Self-induction of Wires. 285 



or (R' 2 + LV)U, at the frequency w/27r, according as current 

 density differences are, or are not, ignorable. The impedance 

 according to the latter formula increases with the speed, but 

 is greater or less than that of the former formula (linear 

 theory) according as the speed is below or above a certain 

 speed. 



But if the speed is sufficiently increased, even on a short 

 line, the formula ceases to represent the impedance, whilst, if 

 the line be long it will not do so at any speed except zero. 

 According to (82) we have 



Vo/0 = l±t +^7 (€»' + e-»'-2oofl 2QQ*, (83) 



as the distant end impedance of the line. That is, we have 

 extended the meaning of impedance, as we must (or else have 

 a new word), since the current-amplitude varies as we pass 

 from beginning to end of the line. (83) will, roughly 

 speaking, on the average, give the greatest value of the 

 impedance. It is what the resistance of the line would have 

 to be in order that when an S.H. impressed force acts at one 

 end, the current-amplitude at the distant end should be, 

 without any electromagnetic and electrostatic induction, what 

 it really is. The distant end impedance may easily be less 

 than the impedance according to the electromagnetic reckon- 

 ing. What is more remarkable, however, is that it may be 

 much less than the steady-flow resistance of the line. This is 

 due to the to-and-fro reflection of the dielectric waves, which 

 is a phenomenon similar to resonance. 



To show this, take B/ = in the first place, which requires 

 the conductors to be of infinite conductivity. Then L' = L , 

 the dielectric inductance. We shall have, by (83) and (78), 



V /C =L vsin(^), (84) 



where v= (L S)-*=(/& 2 c 2 )-*, the speed of waves through the 

 dielectric when undissipated. The sine is to be taken positive 

 always. If nl/v = ir, 27r, &c, the impedance is zero, and the 

 current-amplitude infinite. Here nl/v = ir means that the 

 period of a wave equals the time taken to travel to the distant 

 end and back again, which accounts for the infinite accumu- 

 lation, which is, of course, quite unrealizable. 



Now, giving resistance to the line, it is clear that although 

 the impedance can never vanish, it will be subject to maxima 

 and minima values as the speed increases continuously, itself 

 increasing, on the whole. We may transform (83) to 



