PROBABILITY THEORY AND TELEPHONE ENGINEERING 69 

 That is, 



|Zi + ••• +Z„|2 = |Zi|2 + ... + |Z„|^ (106) 



provided the Z's are independent and not more than one is of non- 

 zero mean value, in accordance with (102). 



Part II: Applications 



The methods, theorems and formulas presented in Part I will now 

 be applied to two important problems in telephone transmission engi- 

 neering.^^ However, in each of these problems the solution is carried 

 no further than to formulate the "leading distribution-parameters" in 

 a form suitable for numerical evaluation in any specific case, since 

 Subsection 1.3 of Part I has furnished the means of solving such prob- 

 lems w^hen once these parameters have been evaluated and when the 

 distribution is known to be approximately " normal." 



The two problems mentioned above are treated separately in the 

 following Sections 5 and 6. Section 5 sketches the solution of the 

 general problem which was outlined in the Introduction (in Part I) 

 in connection with the equations there; Section 6 deals somewhat 

 fully with another problem, which, though specific, is yet of a rather 

 broad type. 



The problem in Section 6 has heretofore been handled by various 

 approximate and less comprehensive methods, as indicated in the first 

 footnote of the Introduction. The relative simplicity of the method 

 described by Crisson in his paper there cited is due to his simplifying 

 assumption (made just after his equations 26 and 27) which amounts 

 to assuming that the scatter-diagram is circular instead of, as actually, 

 elliptical. 



5. DEVIATION OF ANY CHARACTERISTIC OF A TRANSMISSION 

 SYSTEM OR OF A NETWORK 



This Section sketches an approximate solution of the general prob- 

 lem outlined in the Introduction, in connection with equations (I) and 

 (II), which are the general functional formulas for the contemplated 

 characteristic H and its deviation h, respectively; in general H and h 

 are complex. 



The present Section relates chiefly to formulas for the "leading 

 distribution-parameters" of h when this is regarded as a chance- 

 variable. 



In accordance with Section 2 (in Part I) the leading distribution- 

 parameters of h are completely determined by h, h^, \h\^. Evidently 



" An additional problem, crosstalk in a telephone cable, is treated in the un- 

 published Appendix C already mentioned in footnote 3. 



