Lines in Telephonic Transmission. 315 



velocity, and line impedance being respectively — 



r /rr 



a= 2VL' 

 l 



= VLC ' 

 k- ^ 



U 



For a loaded line these formulae apply approximately and 

 the problem taken up in this paper is the determination of 

 the correction factors. This is direct ; the approximation of 

 loaded to uniform line shows only indirectly the performance 

 of the loaded line. 



A summary must be given of the general transmission 

 formulae upon uniform lines which will be required. An 

 harmonic electromagnetic steady state is resolvable into a wave 

 propagation with definite velocity and attenuation, but no dis- 

 tortion, throughout each uniform interval of the line, the 

 wave suffering reflexion at points of non-uniformity. This 

 is mathematically an exact and simple analysis of the steady 

 state, but it conforms only approximately to the physical 

 action. It neglects the diffusion or distortion resulting from 

 dissipation which is. in the steady state, not in evidence, 

 except indirectly, as a variation with the frequency of the 

 velocity and attenuation. This variation of the velocity and 

 attenuation furnishes sufficient measure of the distortion at 

 the head of an advancing wave for most practical applica- 

 tions. Except for this head distortion an harmonic steady 

 state is established by pure wave propagation — the line 

 presents a definite line impedance which determines the 

 initial current at the impressed force; the electromagnetic 

 wave originating at the impressed force travels with a de- 

 finite attenuation and a definite velocity along the line and 

 divides upon reaching a point of non-uniformity into a re- 

 flected wave and a transmitted wave. Repeated reflexions 

 establish the steady state. 



The equation of a simple current wave upon a uniform 

 line is 



i= -j-e-**, (1) 



where the line impedance k, the propagation coefficient y, 



