PROPAGATION OF PERIODIC CURRENTS 509 



mate solution. This process can, theoretically, be repeated indefi- 

 nitely and successively closer approximations thereby obtained. 

 Practically, however, even in a system of only a few wires, the process 

 rapidly becomes prohibitively laborious and complicated, so that 

 only the first and perhaps the second approximate solutions are prac- 

 ticable. Theoretically, however, the process is straight-forward and 

 the successive approximate solutions form a convergent sequence. 

 Fortunately, in engineering applications the allowable amount of 

 crosstalk is so strictly limited that higher approximations than the 

 second at most are not usually required. 



It is an important and valuable property of the solution by successive 

 approximations that the 'datum configuration' is not uniquely fixed, 

 but is at our disposal, within limits. By 'datum configuration' is 

 meant the assumed distribution from which the first approximate 

 solution is derived. In the preceding the datum configuration for the 

 primary wires is taken as 



while in calculating any 7/ {j = ^, • ■ • n) it is assumed that the unbal- 

 ance currents and charges of the other disturbed wires are zero. From 

 the form of the equations this is certainly the natural configuration 

 with which to start. It does not at all follow, however, that this datum 

 configuration results in the optimum first approximate solution. 



Another datum configuration which may be taken and which appears 

 to possess practical advantages in certain cases is the following : ^^ 



/i = /« = - h, 



(39) 



for the primary wires, while in calculating any // (i = 3, • • • w) it is 

 assumed that the unbalance currents and potentials (instead of charges) 

 of the other disturbed wires are zero.^^ Higher successive approximate 

 solutions then follow the same scheme of procedure as in the first case. 

 The foregoing completes the formal analytical theory. The remain- 

 ing sections of the paper will be devoted to the interpretation of the 

 fundamental mathematical theory and its formulation along more 

 physical and engineering lines, together with applications to repre- 

 sentative problems. 



" This is essentially the basis of the crosstalk formulas developed, in terms of a 

 different mathematical treatment, by Dr. G. A. Campbell of the American Telephone 

 and Telegraph Co., in his early and fundamental work on crosstalk and transposition 

 theory, 



"^ See, however, the preceding footnote as to possible modification of the datum 

 configuration. 



