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



cosines of multiples of (i| 6) will vanish when integrated between the limits. 

 In this way we find 



........... (22). 



r 



The other terms containing higher powers of . 



In order to express V in terms of r, and <f> lt as we have assumed in the 

 former investigation, we must put 



r'cosi/ = -r 1 

 r'sini/= - 



From which we find (-T-) = if, 



\drj. a'- 7 ' 



(24). 



These results may be confirmed by the following considerations applicable to 

 any circular ring, and not involving any expansion or integration. Let of be 

 the distance of the centre of gravity from the centre of the ring, and let 

 the ring revolve about its centre with velocity ta. Then the force necessary 

 to keep the ring in that orbit will be Rafoi 1 . 



But let S be a mass fixed at the centre of the ring, then if 



j_8 

 a" 



every portion of the ring will be separately retained in its orbit by the attrac- 

 tion of S, so that the whole ring will be retained in its orbit. The resultant 

 attraction must therefore pass through the centre of gravity, and be 



a 1 ' 



therefore ^- = R , , 



ar a* 



