PROPAGATION OF PERIODIC CURRENTS 527 



impressed field, since this constituent acts on the circuit consisting 

 of the two wires in parallel with each other, with ground return, which 

 is formally the same as a one-wire line with ground return. 



All of the sets of equivalent electromotive forces become applicable 

 to dealing with the mode-a constituent of the impressed field by an 

 appropriate interpretation of the diagrams (Fig. 2, • • • 11), namely, the 

 following interpretation : 



1. In each diagram regard the wire-symbol as representing the out- 

 going wire of the actual two-wire line, and regard the ground-symbol 

 as representing not the ground but the return wire of the two-wire line. 

 (The presence of the earth is then to be regarded as implied, its effects 

 appearing implicitly in the values of the line parameters.) 



2. Hence regard Zo and Zs as denoting the mode-a terminal im- 

 pedances functioning as though connected directly across the two- 

 wire line at its ends x = and x = s respectively. 



3. Regard Y', Y^, Y, Z as denoting the mode-a line constants 

 (including implicitly the efi'ects of the earth). 



4. Regard /(x), F(x), and $(.v) as denoting the mode-a constituents 

 of the impressed field — that is, as denoting the difference of the actual 

 values impressed at the two wires. (In order to maintain the balanced 

 condition of the two-wire line, f(x) is to be regarded as constituted of 

 fix) {2 in the outgoing wire and — f(x)/2 in the return wire; and simi- 

 larly for F(x) and $(x).) 



Tlie Electric Field Due to a System of n Parallel Wires in an Arbitrary 



Impressed Field 



Thus far in the present section of this paper the field impressed on 

 the given physical system has been supposed known and the problem 

 has been to calculate the resulting currents. Actually, however, the 

 impressed field is not usually known but has to be calculated — from a 

 knowledge of the currents and charges producing it. 



The present subsection deals with the problem of calculating the 

 electric field impressed on a secondary system consisting of a single 

 horizontal wire f by a primary system tt consisting of n wires which 

 are parallel to each other and to j. For generality, the primary and 

 secondary systems are supposed to be in an arbitrary impressed field. ^^ 



Consider at first any parallel geometrical line i, not necessarily in 

 any of the wires; and let Vi = Vi{x) and Ei = Ei{x) denote the 



^^ Of course the field produced by any given system is directly due only to the 

 currents and charges of the system, and does not depend directly on any field that 

 may be impressed on the system; but, assuming the system to be energized only by 

 the impressed field, the currents — and thence the charges — are directly due to the 

 impressed field and can (theoretically, at least) be expressed in terms of it. 



