512 BELL SYSTEM TECHNICAL JOURNAL 



A physical verification of the correctness of the foregoing set of 

 equivalent electromotive forces can be obtained by starting with the 

 given system, situated in the specified arbitrary impressed field (but 

 not otherwise energized), and then inserting in the wires and bridges 

 a set of electromotive forces, which will be termed the 'annulling 

 electromotive forces,' such as to annul all currents in the wires and 

 bridges. The resultant axial electric force in the wires will then be 

 zero, and furthermore the wires will be uncharged; hence the inserted 

 axial electric force must be equal and opposite to the impressed axial 

 electric force. Since the wires are uncharged their potentials will be 

 those of the impressed field; hence, since no current flows in the 

 bridges, the electromotive forces inserted in the bridges must be equal 

 and opposite to the voltages of the impressed field at the bridges. 

 Evidently the negatives of the annulling electromotive forces consti- 

 tute a set of electromotive forces equivalent to the impressed field; 

 for, insertion of the negatives of the annulling electromotive forces 

 restores the system to its original state, in which it is acted on by only 

 the original impressed field. 



From the nature of this demonstration it is seen that the 'funda- 

 mental set' of equivalent electromotive forces is not limited to a 

 system of parallel horizontal wires. In the general case, where the 

 wires are neither straight nor parallel nor horizontal, x (and hence u 

 and h) is to be interpreted as being the 'intrinsic coordinate' of a 

 point in the particular wire contemplated, that is, the distance meas- 

 ured along that wire from any arbitrary fixed point therein. Thus, 

 for wires h and k respectively, x becomes Xh and xu, which in general 

 are independent of each other. 



For the case of a one-wire line, an analytical derivation of this set 

 of equivalent electromotive forces is given in a later subsection by 

 interpretation of the fundamental differential equations of the line. 



A One-Wire Line in an Arbitrary Impressed Field 



As indicated by Fig. 1, the line extends from x = to x = s, and 

 is terminated in impedances Zo and Zs respectively. 7 denotes the 

 propagation constant per unit length, and K the characteristic im- 

 pedance.^^ The direct leakage admittance from the wire to ground, 

 per unit length, is denoted by Y' ; this is the generalization of a mere 

 leakage conductance.'^ 



The impressed field is specified by the functions f{x) and F{x) ; 

 f{x) denoting the impressed axial electric force and F{x) the impressed 



" Given by formulas (12) and (11) of Appendix I. 



