526 BELL SYSTEM TECHNICAL JOURNAL 



element, consisting of V^dx in parallel with Y'dx containing the elec- 

 tromotive force F{x), by the equivalent simple shunt element Ydx con- 

 taining the electromotive force {Y'IY)F{x). 



Figs. 7, 8, 9 are derived from Figs. 6, 7, 5 respectively by applying 

 the 'branch point theorem.' 



Fig. 11 is derived from Fig. 7 by applying the 'branch point the- 

 orem' in the manner indicated by Fig. 10, where /'(x)(/x denotes, for 

 brevity, the original axial equivalent electromotive force situated 

 between x and x -\- dx oi Set 2 as indicated by Fig. 7, so that 



/(x)-/(x)+-^'^, (76) 



and a is the coordinate of the contemplated arbitrary point. The 

 E's, of which Ex is typical, are sets of electromotive forces inserted 

 at the branch-points. At first these electromotive forces are arbitrary, 

 except that each set of three accords with the branch-point theorem, 

 so as not to alter the original currents in the system. Next, starting 

 at the ends, it is found that these electromotive forces can be so deter- 

 mined as to annul the original axial electromotive forces /'(x)<ix in all 

 of the differential elements dx except in the one containing the point a; 

 the requisite value of Ex and the resulting value of Ga are found to be 

 as formulated in Set 6. 



Finally it may be remarked that each of the Sets 2, 3, 4, 5, 6 can 

 be verified against Set 1 by formulating the total current produced 

 at any point x by each of the Sets 2, 3, 4, 5, 6 and then comparing 

 the resulting formula with the sum of formulas (41), (42), (43), (44). 

 Evidently it suffices to do this for the relatively simple case where the 

 terminal impedances are equal to the characteristic impedance of 

 the line; for this case, formulas (41), (42), (43), (44) reduce to (50), 

 (51), (52), (53) respectively. 



Sets of Equivalent Electromotive Forces for a Balanced Two-Wire Line 

 in an Arbitrary Impressed Field 



The foregoing six sets of equivalent electromotive forces for a one- 

 wire line can be readily extended to a two-wire line after resolving the 

 impressed field into mode-a and mode-c constituents, which are then 

 dealt with separately. For Set 1 this procedure has been fully out- 

 lined above in the subsection entitled 'A Balanced Two-Wire Line 

 in an Arbitrary Impressed Field,' and it has found a natural applica- 

 tion in the ' Crosstalk Problem ' treated below in Section IV. 



It is clear that all of the sets of equivalent electromotive forces are 

 immediately applicable to dealing with the mode-c constituent of the 



