ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 323 



from the centre recurring m times, and forming m regular waves of trans- 

 verse displacement at equal intervals round the circle. Besides these, there are 

 waves of condensation and rarefaction, the effect of longitudinal displacement. 

 When n is positive the points of greatest distance from the centre are points 

 of greatest condensation, and when n is negative they are- points of greatest 

 rarefaction. 



13. We have next to determine the velocity with which these waves of 

 disturbance are propagated round the ring. We fixed our attention on a par- 

 ticular satellite by making s constant, and on a particular instant by making t 

 constant, and thus we determined the motion of a satellite and the form of the 

 ring. We must now fix our attention on a phase of the x motion, and this we 

 do by making p or <r constant. This implies 



ms + nt + a = constant, 



ds _ n 

 dt~ in' 



So that the particular phase of the disturbance travels round the ring with an 



77i 



angular velocity = relative to the ring itself. Now the ring is revolving 



in space with the velocity ta, so that the angular velocity of the wave in space is 



vr = o-- (36). 



m 



Thus each satellite moves in an ellipse, while the general aspect of the 

 ring is that of a curve of m waves revolving with velocity w. This, however, 

 is only the part of the whole motion, which depends on a single term of the 

 solution. In order to understand the general solution we must shew how to 

 determine the whole motion from the state of the ring at a given instant. 



14. Given the position and motion of every satellite at any one time, to 

 calculate the position and motion of every satellite at any other time, provided 

 that the condition of stability is fulfilled. 



The position of any satellite may be denoted by the values of p and a- for 

 that satellite, and its velocity and direction of motion are then indicated by the 



values of -/- and -j~ at the given instant. 

 at at 



412 



