502 BELL SYSTEM TECHNICAL JOURNAL 



circuit, and the third to ground return transmission. It may be 

 easily shown that the solution for the direct waves is (writing /y and 

 1/ for the currents in the two wires, j and j' , of the jth pair, and Vj, 

 V/ for the corresponding potentials) : 



1/ = - AjC-'^^ + 5ye-T^^ + Ce-'''\ 



with the further condition XI Bj = 0. The corresponding potentials 

 are: 



V/ = - Ki^ye-^'^ + KoBje--'-^^ + InKsCe-^^'. 



The characteristic impedances Ki, K2, K3 correspond to the three 

 modes of propagation; from the foregoing and equation (8) they are 

 found to have the following values: 



Ki = 2{zu — w')yi, 



Ko = (w + w' - 2iu")y2, ^24) 



Ki = -^ {w + iv' + 2[_7l - l]zt'")7:i. 

 2 It 



The importance of the case just considered lies in the fact that the 

 conditions of symmetry which obtain are those of the ideal multi- 

 circuit telephone system. Two of the normal modes of propagation, 

 physical and phantom circuit transmission, are those actually em- 

 ployed in telephone transmission and these modes can exist in any 

 physical or phantom circuit without crosstalk or the induction of 

 current in any other circuit. In fact the problem of crosstalk is the 

 designing of the system, by means of transpositions, to approximate 

 the ideal case, and the calculation of the effect of small departures 

 from the ideal conditions of symmetry. 



In investigating problems of crosstalk and of induction from foreign 

 disturbing sources, the general formulas developed in the preceding 

 pages do not lend themselves readily to the necessary calculations and 

 interpretations. In the first place, calculation by means of the 

 general formulas involves the location of the n roots of an wth order 

 equation and thus presents the same difificulties as those encountered 

 in the calculation of the transient oscillations of a network of n 

 degrees of freedom; in fact the problems are mathematically the same, 

 the space variable x of the present problem corresponding to the time 

 variable / of the transient problem. In the second place the formulas, 

 as they stand, are inapplicable to the important case where the circuit 



