Prof. Perry on Long -Distance Telephony. 223 



the most general problem of long-distance telephony, involving 

 certain terminal conditions, has been taken up by Mr. Heavi- 

 side ; but the ordinary mathematical physicist must find great 

 difficulty in understanding the investigation (Phil. Mag. 

 January 1887). Some of my students have recently obtained 

 numerical results, neglecting the terminal conditions, which 

 seem to me to be very instructive, and I think that the Tables 

 will have a permanent value. 



As a matter of fact, the fine is supposed to be of infinite 

 length, and we consider the state of a signal as it gets further 

 and further away from the origin. 



By comparing the current c at a section x centimetres from 

 the origin with the current at x + dx, and properly disposing 

 of the difference, we arrive at the equation : — 



d"c dp do 



d? =kl 7w + {kr + sl) dt +sro >- ■ ■ ■ (1) 



where (per unit length of conductor) k is the capacity, r the 

 resistance, I the self-induction, and s the leakage conductivity. 

 The solution which suits telephonic conditions is 



c = ae~^sin (pt—gx), 

 where 



gives the value of It if the minus sign is taken, and gives the 

 value of g if the plus sign is taken, and c = a sin pt is the 

 current at the origin. Of course p = 27rf, where / is the 

 frequency. Any number of such functions of any frequencies 

 may exist simultaneously. 



Two conditions must be satisfied in telephony. Taking the 

 shrillest and gravest notes of the human voice as being of 

 frequencies / and/', and taking therefore currents of these 

 frequencies : — let X be the distance at which the ratio of the 

 amplitudes of the shrill and grave currents is increased by 

 1/mth of itself; let Y be the distance at which one of the 

 currents has altered in lag behind the other by 1/nth of the 

 periodic time of the more rapid one ; then it is easy to see that 



X= l/{m(h-h')\*, 



ww(H)}- 



* This is approximate. If m is not large, the true expression ought 

 to he used, 



X=log 6 (l+i)/(A'-A). 



