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



By referring to the original expression for the variable section of the ring, 

 it appears that the effect of the coefficient / is to make the ring thicker on 

 one side and thinner on the other in a uniformly graduated manner. The effect 

 of g is to thicken the ring at two opposite sides, and diminish its section in 

 the parts between. The coefficient h indicates an inequality of the same kind, 

 only not symmetrically disposed about the diameter through the centre of 

 gravity. 



Other terms indicating inequalities recurring three or more times in the 

 circumference of the ring, have no effect on the values of L, M and N. There is 

 one remarkable case, however, in which the irregularity consists of a single 

 heavy particle placed at a point on the circumference of the ring. 



Let P be the mass of the particle, and Q that of the uniform ring on 

 which it is fixed, then R = P + Q, 



(27). 



PROB. VI. To determine the conditions of stability of the motion in terms 

 of the coefficients /, g, h, which indicate the distribution of mass in the ring. 



The quantities which enter into the differential equation of motion (18) 

 are R, S, J?, r,, *, L, M, N. We must observe that S is very large compared 

 with R, and therefore we neglect R in those terms in which it is added to S, 

 and we put 



* = a' (I-/'). 



5? 



~ 



