PROPAGATION OF PERIODIC CURRENTS 511 



The electric constituent of the impressed field is assumed to be 

 specified at every point along the wires by the impressed axial electric 

 force and the impressed potential. At any point x in any wire, h, 

 the impressed axial electric force will be denoted by fh(x) and the 

 impressed potential by Fhix) ; these are to be regarded as arbitrary 

 functions of x, and may even be discontinuous. 



The following set of electromotive forces is easily seen to be equiva- 

 lent to the above-specified arbitrary impressed field, in the sense of 

 producing the same currents and charges. This set will be termed 

 the 'fundamental ' set of equivalent electromotive forces; for, from the 

 physical viewpoint of this paper, it is in fact the fundamental set.^^ 



(A) In each wire a distributed axial electromotive force whose 

 value, per unit length, at each point is equal to the impressed 

 axial electric force there; thus, at any point x in wire h, an 

 electromotive force fh{x)dx in the differential length dx. 



{B) At each point where the impressed potential is discontinuous, 

 an axial electromotive force equal to the decrement in the 

 impressed potential there; thus, at any point of discontinuity 

 X = u in any wire //, an electromotive force equal to 



- AF,(«) = Fh{u -) - Fh{u -f ). 



(C) In each bridge an electromotive force equal to the imipressed 



voltage in that bridge; thus, in a bridge at any point x = b, 

 from wire h to any other wire k (or to ground), an electro- 

 motive force equal to Fkib) — Fk(b). 



(D) In case a point x = b where a bridge is situated coincides with 



a point X ^ 2i where the potential Fk(x) impressed on wire h 

 is discontinuous, the corresponding electromotive forces are 

 as follows: Axial electromotive forces equal to Fh(b — ) and 



— Fh{b -\-) at points b — and b + respectively in wire h; no 

 electromotive force in the bridge itself, which is connected 

 to the point b situated between b — and b -\- in wire h.^^'^ 



^' For a one- wire line and for a balanced two-wire line, five other sets of equivalent 

 electromotive forces are formulated in a later subsection. 



"" By supposing points b and u to be not quite coincident, say b = u — or 

 b = ti+, item (D) can be derived by first applying items (B) and (C) and then 

 applying the 'branch-point theorem' formulated in the second paragraph following 

 equation (75). 



A further application of the 'branch-point theorem' yields for item (D) the 

 following alternative set of electromotive forces: Axial electromotive forces each 

 equal to iFhib — ) — Fh{b +)^/2 at b — and b + in wire h; an electromotive force 

 equal to lFh{b — ) -|- Fhib -|-)]/2 in the bridge at b. Clearly, this set reduces to 

 (C) when Fh{x) is continuous at x = b, and it reduces to (5) when there is no bridge 

 at X = u. 



