﻿584 Mr. S. Butterworth on the Coefficients of 



by integration, to find the mutual induction between two 

 solid coils. 



Certain reduction form alee are required, and these will be 

 dealt with first. If E and K are complete elliptic integrals 

 of the first and second kind to modulus x, then 



<ZE_ E-K <7K = E _K 



dx ' x ' dx x(l-x 2 ) x' ' ' K ' } 



From (21) we readily derive the following reduction 

 formulae, 



(n + 2)(x n Bdx=x n + 1 B+(x n Kdx (22) 



(n + 2) 2 (x n + 2 Kdx=x n + 1 E-(n + 2)x n +\l-x 2 )K 



^ C 



+ (n + l) 2 \x n Kdx . . . (23) 



j>=(H K - 3 ?' ~ ^ 



by means of which I x n Bdx, I x n ~Kdx can be expressed in 



terms of E, K, j Kdx and \ — , i£ n is any integer positive 



or negative. 



In the succeeding work we are particularly concerned 

 with 



f * C 1 K 



h=\ Kdx, and v=\ — dx. . . (25) 



^0 Jx * 



Expressing K in series, 



7T f -/1.3.5 2>*-1\ 2 a?2« -i 



W= 2n 1+ ?(2T476T772^)2^ri.h (2b) 



When *=1, 



7T 



sin0 



f^f 1 ±x nod 



u = u l =\ dV 1 — . = ! — 



Jo Jo (l-^' 2 sin^> l o sin 



= 2(l- jj + vj- ...)=1'83193. . . (27) 



