Inductance of Two Parallel Wires. 257 



In the wire, the current may be regarded as due to ohmic 

 conductivity alone, and writing 



k 2 =—4:7rfiip/a; (5) 



then B 2 H 1BH 1 3 2 H _ 



^ + r^7 + r>W 2 +kR -°> ' ' ' (6) 



whose solution finite at the axis of the wire is 



H= 2* A n J n (kr) cos n0, .... (7) 

 where 



^ n[kr) -Wn~\ L ¥l\n + l + '2^bi + ln + 2'~ J" ( } 



and is the ordinary Bessel function of the first kind. In 



the surroundino- medium, may be neglected, an{ j writing 



l/K/x=C 2 , where is the velocity of propagation in the 

 medium, 



Or- r or r" o#' v ; 



where h=p/Q. 



The solution proper to the space outside the wire, satisfying 

 the condition of being finite at infinity, is 



H= X B n K* (lAr) cos n0, . . (10) 



where 



K» (*/*>■) = I tf- ,A '' cosh 0cosh»0^. . . (11) 



^ 



This function is connected with the more usual Bessel 

 function of the second kind, introduced by Hankel, by the 

 formula* 



K n (thr) = -U-» {Y n (hr) + nr J n (hr)} , . (12) 

 h being real, and, if 7= '577 ... 

 Y, W= 2J. W { fcg | +7 }-(|)-{.-l 1+ ^!(|)'...} 



B"{.Virail6 +S -)G)' + ^W:(I * 1 ♦«■)©*+...}.«> 



where a . 1 1 1 



S»=l+^+ s + ... + -. ... (14) 



* Vide e.g. Somne, Math. Ann. 16, 18S0. 



