ON THE STABILITY OF THE MOTION OF SATURN'S KINGS. 361 



as the sum of a series of sines and cosines of multiples of the variable. This 

 method, due to Fourier, is peculiarly applicable to the case of a ring returning 

 into itself, for the value of Fourier's series is necessarily periodic. We now 

 regard the form of the disturbed ring at any instant as the result of the 

 superposition of a number of separate disturbances, each of which is of the nature 

 of a series of equal waves regularly arranged round the ring. Each of these 

 elementary disturbances is characterised by the number of undulations in it, by 

 their amplitude, and by the position of the first maximum in the ring. 3. 



When we know the form of each elementary disturbance, we may calculate 

 the attraction of the disturbed ring on any given particle in terms of the con- 

 stants belonging to that disturbance, so that as the actual displacement is the 

 resultant of the elementary displacements, the actual attraction will be the 

 resultant of the corresponding elementary attractions, and therefore the actual 

 motion will be the resultant of all the motions arising from the elementary 

 disturbances. We have therefore only to investigate the elementary disturbances 

 one by one, and having established the theory of these, we calculate the actual 

 motion by combining the series of motions so obtained. 



Assuming the motion of the satellites in one of the elementary disturbances 

 to be that of oscillation about a mean position, and the whole motion to be 

 that of a uniformly revolving series of undulations, we find our supposition to 

 be correct, provided a certain biquadratic equation is satisfied by the quantity 

 denoting the rate of oscillation. 6. 



When the four roots of this equation are all real, the motion of each 

 satellite is compounded of four different oscillations of different amplitudes and 

 periods, and the motion of the whole ring consists of four series of undulations, 

 travelling round the ring with different velocities. When any of these roots 

 are impossible, the motion is no longer oscillatory, but tends to the rapid 

 destruction of the ring. 



To determine whether the motion of the ring is permanent, we must assure 

 ourselves that the four roots of this equation are real, whatever be the number 

 of undulations in the ring; for if any one of the possible elementary disturb- 

 ances should lead to destructive oscillations, that disturbance might sooner or 

 later commence, and the ring would be destroyed. 



Now the number of undulations in the ring may be any whole number 

 from one up to half the number of satellites. The forces from which danger 

 VOL. I. 46 



