518 BELL SYSTEM TECHNICAL JOURNAL 



7 denoting the propagation constant of the line per unit length, and 

 K the characteristic impedance. Or they may be regarded as defined 

 by the differential equations 



- dVldx = ZI, - dl/dx = YV, 



characterizing the line when there is no impressed field present. Y' 

 denotes the 'direct leakage admittance' and F" the 'basic shunt 

 admittance,' the latter defined as being the value of F when F' = 0, 

 whence Y = Y'^ -{- Y' . On referring to equations (12), (11), (8), (1) 

 of Appendix I, and also to equations (2) and (7) in Section I, it is 

 seen that ^^ 



Z = z -\- iwL, Y = G -\- ioiC, 



G = GO + Y', F" = GO + ioiC. 



The various sets of equivalent electromotive forces remain valid 

 even when the line parameters are functions Z{x), F(x), etc., of position 

 X along the system. For the 'fundamental' set this fact can be 

 readily seen by reference to the formulation and verification of the 

 fundamental set, in the early part of Section III. 



As indicated by the arrows, the positive axial (longitudinal) direc- 

 tion is the direction of increasing x, and the positive vertical direction 

 is downward. 



Six Different Sets of Equivalent Electromotive Forces 

 Set 1 (Fig. 4) 



{A) In the wire, a distributed electromotive force, f(x)dx in each 

 differential length dx. 



(B) In the distributed direct leakage admittance, a distributed 

 electromotive force, F{x) in each differential element Y'-dx of direct 

 leakage admittance. 



(G) In the termial impedances Zo and Zs, electromotive forces 

 F(0) and F{s) respectively. 



From the physical viewpoint of the present paper, Set 1 is the 

 fundamental set of equivalent electromotive forces. 



This set is particularly simple when there is no direct leakage 

 admittance (F' = 0), for then it reduces to merely the axial con- 

 stituents (A) and the terminal constituents (G). 



i^Thus Z, unsubscripted, includes the internal impedance z = Zw + Zg_ of the 

 circuit, and hence is to be sharply distinguished from the double-subscripted Z 

 occurring frequently in this paper; for, as remarked in connection with equation 

 (2), Zjj does not include the internal impedance z„ of wire j, whence it is seen that 

 Z = Zjj + Zjj for wire j. 



I 



