38 BELL SYSTEM TECHNICAL JOURNAL 



where 



Dr = dF{K„'- ' Kn)/dKr, {r= \,2,-" fl). (IV) 



Before the physical elements are manufactured the ^'s are chance- 

 variables, in the sense already defined; for it is not possible to predict 

 the value which any one, say kr, will have, but only to state the chance 

 that it will lie within any specified range, this chance being calculable 

 from the known (or assumed) probability law pr{kr). Hence h is also 

 a chance-variable, whose probability law p{h) depends on the func- 

 tional formula for h and on the individual probability laws pi{ki), 

 ' • • pn(kn). In the general case, the "direct" problem is to calculate 

 from p(h) the probability that h will have a value lying within any 

 specified region of the /?-plane. 



In the types of problem contemplated in the present paper, the 

 probability law p{h) of h may usually be assumed to be approximately 

 "normal" (Subsection 1.2). Moreover, the specified region in the 

 A-plane will usually be a circle, since in such problems we are usually 

 concerned only with the magnitude of //, not with its angle. For 

 crosstalk, this is obviously true. For the usual type of two-way tele- 

 phone repeater operating between lines whose impedances do not 

 balance each other, it is true as a good approximation when the un- 

 balance is not too large, since then the practicable amplification de- 

 pends (approximately) only on the magnitude of the unbalance, not 

 on its angle. 



Unfortunately the complete solution of the problem for a circular 

 region is sufficiently difficult and laborious, particularly as regards 

 numerical evaluation, that apparently there has not heretofore been 

 sufficient incentive to lead to its being carried through — at least so 

 far as I am aware. ^ The present paper includes the needed solution, 

 in convenient form for practical applications, by means of the com- 

 prehensive set of graphs described in Subsection 1.3, supplemented by 

 Subsection 1.2 defining and formulating the " normal" complex chance- 

 variable, and further supplemented by Section 2 giving general meth- 

 ods and formulas for evaluating the distribution-parameters of the 

 "normal" complex chance-variable; and by Section 3, which applies 

 Section 2 to the case where, as is usual, the contemplated "resulting" 

 complex chance-variable is (at least approximately) a linear function 

 of other complex chance-variables. 



Section 4, which is somewhat in the nature of an appendix, supplies 

 a considerable number of formulas and theorems on "mean values" 



^ As well-known to those familiar with the literature of the subject, the solution 

 is quite easy for regions having certain other shapes, notably for an equiprobability 

 elliijsc and for a rectangle lying iiarallcl to a principal axis of such an ellipse. How- 

 ever, those solutions arc of no help in the case of a circular region. 



