ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 319 



Let - RL = x. then the equation of motion will be 

 P- 



x)=U=Q ....................... (23). 



The conditions of this equation having real roots are 



S>x ....................................... (24), 



(S-x)*>8x(3S-x) ..... ...................... (25). 



The last condition gives the equation 



whence S>2G'642x, or S<0-351a: ........................ (26). 



The last solution is inadmissible because S must be greater than x, so that 

 the true condition is *S>25'649x, 



> 25-649 -S^ '5259, 

 p. if 



.................................... (27). 



So that if there were 100 satellites in the ring, then 



5>4352 R 



is the condition which must be fulfilled in order that the motion arising from 

 every conceivable displacement may be periodic. 



If this condition be not fulfilled, and if S be not sufficient to render the 

 motion perfectly stable, then although the motion depending upon long undu- 

 lations may remain stable, the short undulations will increase in amplitude till 

 some of the neighbouring satellites are brought into collision. 



9. To determine the nature of the motion when the system of satellites 

 is of small mass compared with the central body. 



The equation for the determination of n is 



U= n< - {' + - R (2K+ L - N}} n* - 4o> - RMn 



Q ......... (28). 



When R is very small we may approximate to the values of n by assuming 

 that two of them are nearly <a, and that the other two are small. 



