Self-induction of Wires. 423 



use of (141), with e=0, 



_A(YC) =; |(iSV 2 + iL G 2 ) + CR/'C + CB/C; (142) 



that is, in increasing the electric and magnetic energies in the 

 dielectric, and in transfer of energy into the conductors, to 

 the amounts CR/'C and CR 2 "C per second respectively, which 

 are, in their turn, accounted for by the rate of increase of the 

 magnetic energy, and the dissipativity, or Joule heat per 

 second in the two conductors ; or 



CE 1 ,, C = Q 1 + T 1 , CB 2 "C = Q 2 + T 3 , . . (143) 



Q being the dissipativity and T the magnetic energy per unit 

 length of conductor. 



These equations (143) must therefore contain the enlarged 

 definition of the meaning of the functions ~RJ ! and R 2 ". For 

 it is no longer true that R/'C is, as it was in the tubular case, 

 the longitudinal electric force at the boundary of the conductor 

 to which R/' belongs. It is a sort of mean value of the lon- 

 gitudinal electric force. Thus, we must have 



j , EH/47r.<fc = CR 1 "C, .... (144) 



if E be the longitudinal electric force and H the component 

 of the magnetic force along the line of integration, which is 

 the closed curve boundary of the section of the conductor 

 perpendicular to its length. But no extension of the meaning 

 of V is required from that last stated. 



Let us, then, assume that R/' and R/ can be found, their 

 actual discovery being the subject of independent investiga- 

 tion. We can always fall back upon round wires or tubes if 

 required. They are functions of d/dt and constants, if the 

 line is homogeneous. But, as we have got rid of the radial 

 component of current in the conductors, and its difficulties, 

 the constancy of the constants in R/' and R 2 " (as the conduc- 

 tivity and the inductivity, or the steady-flow resistance, or the 

 diameter) need no longer be preserved. Provided the con- 

 ductors may be regarded as homogeneous along any few yards 

 of length, they may be of widely different resistances &c. at 

 places miles apart.. Then R/', R/ become functions of z as 

 well as of d/dt, and S a function of z. Let our system be 



-g = S'<V, .— -BTO, . . . (145) 



where both R" and S" are functions of d/dt and z. As re- 

 gards S", it is simply &(d/dt) when the dielectric is quite non- 

 conducting. But when leakage is allowed for, it becomes 



2G2 



