ON THE STABILITY OF THE MOTION OP SATURN'S KINGS. 311 



The ring which we consider is therefore small in section, and very nearly 

 circular and uniform, and revolving with nearly uniform velocity. The variations 

 from circular form, uniform section, and uniform velocity must be expressed by a 

 proper notation. 



2. To express the position of an element of a variable ring at a given time 

 in terms of the original position of the element in the ring. 



Let S (fig. 3) be the central body, and SA a direction fixed in space. 



Let SB be a radius, revolving with the mean angular velocity w of the 

 ring, so that ASB = <at. 



Let TT be an element of the ring in its actual position, and let P be the 

 position it would have had if it had moved uniformly with the mean velocity a 

 and had not been displaced, then BSP is a constant angle =s, and the value 

 of s enables us to identify any element of the ring. 



The element may be removed from its mean position P hi three different 

 ways. 



(1) By change of distance from S by a quantity pir = p. 



(2) By change of angular position through a space Pp = a. 



(3) By displacement perpendicular to the plane of the paper by a quantity . 



p, cr and are all functions of s and t. If we could calculate the attrac- 

 tions on. any element as depending on the form of these functions, we might 

 determine the motion of the ring for any given original disturbance. We cannot, 

 however, make any calculations of this kind without knowing the form of the 

 functions, and therefore we must adopt the following method of separating the 

 original disturbance into others of simpler form, first given in Fourier's Traite 

 de Chaleur. 



3. Let U be a function of s, it is required to express U in a series of 

 sines and cosines of multiples of s between the values s = and s = 2jr. 



Assume U=A l cos s + A 3 cos 2s + &c. + A m cos ms + A n cos ns 

 + B l sin s + B t cos 2s + &c. + B m sin ws + B n sin ns. 



