508 BELL SYSTEM TECHNICAL JOURNAL 



// — dFj'Idx along the jth wire ( j = 1, 2, • • • n), and an impressed 

 potential F/ (line integral of impressed field fromjth wire to ground), 

 where 



F,' = P12Q2 + pnQz' + piiQi' + • • •, 



F2' = P21Q1' + p2^Qs' + p2,Q,' + • • • , .3^. 



^/ = {Pn - pj2)Q' + PnQi + Pi2Q2' + /'/a'?/ + • • •, 



(j = 3, 4 • • • w). 



Consequently if // and F/ were known, equations (25), • • • (27) 

 would be immediately applicable to the calculation of the unbalance 

 currents. Inspection of equations (34), • • • (36) shows, however, that 

 while /", F", Q^ are supposed known, the expressions for // and F/ 

 involve the unbalance currents and charges themselves. The solution 

 of the equations calls therefore for a process of successive approxima- 

 tion, now to be discussed. While this method of solution is theoret- 

 ically sound and applicable in all cases, its success in practical applica- 

 tions depends largely on the fact that the unbalance currents must be 

 extremely small, compared with the primary current /", if the cross- 

 talk is to be kept within tolerable limits. 



Returning to equations (34), • • • (36), the first approximate solu- 

 tion is obtained by (1) ignoring the unbalance currents and charges 

 in their effect on the current in the primary wires (No. 1 and No. 2), 

 and (2) replacing fz', • • • fj and F/, • • • F„' by 



// = - (Z,-: - Z,;)I^ 



F/ = {pn - Pi2)Q\ (37) 



(j = 3, 4 • • • n). 



Consequently in the first approximate solution the primary current, 

 charge and potential are /'', (3°, V^, which are calculable in terms of the 

 impressed e.m.f. and the terminal impedances for the circuit composed 

 of the primary wires (No. 1 and No. 2) by ignoring the reaction of the 

 other wires.^° The unbalance currents, charges and potentials of the 

 other wires are then calculable on the supposition that those wires 

 are energized by the known impressed field //, F/, as given by (37), 

 which depends only on /° and Q^. 



The second approximate solution is obtainable by substituting the 

 first approximate values of // and Q/ in the right hand side of equa- 

 tions (34) and (36) and then proceeding precisely as in the first approxi- 



1" It should be clearly understood that this particular procedure is not required 

 and is not always followed in practice. For example, it is customary in calculating 

 the crosstalk induced in a metallic or 'side' circuit to take into account, in the first 

 approximate solution, the reaction between the wires making up the disturbed circuit. 



