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



If p and q denote the components parallel and perpendicular to 00 of the attraction 

 of the body on a unit of matter at S, we have 



X=pcos<f> qain<f> = p, and Y=psm<f> + qcos <j> =p<j> + q, 



since q and <f> are each infinitely small; and if we put V= potential at 8, and 



<TV 



_ 



" d? ' ~ drf ' 



then p =/- af - 717, q = -@r)- yf, 



If we make these substitutions for A' and )'. and take into account that 



/='<* f ................................................ (*), 



the first and second equations of motion become 



Combining equations (a), (c), and (e?), by the same method as that adopted in the text, 

 we find that the differential equation in f , 17, or <, is of the form 



where A = Jf, 



B=a>* (2i* o*) - ^~ {/fc'a + (a" 



C = a> 4 (* - 3a") + w" 5 {(a" + ') (a + /8) - 4a'/9) + (a 1 



In comparing this result with that obtained in the Essay, we must put 



r for a, 

 R for M, 

 B + S for 8, 

 L for o, 

 XrJ for & 

 Mr, for 7. 



