3G4 ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 



ring. In this case the satellites are crowded together when nearest to the centre, 

 so that the case is that of the first series of waves, when m = 5. 



Now suppose the cranked axle C to be turned, and all the small cranks 

 K to turn with it, as before explained, every satellite will then be carried 

 round on its own arm in the same direction ; but, since the direction of the 

 arms of different satellites is different, their phases of revolution will preserve 

 the same difference, and the system of satellites will still be arranged in five 

 undulations, only the undulations will be propagated round the ring in the 

 direction opposite to that of the revolution of the satellites. 



To understand the motion better, let us conceive the centres of the orbits 

 of the satellites to be arranged in a straight line instead of a circle, as in 

 fig. 10. Each satellite is here represented in a different phase of its orbit, so 

 that as we pass from one to another from left to right, we find the position 

 of the satellite in its orbit altering in the direction opposite to that of the 

 hands of a watch. The satellites all lie in a trochoidal curve, indicated by 

 the line through them in the figure. Now conceive every satellite to move in 

 its orbit through a certain angle in the direction of the arrows. The satellites 

 will then lie in the dotted line, the form of which is the same as that of 

 the former curve, only shifted in the direction of the large arrow. It appears, 

 therefore, that as the satellites revolve, the undulation travels, so that any 

 part of it reaches successively each satellite as it comes into the same phase 

 of -rotation. It therefore travels from those satellites which are most advanced 

 in phase to those which are less so, and passes over a complete wave-length 

 in the time of one revolution of a satellite. 



Now if the satellites be arranged as in fig. 8, where each is more advanced 

 in phase as we go round the ring in the direction of rotation, the wave will 

 travel in the direction opposite to that of rotation, but if they are arranged 

 as in fig. 12, where each satellite is less advanced in phase as we go round 

 the ring, the wave will travel in the direction of rotation. Fig. 8 represents 

 the first series of waves where m = 5, and fig. 12 represents the fourth series 

 where m = 7. By arranging the satellites in their sockets before starting, we 

 might make m equal to any whole number, from 1 to 18. If we chose any 

 number above 18 the result would be the same as if we had taken a number 

 as much below 18 and changed the arrangement from the first wave to the 

 fourth. 



