PROPAGATION OF PERIODIC CURRENTS 



501 



m m 



= 0, 



(15) 



and there are only two modes, 7i and 72, corresponding to m — m' = 

 and m + (» — l)w' = respectively. The first mode obviously cor- 

 responds to metallic transmission, the second to ground return trans- 

 mission. It is easily shown that the direct current waves are expressed 

 by 



/; = AiC--'^' + Be-^^-\ (16) 



E^/ = 0, 



(17) 



with corresponding expressions for the reflected waves. The corre- 

 sponding potentials are 



Vj = IKiAje--'^' + nKiBe--'"- 



(18) 



Here Ki is the characteristic impedance of a metallic circuit composed 

 of two wires, and K2 is the characteristic impedance of the n wires in 

 multiple, with the ground for return. 



A case of greater practical importance is that of n balanced pairs, 

 which is the ideal telephone transmission system. To consider this 

 case let 



Wu = W22 = • • • = mnn = ntin, In = m, 



nijk = m' between wires of the same pair, (19) 



mjk — m" between wires of different pairs. 



In this case the determinant becomes 



m 

 m 



m 



m' 

 m' 



m 

 m' 



m' 

 m' 



m' 

 m' 



m 

 m' 



m 



= 0. 



(20) 



There are therefore three modes of propagation 71, 72, 73 correspond- 

 ing respectively to: 



ni — ;;;' = 0, 



m + w' - 2m" = 0, (21) 



m + m' + 2{n ~ \)m" = 0. 



The first mode of propagation corresponds to metallic transmission 

 over a pair, the second to transmission over a four wire phantom 



