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



When U' is positive, A will be of the same sign as P, that is, the parts 

 of the ring will be furthest from the centre where the disturbing force towards 

 the centre is greatest. When U' is negative, the contrary will be the case. 



When v is positive, B will be of the opposite sign to A, and the parts 

 of the ring furthest from the centre will be most crowded. When v is negative, 

 the contrary will be the case. 



Let us now attend only to the tangential force, and let us put q = 0. We 

 find in this case also a = 0, /8 = 0, 



Q (60). 



The tangential displacement is here in the same or in the opposite direc- 

 tion to the tangential force, according as U' is negative or positive. The 

 crowding of satellites is at the points farthest from or nearest to Saturn 

 according as v is positive or negative. 



16. The effect of any disturbing force is to be determined in the following 

 manner. The disturbing force, whether radial or tangential, acting on the ring 

 may be conceived to vary from one satellite to another, and to be different at 

 different times. It is therefore a perfectly arbitrary function of s and t. 



Let Fourier's method be applied to the general disturbing force so as to 

 divide it up into terms depending on periodic functions of s, so that each term 

 is of the form F (t) cos (ras + a), where the function of t is still perfectly arbitrary. 



But it appears from the general theory of the permanent motions of the 

 heavenly bodies that they may all be expressed by periodic functions of t 

 arranged in seriea Let vt be the argument of one of these terms, then the 

 corresponding term of the disturbance will be of the form 



P cos (ms + vt + a). 



This term of the disturbing force indicates an alternately positive and 

 negative action, disposed in m waves round the ring, completing its period 



