112 Prof. G. N. Watson on the 



The expression v — ^S n plays a very important part in the 

 investigation of oscillations about the steady motion, as well 

 as in Bohr's theory of spectra; and an analogous expression 

 obviously occurs in the gravitational problem of a number of 

 equal satellites rotating in a circle about a planet — a simplified 

 form of the problem of Saturn's rings, considered by Clerk- 

 Maxwell. 



Reference mav also be made to Sir J. J. Thomson, 

 Phil. Mag. (6) vii. (1904) p. 237, to G. A. Schott, ' Electro- 

 magnetic Radiation,' and to various recent papers by J. W. 

 Nicholson * in the Monthly Notices of the R.A.S. 



For large values of n, the labour of numerical calculation, 

 in determining S m directly, is obviously severe ; to avoid 

 this labour the need arises for a suitable asymptotic formula 

 for the sum 



n-l 



S n = £ cosec (m/irn) ; 



m=l 



in this paper a complete asymptotic expansion of S„ is 

 obtained, and, by means of this expansion, S« is tabulated 

 for various values of n from 6 to 100 and also for n = 360, 

 w=1000. 



2. By using Euler's well-known formula 



i °° X a ~ Y d0C 7T , n ,. 



\ -T~, =-• , (0<ca<l) 



J 1 + x sin air v 7 



we see at once that 



' S * = [ 7TT x {^ + «^+ ■ • • + « (n - 1)/B }ia? 

 Jo #(! + #) 



{^ n —x)doc 



scil + xyil-x 1 !") 



(l-ry n )(i-y) 

 (l-g-<n-D*)<ft 



.(l + «- f )(«'-"l) 



p (l- g -(-^/ 



Jo (14-^-^(^-1)' 



on summing the geometrical progression and then writing 

 x iin—y — e -t. } ^ e -g na } s ^ e p ? £ bisecting the range of inte- 

 gration, is justified by the fact (which is readily verified) 

 that the integrand is an even function of t. 



* To whom I am indebted for pointing out the desirability of dis- 

 covering the results contained in this paper. 



ITl 



Jo 



^0 



-i 



