524 BELL SYSTEM TECHNICAL JOURNAL 



impedances. Thus, for instance, in the formulations of Set 1 and 

 Set 5, item (C) would read: '(C) In the ends x = and x = s oi the 

 line, axial electromotive forces — F{0) and F{s) respectively.' (Ob- 

 serve, here, the negative sign before F{Q), in contrast to the positive 

 sign in the original formulation.) 



In this way it is readily seen that at a point x = u where the im- 

 pressed potential Fix) is discontinuous, the equivalent electromotive 

 force is an axial electromotive force equal to the decrement of the 

 impressed potential, that is, equal to F{u — ) — F{u -f); this agrees 

 with equation (65), and with item {B) in the fundamental set of equiva- 

 lent electromotive forces formulated in the early part of Section III. 



Derivations of Set 1 



A synthetic derivation of Set 1 has already been furnished in the 

 early part of Section III. An analytical derivation will now be out- 

 lined; it is based on an interpretation of equations (68), (71), (73) 

 below; these equations, in turn, are based on certain equations of 

 Appendix I, as follows: 



Combining equations (1) and (2) of Appendix I gives 



where / denotes fw\ and V is that part of the potential of the wire 

 due to its charges (and the corresponding opposite charges on the 

 surface of the ground), while 4>' is that part of the magnetic flux due 

 to the current in the wire (and the corresponding return current in 

 the ground) ; that is, 



r = V - F= QIC (69) 



0' = 4, - ^ = LI, (70) 



so that V and (f>' do not include the impressed potential and impressed 

 magnetic flux F and $ respectively. 



By (5) and (7) of Appendix I the equation of current continuity 



can be written 



i = f+^e+^'^- (^') 



The actual potential V of the line is of course the resultant of V 

 and F; that is, 



V ^ V(x) = V'ix) + F{x), (72) 



whence, in particular, at the ends x = and x = s, 



V{0) = V'iO) -\- FiO), 

 V{s) = V'{s) + F(s). 



