ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 335 



turn the major axis into an inclined position, so that its fore end points a little 

 inwards, and its hinder end a little outwards. The oscillations of each particle 

 round its mean position are therefore in ellipses, of which both axes increase 

 continually while the eccentricity increases, and the major axis becomes slightly 

 inclined to the tangent, and this goes on till the ring is destroyed. In the 

 mean time the irregularities of the ring do not remain among the same set of 

 particles as in the former case, but travel round the ring with a relative angular 



ar\ 



velocity . Of these waves there are four, two travelling forwards among the 



satellites, and two travelling backwards. One of each of these pairs depends 

 on a negative value of q, and consists of a wave whose amplitude continually 

 decreases. The other depends on a positive value of q, and is the destructive 

 wave whose character we have just described. 



19. We have taken the case of a ring composed of equal satellites, as 

 that with which we may compare other cases in which the ring is constructed 

 of loose materials differently arranged. 



In the first place let us consider what will be the conditions of a ring 

 composed of satellites of unequal mass. We shall find that the motion is of 

 the same kind as when the satellites are equal. 



For by arranging the satellites so that the smaller satellites are closer 

 together than the larger ones, we may form a ring which will revolve uni- 

 formly about Saturn, the resultant force on each satellite being just sufficient 

 to keep it in its orbit. 



To determine the stability of this kind of motion, we must calculate the 

 disturbing forces due to any given displacement of the ring. This calculation 

 will be more complicated than in the former case, but will lead to results of 

 the same general character. Placing these forces in the equations of motion, 

 we shall find a solution of the same general character as in the former case, 

 only instead of regular waves of displacement travelling round the ring, each 

 wave will be split and reflected when it comes to irregularities in the chain of 

 satellites. But if the condition of stability for every kind of wave be fulfilled, 

 the motion of each satellite will consist of small oscillations about its position 

 of dynamical equilibrium, and thus, on the whole, the ring will of itself assume 

 the arrangement necessary for the continuance of its motion, if it be originally 

 in a state not very different from that of equilibrium. 



