PROBABILITY THEORY AND TELEPHONE ENGINEERING 63 



where, however, \pc is arbitrary, so that w is not determined until ypc 

 is assigned. Also let \}/c' be such a value of yj/c that uv = 0\ and let 

 SJ and Sv denote the corresponding values of w^ and v"^ respectively, 

 that is, the particular values taken by u^ and v^ when ^c = ^J , so that 

 uv = 0. 



The formulas (28), (29), (30) for i/'/, Su, S„ can now be established 

 in a purely analytical manner in just the same way as the more gen- 

 eral formulas (23), (25), (26) were established in Subsection 2.1. 



3. FORMULAS FOR THE LEADING DISTRIBUTION-PARAMETERS OF 

 A LINEAR FUNCTION OF COMPLEX CHANCE-VARIABLES 



To meet the needs in dealing with problems of the type handled in 

 Part II, namely problems involving linear functions of complex chance- 

 variables, the present Section furnishes formulas for the "leading 

 distribution-parameters" of any complex chance-variable Z which is 

 a linear function of any number n of complex chance-variables Zi, 

 • • • Zn, so that 



Z = a + h,Z, + • • • + 6nZn, (59) 



where a, hi, • • • bn are any constants, complex in general. 



It will be recalled that the "leading distribution-parameters" of any 

 complex chance-variable Z are the quantities Ze, 4'c', Su, Sv defined 

 and formulated in Section 2. 



Since, in general, Zc = Z, application of Theorem 3 of Subsection 

 4.2 to (59) gives _ _ _ 



Z = a + biZi + • • • + bnZn, (60) 



so that here Z is not zero even when Zi, • • • Z„ are all zero. 



The formulas for \P/, Su, Sv are (28), (29), (30), where z = Z - Z,; 

 or the equivalent formulas (31), (32), (33) or (34), (35), (36). 



With a view to using formulas (28), (29), (30), which have the 

 advantage of compactness, we introduce the quantities z and Zr de- 

 fined by the equations 



z = Z - Zc = Z -Z, (61) 



Zr = Zr- Zr, (^ = 1 , • • • w) , (62) 



which show that z = and that 



z^ = 0, (r = 1, . . . n). (63) 



Subtracting (60) from (59) and then substituting (61) and (62) into 

 the result gives 



z = bxZx + • • • + &nZ„, (64) 



which has the advantage of not involving a. 



