ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 353 



Let M be the mass of the system of rings, and Sm that of one ring 

 whose radius is r, and angular velocity a> = $*r~*. The angular momentum of 

 the ring is 



half its vis viva is ^coVSm = ^Sr' 1 8m. 



The potential energy due to Saturn's attraction on the ring is 



-Sr-i&m. 



The angular momentum of the whole system is invariable, and is 



S*2 (r*Sm) = A .............................. (119). 



/ 

 The whole energy of the system is the sum of half the vis viva and the 



potential energy, and is 



E ........................... (120). 



A is invariable, while E necessarily diminishes. We shall find that as E 

 diminishes, the distribution of the rings must be altered, some of the outer 

 rings moving outwards, while the inner rings move inwards, so as either to 

 spread out the whole system more, both on the outer and on the inner edge 

 of the system, or, without affecting the extreme rings, to diminish the density 

 or number of the rings at the mean distance, and increase it at or near the 

 inner and outer edges. 



Let us put x = i*, then A = S^t (xdm) is constant. 



- T 2 (xdm) 



Now let BI = -*/J /, 



2 (am) 



and x = #! + x' t 



then we may write 





= 1dm (V s - 2 -j+ 8 -7-&C.), 



.Cj .//! 



= -^ 2 (dm) -^2 (x'dm) + t 2 (*8i) -&c ..... (121). 



X, X l X l 



VOL. I. 45 





