ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 345 



but it may be shown that in this case also there will be a maximum value 

 of x, which will require the density of the planet to be several times that of 

 the ring, and that in all cases short waves will give rise to negative values 

 of x, inconsistent with the stability of the ring. 



It appears, therefore, that a ring composed of a continuous liquid mass 

 cannot revolve about a central body without being broken up, but that the 

 parts of such a broken ring may, under certain conditions, form a permanent 

 ring of satellites. 



On the Mutual Perturbations of Two Rings. 



25. We shall assume that the difference of the mean radii of the rings 

 is small compared with the radii themselves, but large compared with the 

 distance of consecutive satellites of the same ring. We shall also assume that 

 each ring separately satisfies the conditions of stability. 



We have seen that the effect of a disturbing force on a ring is to produce 

 a series of waves whose number and period correspond with those of the dis- 

 turbing force which produces them, so that we have only to calculate the 

 coefficient belonging to the wave from that of the disturbing force. 



Hence in investigating the simultaneous motions of two rings, we may 

 assume that the mutually disturbing waves travel with the same absolute 

 angular velocity, and that a maximum in one corresponds either to a maximum 

 or a minimum of the other, according as the coefficients have the same or 

 opposite signs. 



Since the motions of the particles of each ring are affected by the disturbance 

 of the other ring, as well as of that to which they belong, the equations of 

 motion of the two rings will be involved in each other, and the final equation 

 for determining the wave-velocity will have eight roots instead of four. But as 

 each of the rings has four free waves, we may suppose these to originate forced 

 waves in the other ring, so that we may consider the eight waves of each ring 

 as consisting of four free waves and four forced ones. 



In strictness, however, the wave- velocity of the "free" waves will be 

 affected by the existence of the forced waves which they produce in the other 

 ring, so that none of the waves are really " free " in either ring independently, 

 though the whole motion of the system of two rings as a whole is free. 



VOL. i. 44 





