Field of an Electrical Current. 



R 



205 



Now since k' = — the first term in brackets in (11) can be 

 P 



written 



. ft* sin fn (- 1 + ft' 9 sin 2 ^r, ft), 



B 



and we have 



n=29r.-2-T 7 ^.ft , ?sm^n(-l + ft /2 sin 2 ^,ft) 



+ !±£. J-n(cot 2 x^-)), . (H) 



p Sill % ^ J 



in which the value of the first elliptic integral is to be sub- 

 stituted from (13). 



Dealing now with the second elliptic integral, its value is 

 given by the known equation 



t ( x'£ x n (cot2 * *) = J + K < tan X^', X) -E \V, X) > 



+ (K-E).K(*', X ). . (15) 

 But observe that 



sin X=2 a sin(9 ( 1 + - cos6, )> 



so that v is a small quantity of the same order as ¥. Hence, 



E 3 



if in the coefficient of K we neglect quantities of the order — , 



n(cot 2 x ,A) 



sm x 



= cos X {|+(K-E). X } 



= (l-i5sin^)[| + £ S in^(l + ?co S e)(K-E)].(16) 



i ~R2 



The coefficient to the second order is 1 + ^— 2 sin 2 0, so 



that we have P * a 



r + u 



9 sin^ 

 Hence 



^n(cot 2 % ,^=^ + ^sin^fl+-cos^(K-E). (17) 



n = 2 -- 2 {V a [ E f^ + T> K+E ) s!n * 



77 f 



9~t 



^(2K-Ecos2^)}] + | + gsin^(l+?cos^(K--E)}.(18) 



