ON THE STABILITY OF THE MOTION OF SATURN'S RINGS. 357 



potential (V) of the ring at the centre of the planet in terms of r and <$>. Then 

 the work done by any displacement of the system is measured by the change 

 of VS during that displacement. The attraction between the centre of gravity 



dV 

 of the Ring and that of the planet is S-j, and the moment of the couple 



dV 

 tending to turn the ring about its centre of gravity is S-j-r. 



It is proved in Problem V, that if a be the radius of a circular ring, r^ = af 

 the distance of its centre of gravity from the centre of the circle, and R the 



mass of the ring, then, at the centre of the ring, -,- = -- -/, -TJ = 0- 



CtT* Ot Ct(b 



It also appears that r^ = ^, , (1 +5 r ) which is positive when g > 1, 



Cb 



7JTT J3 



and that -y-r- 2 = f (3 g), which is positive when g<3. 



If -j-j- is positive, then the attraction between the centres decreases as the 



distance increases, so that, if the two centres were kept at rest at a given 



d'V 

 distance by a constant force, the equilibrium would be unstable. If -5-75 is positive, 



then the forces tend to increase the angle of libration, in whichever direction 

 the libration takes place, so that if the ring were fixed by an axis through its 

 centre of gravity, its equilibrium round that axis would be unstable. 



In the case of the uniform ring with a heavy particle on its circumference 

 whose weight = '82 of the whole, the direction of the whole attractive force of 

 the ring near the centre will pass through a point lying in the same radius as 

 the centre of gravity, but at a distance from the centre = fa. (Fig. 6.) 



If we call this point O, the line SO will indicate the direction and position 

 of the force acting on the ring, which we may call F. 



It is evident that the force F, acting on the ring in the line OS, will tend 

 to turn it round its centre of gravity R and to increase the angle of libration 

 KRO. The direct action of this force can never reduce the angle of libration 

 to zero again. To understand the indirect action of the force, we must recollect 

 that the centre of gravity (R) of the ring is revolving about Saturn in the 

 direction of the arrows, and that the ring is revolving about its centre of gravity 



