164 BELL SYSTEM TECHNICAL JOURNAL 



and 



Nss = f f P(r) cos e dSds. 



Both Q{A-B){a~h) and N ss are generally complex-valued and thus do 

 not represent resistance and inductance, as ordinarily defined, as 

 might be inferred from the similarity of expression (II) to the 

 usual impedance expression. At zero frequency iooNss vanishes and 

 QiA-B)(a-b) becomes R(A-B)(a-b), a real number, the d.-c. mutual resist- 

 ance of the circuits. For frequencies sufficiently low, such that terms 

 involving higher powers of the frequency in the expansions of the 

 functions in powers of the frequency are negligible, the mutual imped- 

 ance can be expressed in the ordinary form; that is. in the formula 



2ss = Ru-B)(a-b) + ico[iV°(^i_B)(a_6) + N° Ss"] (HI) 



R(A~n)(a-b) is as above the d.-c. mutual resistance; N°(A~B)(a-b) is the 

 coefficient of ico in the expansion of Q(A-B)(a~b), a real number, and 

 generally equal to the sum of the Neumann integrals of the earth 

 flows with the wires and with each other, the earth flows being those 

 for direct current; N°ss is generally the Neumann integral of the 

 wires.^ The bracketed terms thus give the d.-c. mutual inductance of 

 the wires with earth return. 



For infinite distance between all terminal grounds A, B, a, b, taken 

 in pairs, Q(A-B)(a~b) vanishes. 



The physical distinction of Q(^A-B)(.a-b) and Nss may be illustrated 

 by the following two cases: In the first, one wire is supposed straight 

 and of arbitrary length; the second extends at right angles to it from 

 two grounding points and is closed at infinity (that is, by a segment 

 parallel to the first wire and at such distance that its mutual impedance 

 with the first wire is negligibly small). In this case, in the per- 

 pendicular segments cos e = 0, and in the parallel segment P{r) = 0, 

 since r = oo , so that Nss = and the mutual impedance is given 

 entirely by Q(^A-B)(a-b)', that is, the mutual impedance depends only on 

 the grounding points. In the second case, the two perpendicular 

 segments of the second wire extend away from a parallel segment to 

 grounding points at infinity. Here the mutual impedance is given 

 entirely by N'ss, since Q(r) and, therefore, (2(.i-B)(a-6) vanishes for the 

 limit r = CO . 



Table I is a summary of mutual impedance formulas obtained as 

 special cases of the general formula. For each case the first column 

 entries consist of the Q{r) and P(r) functions in the mutual impedance 



- An ambiguity concerning this statement as well as that referring to N°u-B)(a-b), 

 arising in certain particular cases, is discussed below. 



