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



with the waves of the ring (an operation which is always possible), we found 

 that each term of the disturbing force generates a "forced wave" travelling with 

 its own angular velocity. The magnitude of the forced wave depends not only 

 on that of the disturbing force, but on the angular velocity with which the dis- 

 turbance travels round the ring, being greater in proportion as this velocity 

 more nearly coincides with that of one of the "free waves" of the ring. We 

 also found that the displacement of the satellites was sometimes in the direction 

 of the disturbing force, and sometimes in the opposite direction, according to 

 the relative position of the forced wave among the four natural ones, producing 

 in the one case positive, and in the other negative forced waves. In treating 

 the problem generally, we must determine the forced waves belonging to every 

 term of the disturbing force, and combine these with such a system of free 

 waves as shall reproduce the initial state of the ring. The subsequent motion 

 of the ring is that which would result from the free waves and forced waves 

 together. The most important class of forced waves are those which are pro- 

 duced by waves in neighbouring rings. 15. 



We concluded the theory of a ring of satellites by tracing the process by 

 which the ring would be destroyed if the conditions of stability were not 

 fulfilled. We found two cases of instability, depending on the nature of the 

 tangential force due to tangential displacement. If this force be in the direction 

 opposite to the displacement, that is, if the parts of the ring are statically 

 stable, the ring will be destroyed, the irregularities becoming larger and larger 

 without being propagated round the ring. When the tangential force is in the 

 direction, of the tangential displacement, if it is below a certain value, the 

 disturbances will be propagated round the ring without becoming larger, and 

 we have the case of stability treated of at large. If the force exceed this value, 

 the disturbances will still travel round the ring, but they will increase in ampli- 

 tude continually till the ring falls into confusion. 18. 



We then proceeded to extend our method to the case of rings of different 

 constitutions. The first case was that of a ring of satellites of unequal size. 

 If the central body be of sufficient mass, such a ring will be spaced out, so that 

 the larger satellites will be at wider intervals than the smaller ones, and the 

 waves of disturbance will be propagated as before, except that there may be 

 reflected waves when a wave reaches a part of the ring where there is a change 

 in the average size of the satellites. 19. 



