312 ON THE STABILITY OF THE MOTION OF SATUKN's RINGS. 



Multiply by COB msds and integrate, then all terms of the form 



J cos ms cos nsds and / cos ms sin nsds 

 will vanish, if we integrate from 5 = to s = 2ir, and there remains 



U cos msds = irA m , U sin msds = irB n . 



If we can determine the values of these integrals in the given case, we 

 can find the proper coefficients A m , B m , &c., and the series will then represent 

 the values of U from s = to s = 2ir, whether those values be continuous or 

 discontinuous, and when none of those values are infinite the series will be 

 convergent. 



In this way we may separate the most complex disturbances of a ring into 

 parts whose form is that of a circular function of s or its multiples. Each of 

 these partial disturbances may be investigated separately, and its effect on the 

 attractions of the ring ascertained either accurately or approximately. 



4. To find the magnitude and direction of the attraction between two 

 elements of a disturbed ring. 



Let P and Q (fig. 4) be the two elements, and let their original positions 

 be denoted by s, and s,, the values of the arcs BP, BQ before displacement. 

 The displacement consists in the angle BSP being increased by o-j and BSQ 

 by <TJ, while the distance of P from the centre is increased by p t and that of 

 Q by p s . We have to determine the effect of these displacements on the distance 

 PQ and the angle SPQ. 



Let the radius of the ring be unity, and s t s 1 = 20, then the original 

 value of PQ will be 2 sin 6, and the increase due to displacement 



= (p a + pi) sin 6 + (<r a - o-,) cos 0. 

 We may write the complete value of PQ thus, 



} ............... (1). 



The original value of the angle SPQ was n~^ anc ^ ^ ne i ncrease due to 

 displacement is i (p PI) cot (<r, <r,), 



