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



of this action, is one which would prevent the particle from receding from the 

 planet under the influence of the tangential force, or at least prevent the dimi- 

 nution of angular velocity. The transversal force of attraction of the ring is of 

 this kind, and acts in the right direction, but it can never be of sufficient magni- 

 tude to have the required effect. In fact the thing to be done is to render the 

 last term of the equation in n* positive when N is negative, which requires 



and this condition is quite inconsistent with any constitution of the ring which 

 fulfils the other condition of stability which we shall arrive at presently. 



We may observe that the waves belonging to the two real values of n, 

 (a, must be conceived to be travelling round the ring during the whole time 

 of its breaking up, and conducting themselves like ordinary waves, till the 

 excessive irregularities of the ring become inconsistent with their uniform propa- 

 gation. 



The irregularities which depend on the exponential solutions do not travel 

 round the ring by propagation among the satellites, but remain among the same 

 satellites which first began to move irregularly. 



We have seen the fate of the ring when x is negative. When x is small 

 we have two small and two large values of n, which indicate regular waves, 

 as we have already shewn. As x increases, the small values of n increase, and 

 the large values diminish, till they meet and form a pair of positive and a 

 pair of negative equal roots, having values nearly '68w. When x becomes 

 greater than about -^ta', then all the values of n become impossible, of the 

 form p + -J Iq, q being small when x first begins to exceed its limits, and p 

 being nearly '68a>. 



2ir 

 The values of p and a- indicate periodic inequalities having the period - - , 



but increasing in amplitude at a rate depending on the exponential c". At the 

 beginning of the motion the oscillations of the particles are in ellipses as in the 

 case of stability, 'having the ratio of the axes about 1 in the normal direction 

 to 3 in the tangential direction. As the motion continues, these ellipses increase 

 in magnitude, and another motion depending on the second term of <r is com- 

 bined with the former, so as to increase the ellipticity of the oscillations and to 



