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



The forced wave which we have just investigated is that which would main- 

 tain itself in the ring, supposing that it had been set agoing at the commence- 

 ment of the motion. It is in fact the form of dynamical equilibrium of the 

 ring under the influence of the given forces. In order to find the actual motion 

 of the ring we must combine this forced wave with all the free waves, which 

 go on independently of it, and in this way the solution of the problem becomes 

 perfectly complete, and we can determine the whole motion under any given 

 initial circumstances, as we did in the case where no disturbing force acted. 



For instance, if the ring were perfectly uniform and circular at the instant 

 when the disturbing force began to act, we should have to combine with the 

 constant forced wave a system of four free waves so disposed, that at the given 

 epoch, the displacements due to them should exactly neutralize those due to the 

 forced wave. By the combined effect of these four free waves and the forced 

 one the whole motion of the ring would be accounted for, beginning from its 

 undisturbed state. 



The disturbances which are of most importance in the theory of Saturn's 

 rings are those which are produced in one ring by the action of attractive 

 forces arising from waves belonging to another ring. 



The effect of this kind of action is to produce in each ring, besides its 

 own four free waves, four forced waves corresponding to the free waves of the 

 other ring. There will thus be eight waves in each ring, and the corresponding 

 waves in the two rings will act and react on each other, so that, strictly speak- 

 ing, every one of the waves will be in some measure a forced wave, although 

 the system of eight waves will be the free motion of the two rings taken 

 together. The theory of the mutual disturbance and combined motion of two 

 concentric rings of satellites requires special consideration. 



18. On the motion of a ring of satellites when the conditions of stability 

 are not fulfilled. 



We have hitherto been occupied with the case of a ring of satellites, the 

 stability of which was ensured by the smallness of mass of the satellites com- 

 pared with that of the central body. We have seen that the statically unstable 

 condition of each satellite between its two immediate neighbours may be com- 

 pensated by the dynamical effect of its revolution round the planet, and a planet 

 of sufficient mass can not only direct the motion of such satellites round its 



