ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 349 



Putting B = $A, A' = xA, K = 



we have n 4 - {a? ( + 2K) + L - N} ri> - 4col/rc + BtfN\ _ n _ . . 



' 'p) n'x + (//m - v'P] 2o>nx) = 



- . . ................. (108), 



dn 



i (109), 



I*MI 



dn \'n u.'mB'n 2 u'mw + 2 v'B'a /-.- \ 



whence -7-=- - (110). 



cte 4rr 2w 



28. If we were to solve the equation for n, leaving out the terms involving 

 x, we should find the wave-velocities of the four free waves of the first ring, 

 supposing the second ring to be prevented from being disturbed. But in reality 

 the waves in the first ring produce a disturbance in the second, and these in 

 turn react upon the first ring, so that the wave-velocity is somewhat different 

 from that which it would be in the supposed case. Now if x be the ratio 

 of the radial amplitude of displacement in the second ring to that in the first, 

 and if ft be a value of n supposing x = 0, then by Maclaurin's theorem, 



n- 4-n4- x (III} 



/fr ^ T^ /fr T^ 7 (/l-L-L.Lf. 



dx 



The wave- velocity relative to the ring is , and the absolute . angular 



velocity of the wave in space is 



n n I dn t -\-\v\ 



m mm dx 



= +p-qx (113), 



n , I dn 



where = &> , and <7 = -r-. 



m m dx 



Similarly in the second ring we should have 



*'=p'-q- 

 x 



and since the corresponding waves in the two rings must have the same abso- 

 lute angular velocity, 



(114); 



or p qx=p' q' - 



