PROBABILITY THEORY AND TELEPHONE ENGINEERING 73 



Then it can rather easily be shown that ^r is related to fr in accord- 

 ance with the simple but exact equation 



f/ = ^re-'^^ = trQ"-, . (112) 



where 



Q = g-r = g-A^-iB^ (113) 



T = A + iB denoting the propagation constant and Q the propaga- 

 tion factor of the loaded cable, each per periodic interval. It is some- 

 times convenient to call f/ the "propagated value" of fr, though it 

 is to be observed that the apparent propagation constant of fr is 2r 

 not r. Alternatively, f/ may be called the "apparent value" of f, 

 as viewed from the initial end of the system. 



Second, consider the typical coil-irregularity, situated in coil No. r 

 and consisting in the impedance-deviation Xr = Xr — X oi the im- 

 pedance Xr of this coil from its nominal value A^ The impedance- 

 increment Xr will be regarded as situated just beyond the nominal 

 mid-point of the coil; and the corresponding reflection coefficient ^r 

 pertaining to that mid-point will, in accordance with (107), be given 

 by the following formula, in which K denotes the mid-coil iterative 

 impedance of the loaded cable: 



)- ^ -^r Xr/ Zli. {'1 i A\ 



^"^ ~ 2K^Xr~ 1+Xr/2K' ^^^^^ 



Since ^r is situated at a distance of r — 1/2 periodic intervals from the 

 initial end, it appears at that end as a reflection coefficient ^Z such 

 that 



y = ^rQ'^-'. (115) 



Third, consider the terminal-irregularity situated at the junction 

 of the loaded cable with the terminal-admittance T and consisting 

 in the admittance-deviation t = T-H of the admittance T from the 

 mid-section iterative admittance H of the loaded cable. The corre- 

 sponding reflection coefficient r pertaining to that point will be given 

 by the formula 



/ t/2H 



2H + t 1 + t/2H 



(116) 



This will appear at the initial end as a reflection coefficient t' given 

 by the formula 



r' = tQ^^ (117) 



Finally let all of the loading-section admittances differ from their 

 nominal values, all of the loading-coil impedances from their nominal 

 values, and the terminal-admittance T from the mid-section iterative 



