PROPAGATION OF PERIODIC CURRENTS 529 



any contributions from tlie secondary system or from the impressed 

 field. (More fully, Vhr may be termed the 'primary systemic poten- 

 tial' at h and Eh^r the 'primary systemic axial electric force' at h.) 



For explicit use below, we may here note the formulas for the sys- 

 temic potential Vut and the systemic axial electric force £a^ at any 

 wire k of the primary system it : 



Vk.= t PkhQh, {k= \, ■-• n), (83) 



Eu. = - t(z,Hh + Pkh^) , {k= I, '■■ n), (84) 



pkh and Zkh being respectively the mutual potential coefficient and 

 the mutual impedance^ between wires h and k, per unit length. 

 Equation (84) is obtainable by applying the second curl law to a 

 differential rectangle substantially as in deriving equations (1) and 

 (2); see also Appendix I. 



As already stated in connection with equations (79) and (80) the 

 problem to be considered in the present subsection is the calculation 

 of the potential Fjv and the axial electric force £j> produced at the 

 secondary wire j by the primary system tt. The fundamental formulas 

 for Fjv and £,v are : 



V,. = E Pp,Q.h = L V,H, (85) 



h=l Ji=l 



£;. = -t( Zj,h + ^ ) = E £;. (86) 



dQ 



n=\ 



= - L [Z,nh + PiK^ , (87) 



dx 



where Vjh and Ejh are the contributions of wire h to F;^ and £,v 

 respectively. 



With regard to applications of the equations (85) and (87) for the 

 potential Fjv and the axial electric force £,v impressed on the secondary 

 J by the primary tt, it will be supposed that all the primary currents 

 I\, • • • In are known. But the primary charges Qh and their axial 

 gradients dQh/dx (where h = I, • • • n) are usually not known; and 

 therefore ways will now be indicated for expressing them in terms of 

 quantities which may be known. For that purpose, the presence of 

 the secondary will be entirely ignored, in all respects. (This proce- 

 dure may be regarded as the first-approximation step in a solution 

 by successive approximations.) 



