On the Sum of a Series of Cosecants. Ill 



Ii(a* + b K ) »=i + n 

 therefore (Af , -. 



n u(b K -b n ) n (b K -b n ) 



Similarly ,, 2 >. » t> 



II (*» + &*) s* +0n 



B _ <j>(—b n ) 



n u\b K -b n ) n (bc-b^' 



and therefore 



Snft-y n (& B -&„) * T °" 



University of Pennsylvania, 

 Philadelphia, Pa., U.S.A. 



XVI. 77ie £wra o/" a Series of Cosecants. By Gr. N. Watson, 



iyr.JL., Fellow of Trinity College, Cambridge, Assistant 



Professor of Pure Mathematics at University College, 



London *. 



1. T~N modern theories of the structure of the atom, the 



JL usual assumption is that it consists of a ring of 



electrons (repelling one another according to the ordinary 



inverse square law) rotating, in the normal configuration of 



the atom, in a symmetrical manner round a small positive 



charge. 



If n be the number of electrons, each of charge — e and 

 mass m, and the positive charge be of magnitude ve (where 

 v may or may not be equal to ri), while the radius of the 

 atom is a, it is easily shown that, in steady circular motion, 

 the angular velocity a> of the system is given by the 

 formula 



O) 2 



e t 

 = — §( v - 

 mar 



■is„), 











.e 



7T 



= cosec — 

 n 



n-l 



= % cosec 



-r cosec 

 (nnrjn). 



2jr + 

 n 





in- 

 . +cosec^- 



-IV 



n 





m=l 



* Communicated 



by 



the Author. 





