﻿based on Free Electrons. 1079 



§ 12. If we apply the trigonometrical expansion of the 

 cosecant * 



S cosec sir n = 2, \ + 2, - v . . .,.,,, 

 s =i s =i Utt r=i7r(5 2 — Hn 2 )J 



we have 



— 2 cosec sir n = 2, - + 1-2 - h 2, - — ^— L 



h j-! ' s= i Ls n — 5 r== i7*ii+s r =if'n-\-n — sj 



or ... (15 a) 



--r- 2 cosec sir in = 2 2 — , (15 6) 



(15 a) being a general form, (15 b) only suited to the special 

 range of s. 

 The series is 



11 1 1 / 1_ 1 1 1_\ 



I + 2 + ■'* n— 1 2n \2n + n + 1 + **•* 2n-l + hi) 



+ (f+^T+---^-r+j 1 )-' ( 16 ) 



\4w 2n + l 3>i — 1 6>i/ 



Bat 



II 111 1 1 ,m n 



i + 2 + - w -ri + 27 1 = 1 °g' i + f y-r2i? + T2o^-' (17a) 



and the series in the first bracket of (16) is the difference 

 between this formula as written for n and 2n, the next 

 bracket the difference as written for 2n and 3rc, and 

 so on. 



The first bracket is 



1 2 i * f 1 l\ L_/l L\ 



g 1 + 12n 2 V 1 2 " 2 V 120n 4 U 4 "" 2V "" " ' 



* I am indebted to Professor Proudman for the suggestion of this 

 method, and a rough sketch showing the term log 7r/2. 



