CONTRIBUTIONS TO SCIENCE. 509 



it will be carried farther away from the planet, and thus not 

 only will its linear velocity be diminished, but, in virtue of 

 its increased distance, its angular velocity will be still farther 

 diminished and, lagging behind, it will fall back into its 

 proper position and, oscillating through it with a velocity 

 less than its mean, will approach the planet and the reverse 

 action will take place. 



After considering the manner in which the ring will 

 break up if the condition of stability is not fulfilled, Maxwell 

 takes up the question of " the dusky ring," consisting of in- 

 numerable small particles resembling " a shower of rain, hail, 

 or cinders." For the stability of such a ring its average 

 density must not exceed ^^ of that of the planet, and the 

 density of Saturn being only '7505, it follows that the aver- 

 age density of the ring cannot greatly exceed that of air at 

 ordinary pressure. Laplace showed that for a ring to rotate 

 as a whole with uniform velocity about Saturn, the density 

 of the planet cannot exceed 1*3 times that of the ring. 

 Hence the particles must move independently, or in a series 

 of concentric rings. 



In the case of concentric rings of satellites, their mutual 

 disturbances will destroy one another if the velocity of any 

 one of the four free waves of one coincide, or nearly coincide, 

 with that of any free wave in another ; and it is impossible 

 that there should be any great number of rings without this 

 condition frequently recurring. 



In the case of a large number of concentric rings, the sta- 

 bility of each pair must be investigated separately, and if in the 

 case of any two, whether concentric rings or not, there are a 

 pair of conspiring waves, those two rings will be agitated more 

 and more till waves of that kind are rendered impossible by the 

 breaking up of those rings into some different arrangement. 

 The presence of the other rings cannot prevent the mutual de- 

 struction of any pair which bear such relations to each other. 



It appears, therefore, that in a system of many concentric 

 rings there will be continually new cases of mutual interference 

 between different pairs of rings. The forces which excite these 

 disturbances being very small, they will be slow of growth, 

 and it is possible that by the irregularities of each of the rings 



