Lines in Telephonic Transmission. 319 



course back and forth upon the line. The following theorem 

 will, however, lead directly to the solution and avoid the 

 infinite summation. 



Upon an infinite line of periodic recurrent structure a steady 

 forced, harmonic disturbance falls off exponentially from one 

 periodic interval to the next. The theorem is proven by the 

 consideration that as the line is infinite, there are identical 

 circuits beyond points separated by a periodic interval, and 

 the relative effect upon the disturbance of advancing an in- 

 terval must be the same for all portions of the line. 



Consider a uniform line (k, y) with loading coils A, B, &c. 

 at the interval d of impedance Htf, or H per unit length of 



Fig. 1. 



A > Ene "9V B 



VWVW * — - -Xe* d WwWXe rd 



a/VW\Aa Y- - Ye* ?d y*/WVV\Y<s rci - 



line, and we will designate by (K, V) the impedance of the 

 loaded line at the middle of a load and the propagation coeffi- 

 cient of the loaded line. Then if X, Y are the direct and 

 reflected current waves at the further side of the loading- 

 coil (A), at the next coil (B) the direct and reflected waves 

 are Xe~y d , Ye + ? d on the sending side, and Xe~^ d , Ye +rd on 

 the further side. At a coil the reflexion coefficient is 



Jc-CEd+Jc) Hd . 2k 



and 



,k + Hd + k ~ 2J* + Hd' 2k + Ed 



is the transmission coefficient ; the equations of condition at 

 B are therefore 



Ygr'ss--- H< * Xe-y*+ 



2k+ttd 2k + Rd ' 



Xe-M= - Ye~ rd + - — Xe-? d 



Eliminating X and Y, we have 



cosh (Td.)= cosh <yd+^ smhyd, . . .(18) 



which completely determines the propagation coefficient o( 



