516 BELL SYSTEM TECHNICAL JOURNAL 



filament p oi A and filament <^ of B. On substituting (4) into (3) we 

 get for filament p the "voltage equation:" 



Vr, = ZZpJ, + HZj^h = Va. (5) 



Next we multiply this equation through by Ip, add together the w 

 such resulting equations corresponding respectively to the tn filaments 

 of wire A, and introduce the condition that the sum of the filamentary 

 currents in A is equal to I a- Finally, we divide the resulting equation 

 through by I a and denote the current-ratios Ip/Ia, IqJIa, mia by 

 Jp, J^, J^ respectively, each / thus denoting the ratio of a filament 

 current to the total current in the wire to which that filament belongs, 

 so that J may be called a "relative filamentary current." We thus get 

 the equation 



P Q P <t> 



Comparison of this with the equation 



Va = ZaaIa + ZabIb, (6a) 



which is the "voltage equation" for wire ^ as a whole, yields the 

 following formulas for the self impedance Zaa of wire A and the 

 mutual impedance Zab to A from B (Fig. 1) : 



Zaa — ^^Z pqJpJg, (7) Zab = YizlZ,^JpJ^. (8) 



P Q P ')> 



Similarly, for wire B, 



Zbb = Z-j^lZ^eJ^Je, (9) Zba = zlzlZ^pJ^J p, (10) 



<j> d <t> p 



Zba denoting the mutual impedance to B from A. It will be recalled 

 that p, q designate any two typical filaments of wire A, and <^, d any 

 two of B. 



The presence of the relative filamentary currents (the J's) in these 

 equations accounts for the fact that the self impedance of a wire 

 depends on the current-distribution over its cross-section, and the 

 mutual impedance between two wires on the current-distributions 

 ov^er their cross-sections. The self and mutual impedances of two 

 wires, such as A and B, must thus depend on the currents in any other 

 wires that may be present, because the voltages induced in A and B 

 by these other currents will partly determine the current-distributions 

 in A and B. Although the values of the summation expressions in 

 equations (7) to (10) depend on the currents in any other wires that 



