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



(4) If n be of the form bJ-la, the first term being positive and 

 the second impossible, there will be a term in the solution of the form 



which indicates a periodic disturbance, whose amplitude continually increases 

 till it disarranges the system. 



(5) If n be of the form 6/ la, a negative quantity and an im- 

 possible one, the corresponding term of the solution is 



.Dc" 6 * cos (at + a), 

 which indicates a periodic disturbance whose amplitude is constantly diminishing. 



It is manifest that the first and fourth cases are inconsistent with the 

 permanent motion of the system. Now since equation (18) contains only even 

 powers of n, it must have pairs of equal and opposite roots, so that every 

 root coming under the second or fifth cases, implies the existence of another 

 root belonging to the first or fourth. If such a root exists, some disturbance 

 may occur to produce the kind of derangement corresponding to it, so that 

 the system is not safe unless roots of the first and fourth kinds are altogether 

 excluded. This cannot be done without excluding those of the second and fifth 

 kinds, so that, to insure stability, all the four roots must be of the third kind, 

 that is, pure impossible quantities. 



That this may be the case, both values of n* must be real and negative, 

 and the conditions of this are 



1st. That A, B, and C should be of the same sign, 

 2ndly. That & > 1A C. 



When these conditions are fulfilled, the disturbances will be periodic and 

 consistent with stability. When they are not both fulfilled, a small disturbance 

 may produce total derangement of the system. 



PROB. V. To find the centre of gravity, the radius of gyration, and the 

 variations of the potential near the centre of a circular ring of small but variable 

 section. 



Let a be the radius of the ring, and let 6 be the angle subtended at the 

 centre between the radius through the centre of gravity and the line through 

 a given point in the ring. Then if p be the mass of unit of length of the 



