PROPAGATION OF PERIODIC CURRENTS 505 



integration constants of the direct wave, while A' and C are those of 

 the reflected wave. The first component, as regards the impressed 

 force/ along the wires, depends on the difference /i — f^ at the surface 

 of the two wires, while the second depends on the mean value 

 (/i +/2)/2. In the case of interference from external sources the 

 latter is usually much the larger and consequently the induction 

 mainly corresponds to the ground return mode of propagation, 73. 



(2) System of n Balanced Pairs 



We shall now write down the expressions for the currents in a 

 system of n balanced pairs (2w wires) when exposed to an arbitrary 

 impressed field. The properties of this system were discussed briefly 

 in the preceding section and formulated in equations (19), • • • (24). 

 Let /,• and // be the currents in the two wires j and j' respectively of 

 the jth pair, and let /,• and // be the corresponding impressed forces 

 along the surfaces of the two wires, while Fj and F/ are the corre- 

 sponding line integrals of the impressed force to ground. By an 

 extension of the previous formulas it is easy to show that the currents 

 are made up of three components : 



Ij = a, + bj + Cj, 



T r I z, I \^^) 



Ij = — O; + Oj -f- Cj. 



If we write /,• = (/, -\- f/)l2, the components aj, bj, Cj are given by: 



, (30) 



- ^-^^ [^/ + 2^ J' [hiy) - fi{y)]e-''^'dy]^ , 



bj = e-r^^ U/ + 2^ r {My) -li:My)]e'^^'dy~\ 



" , (31) 



- e^^^[^/ +2^ J'{//(3') -li:My)}e-'^^'dy'^ , 



J:Bj=Z B/ = 0, (32) 



•- " -■ (33) 



Here the a component corresponds to transmission over a pair, 



