and Distributed Static Capacity in a Conductor. 391 

 In these equations Im. denotes the impedance, (ft 2 + LW)^; 



p =r y — \/Iwl. — Leo ; and a =y — Vl m - + Loo. 



The solutions in equations (3) and (4) show that the poten- 

 tial and current are propagated in harmonic waves whose am- 

 plitudes decrease with the distance from the origin according to 

 a logarithmic decrement. At any point of the conductor the 

 potential and current vary as simple harmonic functions of the 

 time "with constant amplitudes which are different for every 

 point of the conductor. The current wave is propagated in 

 advance of the potential wave by an angle d such that 



p 

 tan d = — . This phase difference diminishes with increase of 

 a. x 



frequency when there is self-induction, but becomes a constant 



2tt 

 angle of 45° when L=0. The wave length is — and the 



rate of propagation is — . The wave length and rate of propa- 

 gation each become less as the self-induction increases. The 

 wave of higher frequency will have the shorter length and be 

 propagated the faster. This difference in rate of propagation 

 of waves of different frequencies is most marked when there 

 is no self-induction. 



The distance at which the amplitude decreases to — th of its 



£ 



value is — = ~ — ; 7. ; the time for the decrease is 



2> 2tt tan d ' co tan 6' 



The rate of decay is most rapid when there is no self-induc- 

 tion. The waves of higher frequency decay more rapidly 

 than those of lower frequency ; when there is no self-induc- 

 tion this difference in the rate of decay is the greatest. 



The difference in the rates of propagation and decay of 

 waves of high and low frequency doubtless constitutes the 

 limitations to the use of the telephone. As the several har- 

 monic components of a complex tone advance along the con- 

 ductor, they keep shifting their relative phases according to 

 the difference in their rates of propagation, and also change 

 their relative intensities according to the difference in their 

 rates of decay, thus changing the resultant combination tone 

 and materially altering its quality. These effects are always 

 present in circuits containing distributed static capacity but 

 are not so marked when there is also self-induction. 



Physical Laboratory of Cornell University, July, 1S92. 



