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



Recapitulation of tfie Theory of a Ring of equal Satellites. 



In attempting to conceive of the disturbed motion of a ring of unconnected 

 satellites, we have, in the first place, to devise a method of identifying each 

 satellite at any given time, and in the second place, to express the motion of 

 every satellite under the same general formula, in order that the mathematical 

 methods may embrace the whole system of bodies at once. 



By conceiving the ring of satellites arranged regularly in a circle, we may 

 easily identify any satellite, by stating the angular distance between it and a 

 known satellite when so arranged. If the motion of the ring were undisturbed, 

 this angle would remain unchanged during the motion, but, in reality, the 

 satellite has its position altered in three ways : 1st, it may be further from 

 or nearer to Saturn; 2ndly, it may be in advance or in the rear of the position 

 it would have had if undisturbed ; 3rdly, it may be on one side or other of 

 the mean plane of the ring. Each of these displacements may vary in any way 

 whatever as we pass from one satellite to another, so that it is impossible 

 to assign beforehand the place of any satellite by knowing the places of the 

 rest. 2. 



The formula, therefore, by which we are enabled to predict the place of 

 every satellite at any given time, must be such as to allow the initial position 

 of every satellite to be independent of the rest, and must express all future 

 positions of that satellite by inserting the corresponding value of the quantity 

 denoting time, and those of every other satellite by inserting the value of the 

 angular distance of the given satellite from the point of reference. The three 

 displacements of the satellite will therefore be functions of two variables the 

 angular position of the satellite, and the time. When the time alone is made 

 to vary, we trace the complete motion of a single satellite ; and when the time 

 is made constant, and the angle is made to vary, we trace the form of the 

 ring at a given time. 



It is evident that the form of this function, in so far as it indicates the 

 state of the whole ring at a given instant, must be wholly arbitrary, for the 

 form cf the ring and its motion at starting are limited only by the condition 

 that the irregularities must be small. We have, however, the means of breaking 

 up any function, however complicated, into a series of simple functions, so that 

 the value of the function between certain limits may be accurately expressed 



