ON THE STABILITY OF THE MOTION OP SATURN'S RINGS. 327 



we shall find the value of A cos a and B cos y8 ; 



Acoaa = - - Pcos^H -- - Qcosq ............. (54). 



Bcoafi= T Pcosp , Qcosq (55), 



Substituting sines for cosines in equations (51), (52), we may find the 

 values of A sin a and B sin y8. 



Now U' is precisely the same function of v that U is of n, so that if v 

 coincides with one of the four values of n, U' will vanish, the coefficients A 

 and B will become infinite, and the ring will be destrpyed. The disturbing 

 force is supposed to arise from a revolving body, or an undulation of any kind 



which has an angular velocity relatively to the ring, and therefore an 



absolute angular velocity = a . 



If then the absolute angular velocity of the disturbing body is exactly or 

 nearly equal to the absolute angular velocity of any of the free waves of the 

 ring, that wave will increase till the ring be destroyed. 



The velocities of the free waves are nearly 



1\ 1 r~l~ I /~T~ / 1\ 



, o>+ fZ-RN. <a 3-RN, and <a (1 (56). 



m/ m V P- m \ p \ m/ 



When the angular velocity of the disturbing body is greater than that of 

 the first wave, between those of the second and third, or less than that of 

 the fourth, U' is positive. When it is between the first and second, or between 

 the third and fourth, U' is negative. 



Let us now simplify our conception of the disturbance by attending to the 

 central force only, and let us put p = 0, so that P is a maximum when ms + vt 

 is a multiple of 2-n. We find in this case a = 0, and = 0. Also 



1 



