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



have been brought into collision, so as to unite and form a smaller number of 

 larger drops, which may be capable of revolving as a permanent ring. 



We have already investigated the disturbances produced by an external 

 force independent of the ring; but the special case of the mutual perturbations 

 of two concentric rings is considerably more complex, because the existence of a 

 double system of waves changes the character of both, and the waves produced 

 react on those that produced them. 



We determined the attraction of a ring upon a particle of a concentric 

 ring, first, when both rings are in their undisturbed state ; secondly, when the 

 particle is disturbed ; and, thirdly, when the attracting ring is disturbed by a 

 series of waves. 26. 



We then formed the equations of motion of one of the rings, taking in the 

 disturbing forces arising from the existence of a wave in the other ring, and 

 found the small variation of the velocity of a wave in the first ring as dependent 

 on the magnitude of the wave in the second ring, which travels with it. 27. 



The forced wave in the second ring must have the same absolute angular 

 velocity as the free wave of the first which produces it, but this velocity of 

 the free wave is slightly altered by the reaction of the forced wave upon it. 

 We find that if a free wave of the first ring has an absolute angular velocity 

 not very different from that of a free wave of the second ring, then if both 

 free waves be of even orders (that is, of the second or fourth varieties of waves), 

 or both of odd orders (that is, of the first or third), then the swifter of the 

 two free waves has its velocity increased by the forced wave which it produces, 

 and the slower free wave is rendered still slower by its forced wave ; and even 

 when the two free waves have the same angular velocity, their mutual action 

 will make them both split into two, one wave in each ring travelling faster, 

 and the other wave in each ring travelling slower, than the rate with which 

 they would move if they had not acted on each other. 



But if one of the free waves be of an even order and the other of an odd 

 order, the swifter free wave will travel slower, and the slower free wave will 

 travel swifter, on account of the reaction of their respective forced waves. If 

 the two free waves have naturally a certain small difference of velocities, they 

 will be made to travel together, but if the difference is less than this, they 

 will again split into two pairs of waves, one pair continually increasing in 



