ON THE STABILITY OP THE MOTION OF SATURN'S KINGS. 333 



Introducing the exponential notation, these values become 



cr= 



) (69), 



2c*A f p(p z + q s + x)(e qt + e-^)sm(ms+pt + a)\ , , 



'(p* + q i y + 2(p' i -q')x + x a \ + q(p' + q i -x) (e 9 * - e' 9 *) cos (ms +j> + a) J " ' ^ '' 



We have now obtained a solution free from impossible quantities, and applicable 

 to every case. 



When q = 0, the case becomes that of real roots, which we have already 

 discussed. When p = Q, we have the case of pure impossible roots arising from 

 the negative values of n\ The solutions corresponding to these roots are 



(71). 



<r = - (t t -t-'> t ) cos (ms + a). ................ (72). 



The part of the coefficient depending on e~ qt diminishes indefinitely as the 

 time increases, and produces no marked effect. The other part, depending on 

 e 7 ', increases in a geometrical proportion as the time increases arithmetically, and 

 so breaks up the ring. In the case of x being a small negative quantity, (f is 

 nearly 3x, so that the coefficient of <r becomes 



-3 A. , - 



? 



It appears therefore that the motion of each particle is either outwards and 

 backwards or inwards and forwards, but that the tangential part of the motion 

 greatly exceeds the normal part. 



It may seem paradoxical that a tangential force, acting towards a position 

 of equilibrium, should produce instability, while a small tangential force from that 

 position ensures stability, but it is easy to trace the destructive tendency of 

 this apparently conservative force. 



Suppose a particle slightly in front of a crowded part of the ring, then 

 if x is negative there will be a tangential force pushing it forwards, and this 

 force will cause its distance from the planet to increase, its angular velocity to 

 diminish, and the particle itself to fall back on the crowded part, thereby 

 increasing the irregularity of the ring, till the whole ring is broken up. In 

 the same way it may be shewn that a particle behind a crowded part will be 

 pushed into it. The only force which could preserve the ring from the effect 



