Resistance and Inductance of a Helical Coil. 87 



Let 16S=cos 2 *(5sin 2 a-l), .... (33) 



Then . ,, . /., , hr\ T n s 



vw = (i + 5) j o /( ^ + S J ° (Ar) ' 



and to order l/k 2 a 2 



A (kr) _ J. (*r) / _ 2Sr J, (£>•)! 

 A '(*r) _ J '(*r) t /( « 2 J«'(kr)f- 



Finally, if 



J«. (kr) _p , ,o K (t/w) _.„, Q , ,„.. 



5 = 2^(P+«Q){i-^(P+*Q)} 



- 2g, (P'+iQ') {l- -J" (P' + »Q')} • (35) 



In determining the effect of the concentration of current 

 upon the inductance and resistance, which in the former 

 case is all that is sought, only the terms involving P and Q 

 (and thus the resistivity) are needed. The effective resistance 

 is altered from its steady current value of ajirr 2 to the real 

 part of E/«r or 



R=-2/^Q+ / ^cos 2 a(l-5sin 2 a)PQ, . (36) 



and the self-induction is 



L = 2/.P-^cos 2 a(l-5sin 2 «)(P 2 -Q 2 ). . (37) 



The values of P and Q are well known. With Lord Kelvin's 

 definition of the functions her x, hei *, where 



#= *~i kr = r (4zir/jLp/a) \ 

 J (kr) = her .v + ibeix, .... (38) 

 and on reduction, 



-^Q=(ber.fbei'.r-bei^ber / .f)/{(ber , .i') 2 4-(bei^) 2 }l 

 .v? = (ber x ber' x -«-bei * bei' x)j{ (ber' xf + (bei' J) 2 } J * ( j 



