366 ON THE STABILITY OF THE MOTION OP SATURN'S RINGS. 



circle, which in strictness ought to be an ellipse not differing much from the 

 circle. 



As the satellites move in their orbits in the direction of the arrows, the 

 curve which they form revolves in the same direction with a velocity 1 times 

 that of the ring. 



By considering these figures, and still more by watching the actual motion 

 of the ivory balls in the model, we may form a distinct notion of the motions 

 of the particles of a discontinuous ring, although the motions of the model are 

 circular and not elliptic. The model, represented on a scale of one-third in figs. 

 7 and 8, was made in brass by Messrs. Smith and Ramage of Aberdeen. 



We are now able to understand the mechanical principle, on account of 

 which a massive central body is enabled to govern a numerous assemblage of 

 satellites, and to space them out into a regular ring ; while a smaller central 

 body would allow disturbances to arise among the individual satellites, and 

 collisions to take place. 



When we calculated the attractions among the satellites composing the ring, 

 we found that if any satellite be displaced tangentially, the resultant attraction 

 will draw it away from its mean position, for the attraction of the satellites it 

 approaches will increase, while that of those it recedes from will diminish, so that 

 its equilibrium when in the mean position is unstable with respect to tangential 

 displacements ; and therefore, since every satellite of the ring is statically unstable 

 between its neighbours, the slightest disturbance would tend to produce collisions 

 among the satellites, and to break up the ring into groups of conglomerated 

 satellites. 



But if we consider the dynamics of the problem, we shall find that this 

 effect need not necessarily take place, and that this very force which tends 

 towards destruction may become the condition of the preservation of the ring. 

 Suppose the whole ring to be revolving round a central body, and that one 

 satellite gets in advance of its mean position. It will then be attracted forwards, 

 its path will become less concave towards the attracting body, so that its distance 

 from that body will increase. At this increased distance its angular velocity 

 will be less, so that instead of overtaking those in front, it may by this means 

 be made to fall back to its original position. Whether it does so or not must 

 depend on the actual values of the attractive forces and on the angular velocity 

 of the ring. When the angular velocity is great and the attractive forces small, 



