ON THE STABILITY OP THE MOTION OF SATURN'S RINGS. 299 



In this case r and <j) are constant, and therefore V and its differential 

 coefficients are given. Equation (7) becomes, 



dV 



j~ = Q > 

 a/- 

 which shews that the angular velocity is constant, and that 



R + SdV 



Hence, -5-5 = 0, and therefore by equation (8), 



%= e* 



Equations (9) and (10) are the conditions under which the uniform motion 

 is possible, and if they were exactly fulfilled, the uniform motion would go on 

 for ever if not disturbed. But it does not follow that if these conditions were 

 nearly fulfilled, or that if when accurately adjusted, the motion were slightly 

 disturbed, the motion would go on for ever nearly uniform. The effect of the 

 disturbance might be either to produce a periodic variation in the elements 

 of the motion, the amplitude of the variation being small, or to produce a 

 displacement which would increase indefinitely, and derange the system altogether. 

 In the one case the motion would be dynamically stable, and in the other it 

 would be dynamically unstable. The investigation of these displacements while 

 still very small will form the next subject of inquiry. 



PROB. II. To find the equations of the motion when slightly disturbed. 

 Let r = r a , & = tat and <j> = <j> in the case of uniform motion, and let 



r=r + r lt 



when the motion is slightly disturbed, where r lt lt and ^, are to be treated 

 as small quantities of the first order, and their powers and products are to be 



dV dV 

 neglected. We may expand -5 and -j-r by Taylor's Theorem, 



dV = dV 



dV^dV 



382 



