PROBABILITY THEORY AND TELEPHONE ENGINEERING 75 



odic inter\al, that is, 



q = \Q\ = e-\ (125) 



A denoting the attenuation constant of the loaded cable per periodic 

 interval, in accordance with equation (113). 



When g-*" is small compared to unity, formulas (123) and (124) 

 reduce approximately to 



_ ^. -L. T2Q2 _ 



P' = i_q: + ^'<2^". (126) 



7772 I I t I 2^2 



\P\'- \_[J + MV". (127) 



When, further, q is nearly equal to unity, which by (125) will be the 

 case when 2 A is small compared to unity, then formula (127) reduces 

 approximately to 



TTT2 I I t I 2^2 



4A 



Returning to the formulas (110) and (114), which give fr and ^r in 

 terms of yr/2H and Xr/2K respectively, it may be said that for prac- 

 tical applications it is more convenient to express ^r and ^r in terms 

 of the fractional deviations 8r and er and the coefficients D and G, 

 defined by the following four equations: 



5r = yr/Y, (129) er = Xr/X, (130) 



D = Y/2II, (131) G = X/2K. (132) 



With these substitutions, formulas (110) and (114) become 



It can be shown that D and G, defined by equations (131) and (132), 

 are approximately equal and may be expressed approximately in each 

 of the forms appearing in the equation 



^ = ^= V l + A^F/4 = ^1 - V^^i^ = tanh (r/2). (135) 



with //, K, r already defined in connection with equations (108), 

 (114), (113) respectively. Equation (135) would be exact if the cable 

 wires were perfectly conducting, since then each section-admittance 

 Y could be regarded as effectively localized, so that the loaded cable 

 would be effectively a ladder-type structure, for which equation (135) 

 is known to be rigorously exact. 



