530 BELL SYSTEM TECHNICAL JOURNAL 



The charges Qh can be expressed in terms of the systemic potentials 

 Vkw by solving the set of n equations (83). Thus 



Qk= i: qhkV,., Qi= \, '" n), (88) 



where qhk is the Maxwell capacity coefficient between wires h and k; 

 in terms of the potential coefficients, its value is 



g« = Dkh{p)lD{p), (89) 



D{p) being the determinant of all the potential coefficients (the p's) 

 in the set of n equations (83) and Dkhip) the cofactor of pkh in D{p). 

 The systemic potentials Vk-r, occurring in (88), can be obtained by 

 solving the equations of current continuity, namely the set of n 

 equations ^^ 



-^'=J: (YnkVk. + X.kFk), (A = 1, • • • n), (90) 



ax fc=i 



Yhk and Xhk being of the nature of admittances (per unit length), and 

 Fk the impressed potential at wire k; it is thus found that 



Vk. = -i:Wkk(^^+Z XhrFr y {k=\, ■■■ 71), (91) 



where the coefficient Wkh is the same function of the F's that qkh is of 

 the ^'s, that is, 



Wkh = Dnk{Y)ID{Y). (92) 



It is seen that Wkh is of the nature of an impedance (per unit length), 

 though it is not a simple impedance. 



The charges can now be expressed in terms of the impressed poten- 

 tials Fr and the axial gradients of the currents by substituting (91) 

 in (88). 



The axial gradients of the charges can be expressed in various ways. 

 They can be immediately expressed in terms of the axial gradients of 

 the systemic potentials Vk by merely difTerentiating (88) with respect 

 to X. Also, they can be expressed in terms of the currents Ir and the 

 systemic axial electric forces Ek^ at the wires, by solving the set of n 

 equations (84); thus 



'^= -t QhkiEk. + L ZkJr), (h = 1, • • • n), (93) 



ax t=l r=l 



21 Derived in the latter part of Appendix I. 



