ON THE STABILITY OF THE MOTION OF SATURN'S KINGS. 313 



so that we may write the values of sin SPQ and cos SPQ, 



(2), 

 } ...... (3). 



If we assume the masses of P and Q each equal to - R, where R is the 



V- 



mass of the ring, and /x the number of satellites of which it is composed, the 

 accelerating effect of the radial force on P is 



and the tangential force 



1 sin SPQ _ 1 R cos , _^ _, , , , ., , , 



1 L 



The normal force is - R r* , ,' . 

 p. 8 sin 



5. Let us substitute for p, a- and their values expressed in a series of 

 sines and cosines of multiples of s, the terms involving ms being 



P! = A cos (ms + a), p a = A cos (ms + a 



cr, = B sin (ms + ft), o-,= 



= C cos (ms + y), 3 = 



The radial force now becomes 



1 A cos (ms + a) (1+ cos 2m0) + A sin (ms + a) sin 2m0 T 



. - n > -\-\A cos(ms + a) (1 cos2i0) cot 2 \A sin (ms + a)sin 2m#cot 2 \ (6). 



/A 4 sm 



. + \B sin (ms + ft) ( 1 cos 2m0) cot %B cos (ms + ft) sin 2m0 cot J 



The radial component of the attraction of a corresponding particle on the 

 other side of P may be found by changing the sign of 0. Adding the two 

 together, we have for the effect of the pair 



1 J? 



{l A cos (ms + a) (2 cos 5 m0 sin 2 m0 cot 2 0) 



} (7). 



VOL. I. 40 



