Forced Vibrations of Electromagnetic Systems. 137 



is harmful* to long-distance telephony. The precise contrary 

 is the case. It is the very life and soul of it, as is proved both 

 by practical experience in America and the Continent on very 

 long copper circuits, and by examining the theory of the 

 matter. I have proved this in considerable detail f ; but they 

 will not believe it. So far does the misconception extend that 

 it has perhaps contributed to leading Mr. W. H. Preece to 

 conclude that the coefficient of self-induction in copper circuits 

 is negligible (several hundred times smaller than it can pos- 

 sibly be), on the basis of his recent remarkable experimental 

 researches. 



The following formula, derived from my general formulae if, 

 will show the role played by self-induction. Let R and L be 

 the resistance and inductance per unit length of a perfectly 

 insulated circuit of length I, short-circuited at both ends. 

 Let a rapidly sinusoidal impressed force act at one end of 

 amplitude e , and let C be the amplitude of the current at the 

 distant end. Then, if the circuit be very long, 



C =p e -™r^, (38) 



where v is the speed (LS) -5 = (//.c)~ J , provided E/Ln be 

 small, say £. It may be considerably greater, and yet allow 

 (38) to be nearly true. We can include nearly the whole 

 range of telephonic frequencies by using suspended copper 

 wires of low resistance§. 



It is resistance that is so harmful, not self-induction ; as, in 

 combination with the electrostatic permittance, it causes 

 immense distortion of waves, unless counteracted by increasing 

 the inductance, which is not often practicable (see Note B, 

 p. 152). 



7. Distorted Plane Waves in a Conducting Dielectric. — 

 Owing to the fact that, as above shown, we can fully utilize 

 solutions involving the unreal g, by changing the meaning of 

 the symbols, whilst still keeping to plane electromagnetic 

 waves, we may preserve g in our equations (28) and (29), 

 remembering that H has to become 0, E become V, 4:irk 



* W. H. Preece, F.R.S., « On the Coefficient of Self-induction of 

 Copper Wires," B. A. Meeting, 1887. 



■f " El. Mag. Ind. and its Propagation," Electrician, Arts. xl. to 1. (1887). 



X See the sinusoidal solutions in Part II. and Part V. of " On the 

 Self-induction of Wires," Phil. Mag. Sept. 1886 and Jan. 1887. 



§ The explanation of the \~Lv dividing e in (38), instead of the ~Lv we 

 might expect from the /j.v resistance-operator of a tube of unit section 

 infinitely long one way only, is that, on arrival at the distant end of the 

 line, the current is immediately doubled in amplitude by the reflected 

 wave. The second and following reflected waves are negligible, on 

 account of the length of the line. 



