PROPAGATION OF PERIODIC CURRENTS 517 



current (the current circulating in the metallic loop), which is equal 

 to the mode-a current in one of the wires and hence to the negative 

 of the mode-a current in the other wire. 



Six Different Sets of Equivalent Electromotive Forces for a One-wire ^* 

 Line in an Arbitrary Impressed Field 



The physical system here contemplated (Fig. 2 below) is a one-wire 

 transmission line consisting of a uniform horizontal straight wire 

 situated in an arbitrary impressed field and terminated" in any arbi- 

 trary impedances to ground. The wire extends from x = to .r = 5; 

 the arbitrary terminal impedances ^^ are denoted by Zo and Z^. The 

 arbitrary impressed field is specified, at each point of the wire, by the 

 impressed axial electric force /(x) and the impressed potential F{x), as 

 previously. 



Six different sets of 'equivalent electromotive forces' are formulated 

 in the early part of this subsection ; while their derivations are briefly 

 outlined toward the latter part. Set 1 will be recognized as a par- 

 ticular case of the 'fundamental' set already formulated in the early 

 part of Section III. The five remaining sets are derived from Set 1. 

 In the actual formulations of these various sets of equivalent electro- 

 motive forces, the impressed potential F{x) is assumed to be a con- 

 tinuous function of x; the extension to the case where F{x) is dis- 

 continuous is a simple matter and is formulated in connection with 

 equations (65) and (66). 



In the following diagrams (Figs. 2, • • • 11) it is found convenient 

 to represent any localized electromotive force by the conventional 

 battery-symbol. This symbol is intrinsically directional; the longer 

 of the two plates is to be regarded as at the higher potential, so that 

 there is an internal rise of potential in passing through the symbol 

 from the shorter to the longer plate. 



In some of the figures the actual line is represented as replaced by 

 the corresponding artificial line composed of differential elements, each 

 of length dx. (For clearness, the line is represented as composed of 

 only a small number of such elements.) 



The letters Z, Y, Y', F" denote certain line parameters per unit 

 length, as follows: Z and Y respectively denote the 'complete series 

 impedance' and the 'complete shunt admittance' or, briefly, the 

 'series impedance' and the 'shunt admittance.' These may be re- 

 garded as defined by the equations 



Z= yK, Y = ylK, 



^^ The case of a balanced two-wire line is outlined in the next subsection. 



" See also the remarks under the subheading following shortly after equation (67). 



