PROPAGATION OF PERIODIC CURRENTS 525 



Returning, now, to a consideration of equations (68), (71), (73), it 

 is seen that they are identically the same as the equations for the same 

 line without any impressed field but containing the set of electromotive 

 forces formulated above under the heading 'Set 1 (Fig. 4)'; for an 

 interpretation of equations (68), (71), (73) yields respectively {A), 

 (B), (Oof Set 1. 



It may be noted that equations (68) and (71) can be written in 

 the following more compact forms: 



- dV'/dx = ZI -f, (74) 



- dl/dx = YV + Y'F, (75) 



whose interpretation yields immediately items (A) and (B) of Set 1. 



Outline of Derivations of Sets 2, 3, 4, 5, 6 



Synthetic derivations of Sets 2, 3, 4, 5, 6 from Set 1 will now be 

 briefly outlined by aid of the diagrams in Figs. 2, ••• 11. The 

 physical systems represented by these diagrams are all equivalent in 

 the sense that the currents at corresponding points in all of them 

 are equal. 



In the derivation-work extensive use is made of an artifice which, 

 for convenience, will be formulated in what may be termed the 

 'branch-point theorem,' as follows: In any network of any number of 

 branches the currents will not be affected by inserting at any branch-point 

 a set of equal electromotive forces, one in each branch, directed either all 

 toward or all from the branch-point. 



Fig. 2 represents the given one-wire line in an arbitrary impressed 

 field, as already specified. For generality the line is assumed to have 

 uniformly distributed leakage admittance of amount Y' per unit 

 length. 



Fig. 3 is derived from Fig. 2 by lumping the distributed direct leak- 

 age admittance into localized admittances each of amount Y'-dx at 

 intervals of length dx. 



Fig. 4 is derived from Fig. 3 by replacing the arbitrary impressed 

 field by Set 1 of equivalent electromotive forces. 



Fig. 5 is derived from Fig. 4 by replacing the line, exclusive of the 

 direct leakage admittance Y' , by its equivalent artificial line having 

 'complete series impedance' Z and 'basic shunt admittance' F" per 

 unit length. This replacement of the actual line by the corresponding 

 artificial line is permissible now that the impressed field has been re- 

 placed by a set of equivalent electromotive forces (Fig. 4). 



Fig. 6 is derived from Fig. 5 by replacing the compound shunt 



