ON THE STABILITY OF THE MOTION OP SATURN'S RINGS. 293 



and p its density ; and X the ratio of the breadth of the ring to its thick- 

 ness. The equation for determining X when e is given has one negative root 

 which must be rejected, and two roots which are positive while e<0'0543, and 

 impossible when e has a greater value. At the critical value of e, X = 2'594 

 nearly. 



The fact that X is impossible when e is above this value, shews that the 

 ring cannot hold together if the ratio of the density of the planet to that of 

 the ring exceeds a certain value. This value is estimated by Laplace at 1*3, 

 assuming a = 2R. 



We may easily follow the physical interpretation of this result, if we observe 

 that the forces which act on the ring may be reduced to 



(1) The attraction of Saturn, varying inversely as the square of the dis- 

 tance from his centre. 



(2) The centrifugal force of the particles of the ring, acting outwards, and 

 varying directly as the distance from Saturn's polar axis. 



(3) The attraction of the ring itself, depending on its form and density, 

 and directed, roughly speaking, towards the centre of its section. 



The first of these forces must balance the second somewhere near the mean 

 distance of the ring. Beyond this distance their resultant will be outwards, 

 within this distance it will act inwards. 



If the attraction of the ring itself is not sufficient to balance these residual 

 forces, the outer and inner portions of the ring will tend to separate, and the 

 ring will be split up; and it appears from Laplace's result that this will be 

 the case if the density of the ring is less than $ of that of the planet. 



This condition applies to all rings whether broad or narrow, of which the 

 parts are separable, and of which the outer and inner parts revolve with the 

 same angular velocity. 



Laplace has also shewn (Liv. v. Chap, in.), that on account of the oblate- 

 ness of the figure of Saturn, the planes of the rings will follow that of Saturn's 

 equator through every change of its position due to the disturbing action of 

 other heavenly bodies. 



Besides this, he proves most distinctly (Liv. in. Chap, vi.), that a solid uni- 

 form ring cannot possibly revolve about a central body in a permanent manner, 

 for the slightest displacement of the centre of the ring from the centre of the 

 planet would originate a motion which would never be checked, and would 



