MUTUAL IMPEDANCES OF PARALLEL WIRES 527 



Inside of the conductors the axial electric force Ej must satisfy the 

 wave equation in two dimensions. It may, therefore, be expressed 

 as the Fourier-Bessel series, 



00 



Ej = gojJoirjz/a) + E Jn{rjzla){gnj cos nSj + //„; sin ndj). (29) 



n = I 



The constants go; are given by the relations 



r'^^ / dE\ 

 ATTfxJo}!]- = I ( -^ I addj, 



or 



Jo{rjz/a) 



Zj being the internal impedance per unit length of the jth. conductor 

 with concentric return. Here iic is the permeability of the conductor, 

 Jn{z) is the Bessel function of the first kind of wth order and argument 

 s = aiViTTOTMw and the arbitrary constants g„/ and /?„;• are to be 

 determined by boundary conditions; it is evident, however, that we 

 must have 



gnl — ~ gni, hnl — «n4» /t<\ 



gn2 = — gn3, /?«2 = ^n3. 



At the surfaces fj = a, the boundary relations are 



dF _ _^dE 



drj Mc drj 



and (32) 



dF _ _dE 



dej~ ddj' 



Hence, introducing (25) and (29) in (32), applying (32) at the two 

 surfaces n = a and r2 = a and equating harmonic coefficients, gives 8« 

 equations in the 8w arbitrary constants An, Bn, C„, J9„, g„i, g„2, //„i, 

 hn2- This procedure requires that F be expressed in terms of ri, 9i and 

 of r2, 02 by suitable transformations of coordinates. ^^ Thus, for all 

 points in the neighborhood of ri = a, for example, F may be written 



F = Ifiiouli log — + liiiwh log — — Y.\ ~- -t) ^^ 



Ti C'l n=l \ ^C / 



w=l \ C2 



cos'-^) (/!„ - C„) -f ( sin"^) (5„ - Z)„)j 



+ Z (cosw0i)(^„Vi" + 5„Vi-") + E (sin«0i)(C„Vi"+Z)„'rr"), (33) 



n=\ n=l 



1' The necessary formulas for these transformations are derived in Note II of 

 the paper "Transmission Characteristics of the Submarine Cable" by John R. 

 Carson and J. J. Gilbert, Journal Franklin Institute, December, 1921. 



