318 ON THE STABILITY OF THE MOTION OF SATURN'S KINGS. 



Now if we put n= 



which is negative if S be large compared to R. 



So that a ring of satellites can always be rendered stable by increasing 

 the mass of the central body and the angular velocity of the ring. 



The values of L, Af, and N depend on TO, the number of undulations in 

 the ring. When m = ~, the values of L and N will be at their maximum 



t 



and M=0. If we determine the relation between S and R in this case so 

 that the system may be stable, the stability of the system for every other 

 displacement will be secured. 



8. To find the mass which must be given to the central body in order 

 that a ring of satellites may permanently revolve round it. 



We have seen that when the attraction of the central body is sufficiently 

 great compared with the forces arising from the mutual action of the satellites, 

 a permanent ring is possible. Now the forces between the satellites depend on 

 the manner in which the displacement of each satellite takes place. The con- 

 ception of a perfectly arbitrary displacement of all the satellites may be rendered 

 manageable by separating it into a number of partial displacements depending 

 on periodic functions. The motions arising from these small displacements will 

 take place independently, so that we have to consider only one at a time. 



Of all these displacements, that which produces the greatest disturbing 

 forces is that in which consecutive satellites are oppositely displaced, that is, 



when in = ^, for then the nearest satellites are displaced so as to increase as 



A 



much as possible the effects of the displacement of the satellite between them. 

 If we make fi a large quantity, we shall have 





-5259, M=0, N=2L, K very small. 



7T 



