PROBABILITY THEORY AND TELEPHONE ENGINEERING 37 



nected together when installed. Assuming that the elements them- 

 selves are known, from previous individual measurements on them, 

 to fall within their specified ranges of allowable variation, a compari- 

 son of the measured value of the contemplated characteristic with the 

 calculated value to be expected on the basis of probability theory will 

 give some indication as to whether some of the elements are incorrectly 

 connected. Further, when there is present not merely a single system 

 but a large number of systems which are nominally alike (for instance, 

 the various pairs in a multi-pair telephone cable), measurement of 

 the contemplated transmission characteristic of each of the systems 

 and comparison of the statistical distribution of these measured values 

 with their calculated theoretical distribution will give a more con- 

 clusive indication as to whether some of the elements are incorrectly 

 connected. 



Any particular problem to be solved can be handled most conveni- 

 ently and advantageously if the general problem is first formulated 

 analytically. Let us suppose, therefore, that // denotes the specific 

 transmission characteristic under consideration (for instance, a trans- 

 fer admittance, or a driving-point impedance, or a current-ratio), and 

 Ki, ■ • • Kn the internal parameters on which H depends; and let the 

 functional formula for // be 



H= F{K„ ■■' Kn), (I) 



where, of course, H and the K's are in general complex (on the suppo- 

 sition that the usual complex quantity method of treating alternating- 

 current problems is being employed). As we shall be particularly con- 

 cerned with the deviations of the various quantities from their nominal 

 values it will be convenient to suppose that // and the K's denote the 

 nominal values of the corresponding quantities, and that any actual 

 set of values are denoted by H -\- Ji and Ki-\- ki, • • • Kn + kn, so 

 that h and ^i, • • • kn will denote the corresponding complex deviations 

 of these quantities from their nominal values. Then the general 

 functional formula for // will of course be 



h = F{K, + ^1, • • • X„ -f kn) - FiKu • . . Kn). (II) 



Since h may be regarded as causally dependent on the ^'s, it may 

 naturally be called the "resulting" chance-variable. 



Usually the ^'s will be so small compared with the K's that the right 

 side of (II) can be replaced, as a good approximation, by the first- 

 order terms of a Taylor expansion ; thus, approximately, 



h = D,k, + • • • + Dnk„, (III) 



