ELECTRICAL TRANSMISSION ALONG WIRES 23 



and 



EJ - e,i{x, y)h + • • • + eni{x, y)/„, ( j = 1, 2, • • • w) (37) 



the / and e functions depending on the geometry of the conducting 

 system and, through the parameter v-, on its electrical constants. 

 They are uniquely determined by the differential equations. 



The actual solution of the differential equations (32)-(35) is essen- 

 tially more difficult than the solution of (24)-(26) involved in the deter- 

 mination of $, and they have not been subjected to the exhaustive 

 study accorded to the potential problem. On the other hand, the 

 analogy with the potential problem suggests that extension and 

 modifications of the general and well-known methods of solution 

 available for that problem should be possible. 



To summarize the foregoing we have succeeded in expressing 

 the potential function <l> (and therefore E^, Ey) in the dielectric as a 

 linear function of the conductor charges, the coefficients of the con- 

 ductor charges Q\ • - • Qn being spatial functions of x and y which 

 we determined by the usual methods of two-dimensional potential 

 theory. 



Similarly it has been shown that E^ (and therefore the current 

 distribution) in the conductors, and the potential function F (and 

 therefore the magnetic field) in the dielectric, are expressible as linear 

 functions of the conductor currents /i • • • /„, the determination of the 

 coefhcients depending on the solution of a generalization of the two- 

 dimensional potential problem. 



3. To complete the solution of the problem, recourse is had to the 

 fact that Ez is continuous at the surfaces of the conductors. At the 

 surface of each conductor we therefore equate Ez, as given by (37), 

 in terms of the currents Ii ■ ■ • In with 'y^ — F (see (22)), $ being 

 given by (27) in terms oi Q\ • • • Qn and F by (36) in terms of /i • • • /„. 

 This gives n equations of the form 



Zuh + • • • + ZnJn = 7*1 = yiPuQl + • • • + PlnQn), 



ZnJl + • • • + Znnin = J^n = lipnlQl + " • ' + pnnQn) ■ 



(38) 



Here $i • • • $„ are the values of 4> at the surfaces of the n conductors 

 respectively; the p coefficients are Maxwell's potential coefficients of 

 the system, while the Z coefficients are the self and mutual "im- 

 pedances" of the conductors. 



