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



30. Now each of the four values of p is a function of m, the number 

 of undulations in the ring, and of a the radius of the ring, varying nearly 

 as a"'. Hence m being given, we may alter the radius of the ring till any 

 one of the four values of p becomes equal to a given quantity, say a given 

 value of p', so that if an indefinite number of rings coexisted, so as to form 

 a sheet of rings, it would be always possible to discover instances of the 

 equality of p and p' among them. If such a case of equality belongs to the 

 first column given above, two constant waves will arise in both rings, one 

 travelling a little faster, and the other a little slower than the free waves. 

 If the case belongs to the second column, two waves will also arise in each 

 ring, but the one pair will gradually die away, and the other pair will increase 

 in amplitude indefinitely, the one wave strengthening the other till at last both 

 rings are thrown into confusion. 



The only way in which such an occurrence can be avoided is by placing 

 the rings at such a distance that no value of m shall give coincident values 

 of p and p 1 . For instance, if &/ > 2o, but <a < 3<u, no such coincidence is possible. 

 For p l is always less than pf, it is greater than p t when m = 1 or 2, and less 

 than pi when m is 3 or a greater number. There are of course an infinite 

 number of ways in which this noncoincidence might be secured, but it is plain 

 that if a number of concentric rings were placed at small intervals from each 

 other, such coincidences must occur accurately or approximately between some 

 pairs of rings, and if the value of (pp'Y is brought lower than 4qq', there 

 will be destructive interference. 



This investigation is applicable to any number of concentric rings, for, by 

 the principle of superposition of small displacements, the reciprocal actions of 

 any pair of rings are independent of all the rest. 



31. On the effect of long-continued disturbances on a system of rings. 



The result of our previous investigations has been to point out several 

 ways in which disturbances may accumulate till collisions of the different par- 

 ticles of the rings take place. After such a collision the particles will still 

 continue to revolve about the planet, but there will be a loss of energy in 

 the system during the collision which can never be restored. Such collisions 

 however will not affect what is called the Angular Momentum of the system 

 about the planet, which will therefore remain constant. 



