the rest of the ring. 



In order to bring the problem within the reach of our mathematical methods, 

 we limit it to the case in which the ring is nearly circular and uniform, and has 

 a transverse section very small compared with the radius of the ring. By 

 analysing the difficulties of the theory of a linear ring, we shall be better able 

 to appreciate those which occur in the theory of the actual rings. 



