PROPAGATION OF PERIODIC CURRENTS 497 



The first equations of the problem are derived by applying the law 



curl E = - fiidHfdt) 



to a contour bounded by a length dx in the surface of the jth wire, a 

 corresponding length dx in the surface of the earth, and two lines nor- 

 mal to the axis of the wire and joining the corresponding ends of the 

 two line elements dx. This gives 



^»I,-E.,= -^-^. (i= 1.2 •■■«). (1) 



In this set of equations, zjj denotes the 'internal' impedance per unit 

 length of the jth wire, that is, the ratio of the axial electric force at the 

 surface of the wire to the current //. Egj is the electric force, parallel 

 to the axis of the wire, in the earth's surface. Vj is the line integral 

 of the electric force from the wire to the surface of the earth, that is, 

 the 'potential,' or 'voltage,' of the wire. Finally, 0/ is the magnetic 

 flux,- per unit length, threading the contour. 



Now, both (f)j and Egj are linear functions of the « currents /i, • • • /„ ; 

 consequently (1) is reducible to the form ^ 



sy.// = -^-^^ ^/^^^. (j = 1, 2 • . • n). (2) 



The calculations of the impedance functions Zjk and, in particular, 

 the effect of the finite conductivity of the earth are dealt with in 

 detail in an earlier paper.^ The internal impedance, Zjj, is of the 

 general form r/,- + icoljj, where r^y is the resistance of the jth wire and 

 /,-,• its 'internal inductance.' In the ideal non-dissipative system the 

 mutual impedance Zjk is a pure imaginary of the form icoLju, where 

 Ljk is the mutual inductance between the jth and ^th wires; actually, 

 however, due to the finite conductivity of the earth and to 'proximity 

 effect' between the wires, it is always complex and of the form 

 Rjk + iwLjk. A similar statement holds for the self impedance Zjj. 

 The 'proximity effect' ^ is, of course, the increased internal impedance 

 of the wire due to the currents in the neighboring wires. It may be 

 taken as negligible in open wire lines but is quite appreciable, at 

 telephonic frequencies, in cable circuits. 



2 Expressed in 10"^ maxwells if the remaining quantities are in 'practical units.' 



' It is to be noted that Zjj does not include the internal impedance Zjj of wire j. 



* 'Wave Propagation in Overhead Wires with Ground Return,' John R. Carson 

 B. S. T. J., October, 1926. 



^ See 'Wave Propagation over Parallel Wires: The Proximity Effect,' John R. 

 Carson, Phil. Mag., April, 1921. Rigorously the term Zjjl, of eqiiations (1) and (2) 

 should be replaced by "Lziklk, the additional terms formulating the proximity effect. 

 This effect will not be explicitly included in the following analysis and the term 

 Zjklk may be regarded as incorporated with Z,klk- 



