Mutual Induction of a Circle and a Coaxal Helix. 57 



Then 



CC dx dx' + dy dy' + dz d^ 



_ p^p AaG0^{e-d')ded6' 



If we change the variables in this integral, putting 

 e-6' = (^, 



we find 



f 2-07-0 /'e fo C'2w-d> 



M= I i V##'+j \ Yd^d(^' 



J— e J— J2-S7-— ej— ^ 



r2'w/^2'o--0 



+ \ I Yd(f>d(P', . (I.) 

 «^o Jo 

 where 



Aacos^ 



~ \^A^ + a^-2Aacos(p + ¥^^ 

 _ Art cos (j) 



if «2=A2 + a2-2Artcos(^, 



Now 





Art cos ^ f ^ _ 1 P<^^ _^ 1_^ F</)'* 



2.4 



Cttj,, Art cos 0r,, 1- 1 ^^ ,/3 1 3 1 ^* J,- . -| 

 Artcos^PQ -, 



= ^ L^<P'J5 



where S^' = the series in brackets. 

 [It may be noticed that 



S,,= llog (^■<^+ ^^qil^^)- 1 log «.] 



It may be shown that S(p' is convergent if kef)' < «. Now 

 the maximum value of (j)' is 0, and the minimum value of a 

 is A — a. Hence the series is convergent for all values of cf)' 

 and a that occur in the second integration if ^©< A— a. So 

 long as this is the case we shall arrive, after performing the 

 second integration, at a convergent series. 



