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



We shall find, however, that it is best to consider the waves first as free, 

 and then to determine the reaction of the other ring upon them, which is such 

 as to alter the wave-velocity of both, as we shall see. 



The forces due to the second ring may be separated into three parts. 



1st The constant attraction when both rings are at rest. 



2nd. The variation of the attraction on the first ring, due to its own 

 disturbances. 



3rd. The variation of the attraction due to the disturbances of the second 

 ring. 



The first of these affects only the angular velocity. The second affects the 

 waves of each ring independently, and the mutual action of the waves depends 

 entirely on the third class of forces. 



26. To determine the attractions between two rings. 



Let R and a be the mass and radius of the exterior ring, R and a' those 

 of the interior, and let all quantities belonging to the interior ring be marked 

 with accented letters. (Fig. 5.) 



1st. Attraction between the rings when at rest. 



Since the rings are at a distance small compared with their radii, we may 

 calculate the attraction on a particle of the first ring as if the second were an 

 infinite straight line at distance a' a from the first. 



TV 



The mass of unit of length of the second ring is - ,, and the accelerating 



effect of the attraction of such a filament on an element of the first ring is 



jy 



. - A inwards .............................. (97). 



ira (a - a) 



The attraction of the first ring on the second may be found by transposing 

 accented and unaccented letters. 



In consequence of these forces, the outer ring will revolve faster, and the 



inner ring slower than would otherwise be the case. These forces enter into 



the constant terms of the equations of motion, and may be included in the 

 value of K. 



