612 TRANSACTIOMS OF SECTION A. 



To prove this it is only necessary to write the equation of reciprocity 

 S{(t>\Y^iH(pi^) - mSXY^v = 

 or 



It was shown by Joly tliat the screws reciprocal to c/jX, X must form the 

 three system — <^'X, X. But we have also shown that — mV/i;', Vc^/LK^y by varying 

 /i and V will form the system reciprocal to ^X. 



Hence if we operate on \(})fi<pv with - (p' we must come to —mVfiv, or 



(f)'y<plj.<f>v = wtV/iK 



which may be written 



(})Y(j)'X(f)'fi = HiVX^. 



This shows that V(^'X(/)V is a linear vector function of VX/j. 



2. The Inductance of Two Parallel Wires. 

 By J. AV. Nicholson, D.Sc, B.A. 



AVhen direct and return currents flow in two wires of great length, and the 

 alternation is not rapid, the effective self-induction L per unit length of the 

 system may be calculated readily by simple integration.' 



If the wires have radii (a, h) and permeabilities {fi, v), and if c be the distance 

 between their axes, 



L = 2 log |!. !(,..). 



But this formula is generally of little practical use when the frequency of alter- 

 nation is several thousands per second. Such frequencies are of constant use iu 

 practical work, where pairs of long wire leads are continually employed, whose 

 self-induction must be small. A knowledge of the effective self-induction of such 

 leads is therefore very desirable. The general case presents considerable mathe- 

 matical difficulty, but the solutions given below appear to include most cases of 

 practical utility. , _ 



The mode of proof depends chiefly upon a transformation of a series of Bessel 

 functions of type K,, finite at infinity, to another series similarly related with 

 respect to a new origin. The disturbances caused by one wire in the vector 

 potential due to the other may be summed to a high order of accuracy. From a 

 knowledge of the vector potential, the total current in each wire, and thus the 

 self-induction, may be deduced. 



The result for wires not too close together is 



T _cy i.~. c" 2/1 bera; ber'.i+bei.r bei'j," 

 "" ^ ah ^ ■ " (ber'.r)^T(bli'j^)' 



2v ber y ber'y + beiy bei'y 

 7 • ~(ber'2/)'^+(bei'l^^~ ' 



where (/i, i') are the permeabilities of the wires, {it, a') their resistivities. 



and ber .r, bei x are the functions introduced by Lord Kelvin, and afterwards 

 tabulated. Their main properties are well known to mathematical electricians. 



In this result, the greatest possible error is of magnitude »i-c-/y* relative to 

 luiity, where v= 3.10'". For two wires one centimetre apart, and with a frequency 



' Kussell, Alternating Currents, i. p. 56. 



