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



with nearly the same velocity. If the angular velocity of the centre of gravity 

 about Saturn were always equal to the rotatory velocity of the ring, there 

 would be no libration. 



Now suppose that the angle of rotation of the ring is in advance of the 

 longitude of its centre of gravity, so that the line RO has got in advance of 

 SRK by the angle of libration KRO. The attraction between the planet and 

 the ring is a force F acting in SO. We resolve this force into a couple, whose 

 moment is F'RN, and a force F acting through R the centre of gravity of the 

 ring. 



The couple affects the rotation of the ring, but not the position of its centre 

 of gravity, and the force RF acts on the centre of gravity without affecting the 

 rotation. 



Now the couple, in the case represented in the figure, acts in the positive 

 direction, so as to increase the angular velocity of the ring, which was already 

 greater than the velocity of revolution of R about S, so that the angle of 

 libration would increase, and never be reduced to zero. 



The force RF does not act in the direction of S, but behind it, so that it 

 becomes a retarding force acting upon the centre of gravity of the ring. Now 

 the effect of a retarding force is to cause the distance of the revolving body to 

 decrease and the angular velocity to increase, so that a retarding force increases 

 the angular velocity of R about S. 



The effect of the attraction along SO in the case of the figure is, first, to 

 increase the rate of rotation of the ring round R, and secondly, to increase the 

 angular velocity of R about & If the second effect is greater than the first, 

 then, although the line RO increases its angular velocity, SR will increase its 

 angular velocity more, and will overtake RO, and restore the ring to its original 

 position, so that SRO will be made a straight line as at first. If this accelerat- 

 ing effect is not greater than the acceleration of rotation about R due to the 

 couple, then no compensation will take place, and the motion will be essentially 

 unstable. 



If in the figure we had drawn < negative instead of positive, then the 

 couple would have been negative, the tangential force on R accelerative, r would 

 have increased, and in the cases of stability the retardation of 6 would be greater 

 than that of (6 +<), and the normal position would be restored, as before. 



