204 Prof. G. M. Minchin on the Magnetic 



and for the second, let 



=COt 2 V, • • \/ i =- • • • (10) 



1 — v A ' V 1 — v sin^ 



Hence we have 



12 = 2tt- 2 | — . k! sin ^II( - 1 + k' 2 sin 2 ^, £) 



+ ? :±^_i_n (cot 2 *,£)}. . (ii) 



The values of these complete elliptic integrals of the third 

 kind are well known in terms of integrals of the first and 

 second kinds. Thus (see Hymers's ' Integral Calculus/ 

 section ix., or any treatise on Elliptic Functions) we have the 

 equation 



^^ { n(-l + ^sin 2 ^.)-K } . 



= |-K.E(^^)+(K-E).K(//,^). . (12) 

 Now 



E {¥, f ) = E' - f ^l-£' 2 sin 2 ^ . df, 



a ^ 



and 



'*/ 2 dyjr 



K ( F^)=K'-j;- 7T 3 



k' 2 sin 2 -yjr 



where E' and K' are the complete integrals with modulus ¥. 



Also it is well known that KE' + K'E - KK ; - £ = 0. Hence 



the right-hand side of (12) becomes, by expanding and 

 neglecting powers of k! beyond the second, 



so that 



k' 2 sin f n ( - 1 + k' 2 sin 2 f, k) 



7T 7T 



= & /2 sin,/r.K + E- ■ *-{ (2K-E)sin^ + - — - (2K-Ecos2^) 



7 cos y 4 I v r cos ijr K r ' 



7T . 7T , 



o -+ 1/9 r O "^ 1 



;=E- r + ^-4 (2K+E)sin^-- — r f2K-Ecos2^) I. (13) 



COS t/t 4 I. r COS i/r v T/ j \ J 



