498 BELL SYSTEM TECHNICAL JOURNAL 



Let Qi, • • • Qn denote the charges per unit length on the n wires; 

 the potentials and charges are then related by the set of linear equa- 

 tions 



Qi=f:qjf.V,, U=i,2---n), (3) 



fc=i 



Vi=tpikQk, (7=1,2 •••«), (4) 



in which the q and p coefificients are Maxwell's capacity and potential 

 coefficients. They are calculable by the usual methods of electro- 

 statics, on the assumption that all the conductors, including the earth, 

 are of perfect conductivity. 



To complete the specification of the system we have the further 

 set of relations 



icoQ,- + // = - ^^ (j=l,2-.-n). (5) 



Here // is the 'leakage' current from the jth wire; it is, in general, 

 a linear function of the n potentials, that is, 



// = tg^kV,, ij = 1,2 ■■- n), (6) 



fe=i 



where the coefficients gjk depend on the geometry of the system and 

 the conductivity of the dielectric medium. From (3), (5) and (6) 

 we have 



- ^' = E {io:q,k + gik) n, (i = 1, 2 • . . «). (7) 



ax 4=1 



This system of linear equations, when solved for the potentials, gives 



Fy= -fZw.J,, (i= 1,2 ••• n). (8) 



ax t=i 



For the special case where the dielectric medium surrounding the con- 

 ductors is homogeneous and isotropic, the coefficient Wjk, which in 

 general is obtained by solving (7), is given by 



Wjk = pikl{i(^ + 5), (9) « 



where 5 = Airal €^, a and e being the conductivity and specific inductive 

 capacity of the dielectric, and ^ a constant whose value depends only 

 on the units.^ In many cases Wjk is calculable with sufficient accuracy 

 from equation (9), so that the solution of (7) is then unnecessary. 



^A derivation of the formula for b is outlined shortly after equation (18) of 

 Appendix I. 



