PROBABILITY THEORY AND TELEPHONE ENGINEERING 71 



a restriction would not correspond to the conditions usually existing 

 in practice. 



The resulting deviation in the impedance Z of the initial end of the 

 system (Fig. 13) from the iterative impedance of the loaded cable is 

 a complex chance-variable which is of much engineering importance 

 in case the loaded cable is to constitute part of a transmission system 

 containing a 2-way repeater, of the 22-type, connected between the 

 initial end of the loaded cable and the remainder of the transmission 

 system (not shown in Fig. 13) ; for, so far as the loaded cable is con- 

 cerned, the practicable amplification obtainalile from the repeater will 

 depend approximately inversely on the impedance-deviation of the 

 loaded cable; more precisely, it will depend inversely on the reflection 

 coefficient defined, in terms of the impedance-deviation, by equation 

 (107) below. 



In Fig. 13 the loaded cable is represented as beginning with a half- 

 section, and as ending with a half-section, and the latter as terminated 

 with an admittance T. The formulas herein established are for this 

 system. Analogous formulas for a system beginning and ending with 

 half-coils, instead of with half-sections, can be obtained in an analo- 

 gous manner, or even written down directly by analogy. 



The important reflection coefficient mentioned at the end of the 

 second paragraph, and to be denoted by p, is defined by the equation 



Z-h _ (Z-h) _ (Z-h)/2h 



^ Z + h 2h + {Z-h) 1 + {Z - h)/2}r ^ ^ 



Z denoting the impedance of the system in Fig. 13, and h the mid- 

 section iterative impedance of the loaded cable. Each of the forms in 

 (107) is useful and significant. However, ii W = l/Z denotes the 

 admittance of the system, and H = \/h the mid-section iterative ad- 

 mittance of the loaded cable, the equation for p can be written in the 

 equivalent forms 



W-H _ (W-H) _ {W-m/lH 

 ^ W + H 2H + {W - H) I -\- {W - H)/2H' ^ ^ 



and these forms, instead of those in (107), will be the ones mostly 

 used herein, because of their simpler and more direct relations to the 

 corresponding current deviations. For, if an electromotive force E is 

 impressed between the terminals of the system in Fig. 13, the current I 

 there will be WE; and if /" denotes the value that / would have if 

 W were equal to H, then P = HE. Thus the reflection coefficient p 

 defined in terms of W and // by equation (108) can be expressed in 



