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



For the planet 



Rr cW R 



df 



, . 



dt ^R di 



Rr \ Rr dff* 



For the centre of gravity of the ring, 



Sr V dff\ S dV 



- * ' 



d (I 



J{ dt\\ 



s+R 



Sr \ & AM dV 



s+R di 



For the rotation of the ring about its centre of gravity, 



<+- a? .............................. < 5 >- 



where A; ia the radius of gyration of the ring about its centre of gravity. 



Equation (3) and (4) are necessarily identical with (1) and (2), and shew 

 that the orbit of the centre of gravity of the ring must be similar to that 

 of the Planet. Equations (l) and (3) are equations of areas, (2) and (4) are 

 those of the radius vector. 



Equations (3), (4) and (5) may be thus written, 



These are the necessary and sufficient data for determining the motion of 

 the ring, the initial circumstances being given. 



PROB. I. To find the conditions under which a uniform motion of the 

 ring is possible. 



By a uniform motion is here meant a motion of uniform rotation, during 

 which the position of the centre of the Planet with respect to the ring does 

 not change. 



