ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 369 



The next case was that of an annular cloud of meteoric stones, revolving 

 uniformly about the planet. The average density of the space through which 

 these small bodies are scattered will vary with every irregularity of the motion, 

 and this variation of density will produce variations in the forces acting upon 

 the other parts of the cloud, and so disturbances will be propagated in this 

 ring, as in a ring of a finite number of satellites. The condition that such a 

 ring should be free from destructive oscillations is, that the density of the 

 planet should be more than three hundred times that of the ring. This would 

 make the ring much rarer than common air, as regards its average density, 

 though the density of the particles of which it is composed may be great. 

 Comparing this result with Laplace's minimum density of a ring revolving as 

 a whole, we find that such a ring cannot revolve as a whole, but that the inner 

 parts must have a greater angular velocity than the outer parts. 20. 



We next took up the case of a flattened ring, composed of incompressible 

 fluid, and moving with uniform angular velocity. The internal forces here arise 

 partly from attraction and partly from fluid pressure. We began by taking the 

 case of an infinite stratum of fluid affected by regular waves, and found the accurate 

 values of the forces in this case. For long waves the resultant force is in the 

 same direction as the displacement, reaching a maximum for waves whose 

 length is about ten times the thickness of the stratum. For waves about five 

 times as long as the stratum is thick there is no resultant force, and for shorter 

 waves the force is in the opposite direction to the displacement. 23. 



Applying these results to the case of the ring, we find that it will be 

 destroyed by the long waves unless the fluid is less than ^ of the density of 

 the planet, and that in all cases the short waves will break up the ring into 

 small satellites. 



Passing to the case of narrow rings, we should find a somewhat larger 

 maximum density, but we should still find that very short waves produce forces 

 in the direction opposite to the displacement, and that therefore, as already 

 explained (page 333), these short undulations would increase in magnitude without 

 being propagated along the ring, till they had broken up the fluid filament into 

 drops. These drops may or may not fulfil the condition formerly given for the 

 stability of a ring of equal satellites. If they fulfil it, they will move as a 

 permanent ring. If they do not, short waves will arise and be propagated among 

 the satellites, with ever increasing magnitude, till a sufficient number of drops 



VOL. I. 47 



