362 ON THE STABILITY OF THE MOTION OF SATURN S RINGS. 



is to be apprehended are greatest when the number of undulations is greatest, 

 and by taking that number equal to half the number of satellites, we find the 

 condition of stability to be 



where S is the mass of the central body, R that of the ring, and p the number 

 of satellites of which it is composed. 8. If the number of satellites be too 

 great, destructive oscillations will commence, and finally some of the satellites 

 will come into collision with each other and unite, so that the number of 

 independent satellites will be reduced to that which the central body can retain 

 and keep in discipline. When this has taken place, the satellites will not only 

 be kept at the proper distance from the primary, but will be prevented by its 

 preponderating mass from interfering with each other. 



We next considered more carefully the case in which the mass of the ring 

 is very small, so that the forces arising from the attraction of the ring are 

 small compared with that due to the central body. In this case the values 

 of the roots of the biquadratic are all real, and easily estimated. 9. 



If we consider the motion of any satellite about its mean position, as 

 referred to axes fixed in the plane of the ring, we shall find that it describes 

 an ellipse in the direction opposite to that of the revolution of the ring, the 

 periodic time being to that of the ring as <a to n, and the tangential ampli- 

 tude of oscillation being to the radial as 2o> to n. 10. 



The absolute motion of each satellite in space is nearly elliptic for the large 

 values of n, the axis of the ellipse always advancing slowly in the direction of 

 rotation. The path of a satellite corresponding to one of the small values of 

 n is nearly circular, but the radius slowly increases and diminishes during a 

 period of many revolutions. 11. 



The form of the ring at any instant is that of a re-entering curve, having 

 m alternations of distance from the centre, symmetrically arranged, and m points 

 of condensation, or crowding of the satellites, which coincide with the points of 

 greatest distance when n is positive, and with the points nearest the centre 

 when n is negative. 12. 



71 



This system of undulations travels with an angular velocity -- relative to 



71 



the ring, and at - in space, so that during each oscillation of a satellite a 



771 



complete wave passes over it. 14. 



