MUTUAL IMPEDANCES OF PARALLEL WIRES 525 



has proved valuable in investigating a variety of problems in wave 

 propagation along parallel conductors. Reference may be made to 

 the paper itself for a fuller exposition of the underlying theory, much 

 of which is omitted from the present analysis. 



In terms of a vector potential A and a scalar potential V, the elec- 

 tric and magnetic forces E and // per unit length in the dielectric are 

 given by the relations, 



E = — grad V — iuA 

 and (19) 



nH = curl A , 



where n is the permeability of the dielectric. Assuming that the wave 

 varies as exp {io:t — yz) and putting F = iwA,, the axial electric force 

 E^ (omitting the subscript 2) may be written 



E= -yV - F, (20) 



where 7 is the propagation constant per unit length of the system. 



Now, as we shall show below, the electric force E inside of the con- 

 ductors and the wave function F in the dielectric may be expressed as 

 linear functions of the conductor currents. That is, at the surface of 

 the jth conductor, for example, we may write 



E; = eijli + eijh + ezji + e^jU 

 and (21) 



where ejk and fjk are determined by the geometry and electrical con- 

 stants of the system. (F,-, the potential at the surface of the 7th 

 conductor, is determinable from the geometry of the conductors as a 

 linear function of the conductor charges, Qu Qi, • • • ; that is, F, may 

 be written 



Vi = puQi + p2,Q2 + ps^Qs + PM, 



the p coefificients being the Maxwell potential coefficients of the system. 

 These, however, are not required in the present problem.) 



But relation (20) must hold at the surface of the conductors. Thus, 

 since the electric force is continuous at the surface of the conductors, 

 relations (20) and (21) give 



Zii/i + Z21/2 + Z31/3 + ZiJi = 7F1 ^ El -\- F, ri = a, 



'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'. (22) 

 Z14/1 + Z24/2 + ZsJi -f Z44/4 = 7F4 = Ei -\- F, ri = a, 



