506 BELL SYSTEM TECHNICAL JOURNAL 



the h component to transmission over a phantom circuit, and the c 

 component to ground return transmission. It will be observed that, 

 as regards the impressed field, the a component depends on the 

 difference / — /' of the impressed force at the two sides of the circuit, 

 while the c component depends on the mean impressed electric force 

 averaged over the 2n conductors of the system; the h, or phantom 

 component, involves the impressed field in a slightly more complicated 

 way, depending on both the mean impressed force averaged for the 

 two conductors of a pair and also averaged over all the 2n conductors 

 of the system. 



The extension of the preceding analysis to the general case of n 

 parallel wires, in general dissimilar, is straightforward. The resulting 

 formulas are, however, extremely complicated and for this reason, as 

 well as their small practical utility, they will not be written down. 



Formulas (25), • • • {2>Z) are immediately applicable to the wave 

 antenna and to interference problems in general where the impressed 

 disturbance is supposed to be known. Their application to the 

 problem of crosstalk, which will now be taken up, is not immediate in 

 the same sense because here the primary disturbance which sets up 

 crosstalk is itself a function of the unbalances among the wires com- 

 posing the system. That is to say, the primary disturbance or im- 

 pressed field causing the crosstalk is implicitly rather than explicitly 

 given. 



I la 



In discussing the theory of crosstalk a representative problem will 

 be dealt with rather than a formulation of the general problem. The 

 types of problem encountered in practice are extremely varied, de- 

 pending on whether we have to do with 'side-to-side,' 'side-to-phan- 

 tom' or 'phantom-to-phantom' crosstalk, etc.; and each problem may 

 call for special treatment. The representative problem, however, be- 

 sides showing the underlying mathematical theory should serve to in- 

 dicate the correct procedure in other specific problems. 



Let us return to the general system of n parallel wires, dealt with 

 in Section I, and let us suppose that two of them, say wires No. 1 and 

 No. 2, constitute a metallic circuit which, for convenience, we shall 

 suppose would be balanced with respect to ground if the other wires 

 were removed. We now suppose that this metallic circuit is energized 

 by an electromotive force impressed at x = 0, which in the absence of 

 the other wires would produce a current P in wire No. 1 and an equal 

 and opposite current — 1° in wire No. 2. Our problem is now to cal- 

 culate the currents induced in the neighboring wires and the additional 



