ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 367 



the compensating process will go on vigorously, and the ring will be preserved. 

 When the angular velocity is small and the attractive forces of the ring great, 

 the dynamical effect will not compensate for the disturbing action of the forces 

 and the ring will be destroyed. 



If the satellite, instead of being displaced forwards, had been originally 

 behind its mean position in the ring, the forces would have pulled it backwards, 

 its path would have become more concave towards the centre, its distance from 

 the centre would diminish, its angular velocity would increase, and it would 

 gain upon the rest of the ring till it got in front of its mean position. This 

 effect is of course dependent on the very same conditions as in the former case, 

 and the actual effect on a disturbed satellite would be to make it describe an 

 orbit about its mean position in the ring, so that if in' advance of its mean 

 position, it first recedes from the centre, then falls behind its mean position in 

 the ring, then approaches the centre within the mean distance, then advances 

 beyond its mean position, and, lastly, recedes from the centre till it reaches its 

 starting-point, after which the process is repeated indefinitely, the orbit being 

 always described in the direction opposite to that of the revolution of the 

 ring. 



We now understand what would happen to a disturbed satellite, if all the 

 others were preserved from disturbance. But, since all the satellites are equally 

 free, the motion of one will produce changes in the forces acting on the rest, 

 and this will set them in motion, and this motion will be propagated from one 

 satellite to another round the ring. Now propagated disturbances constitute 

 waves, and all waves, however complicated, may be reduced to combinations of 

 simple and regular waves ; and therefore all the disturbances of the ring may 

 be considered as the resultant of many series of waves, of different lengths, and 

 travelling with different velocities. The investigation of the relation between 

 the length and velocity of these waves forms the essential part of the problem, 

 after which we have only to split up the original disturbance into its simple 

 elements, to calculate the effect of each of these separately, and then to combine 

 the results. The solution thus obtained will be perfectly general, and quite 

 independent of the particular form of the ring, whether regular or irregular at 

 starting. 14. 



We next investigated the effect upon the ring of an external disturbing 

 force. Having split up the disturbing force into components of the same type 



