ELECTROMAGNETIC THEORY • 13 



and may be written as //coC, whence 



RI + io^LI + -X^I = V, (16) 



which is the usual circuit equation for series resistance, inductance 

 and capacity. 



Extension of the foregoing to networks containing a plurality of 

 circuits or meshes is straightforward and involves no conceptual or 

 physical difficulties, although branch points may be analytically 

 troublesome. These questions will not be taken up, however, as the 

 foregoing is sufficient to show the connection between general electro- 

 magnetic theory and circuit theory and to show how circuit equations 

 may be rigorously derived and their limitations explicitly recognized. 



The Telegraph Equation 



A particularly interesting and instructive application of the pre- 

 ceding is to the problem of transmission along parallel wires and the 

 assumptions underlying the engineering theory of transmission.^ 



Consider two equal and parallel straight wires so related to the 

 impressed field that equal and opposite currents flow in the wires. 

 Here, corresponding to equation (11), we have 



rl = E - io:fl{\{s, s') - \'(s, s')}ds' 



s r (17) 



In this equation \{s, s') is the "mutual inductance" between points 

 s, s' in the same wire while X'(5, s') is the corresponding mutual induc- 

 tance between point 5 in one wire and point s' in the other, fx and /x' 

 have a corresponding significance as "mutual potential coefficients." 

 Now \(s, s') — \'{s, s') is a rapidly decreasing monotonic function 

 of |5 — 5'| and the same statement holds for /j. — fx'. In view of 

 this property and further assuming the variation of / and Q with 

 respect to 5 as small, (17) to a first approximation may be replaced by 



rl = E - io^I MX - X')^^' - |- <2 r (m - l^')ds'. 



(18) 



At a sufficient distance from the physical terminals of the wires the 



* For an entirely different treatment of this problem, reference may be made to 

 "The Guided and Radiated Energy in Wire Transmission," Trans, A. J, E. E., 1924, 



