74 BELL SYSTEM TECHNICAL JOURNAL 



admittance H. Then, when these deviations are not too large, the 

 resulting reflection coefficient p at the initial end of the system will 

 be approximately equal to the sum of the " propagated " or " apparent 

 values of the reflection coefficients arising from all of the individual 

 irregularities, that is, 



P = Efr' + Z^ + r', (118) 



whence, by substitution of (112), (115), (117), 



P = E fr<2'^ + E irQ''-' + rQ^"- (119) 



r=0 r=l 



Since Tr* ^r, t are chance-variables, p is a complex chance-variable. 

 In accordance with Section 2 (in Part I) the leadi ng d istribution- 

 parameters of p are completely determined by p, p^^ \p\'^; and these 

 will completely determine the distribution of p if it is "normal." In 

 the present problem, owing to the presence of r in equation (119), p 

 is not to be taken as zero; for, in accordance with the second half of 

 the first paragraph of this Section, 7 would usually not be zero in 

 practice. However, ^r and |^ would usually be zero and will here be 

 so taken. Hence, from (119), 



p = tQ"-. (120) 



Since the chance-variables fr, ?r, r are independent, and since only one 

 of them, namely r, has a non-zero mean value. Theorems 4 and 5 of 

 Subsection 4.3 (in Part I) are applicable to (119). Assuming all of 

 the loading-section deviations to be statistically alike, so that ^^ 



^2 = r^ IM^= lrl^ (r = 0,1,2, •••«), (121) 



and all of the loading-coil deviations to be statistically alike, so that 



^r' = e, ll--l^= l^l^ {r= 1,2, •••«), (122) 



application of Theorems 4 and 5 to (119), followed by the execution 

 of the indicated summations, gives the formulas 



. 1 _ .;4(«+l) 1 _ Q-i" _ 



ipi^= \^ r i!g. + \^rjrr^<f+ iHV". (124) 



where q denotes the attenuation factor of the loaded cable per peri- 

 ls The assumption represented by (121) is an approximation to the extent that, sta- 

 tistically, fo and f„ would usually differ somewhat from f ,-, where j = l,2,---n — 1. 



