62 BELL SYSTEM TECHNICAL JOURNAL 



Hence 



2<ifA' = ag Z2 - WTT = ag ( ± Z^), (54) 



which is (23). Evidently there are only two geometrically distinct 

 values of ■^^i', namely that for even n and that for odd n; and even 

 this duality is a triviality, in the sense indicated in the latter part of 

 the paragraph containing equations (25) and (26). 



To prove (25) and (26) and at the same time to show that they are 

 extrema, we substitute (51) into (49) and (50) and take the mean 

 value of each result, thus getting 



2t/2 = |Z|2 + |Z2| COS (agZ2 - 2<lfA), (55) 



272 = |Z|2 - |Z2| COS (agZ2 - 2^a). (56) 



For the general case in which \Z'^\ is not zero, these*. two equations 

 show that when ^a is varied, U^ and V^ have extremum values when 

 ^A has any of the special values ^a satisfying (53) and hence satis- 

 fying (23). Substitution of (53) into (55) and (56) gives (25) and 



(26), which are thus proved. 



In the degenerate case characterized by Z^ = 0, the unrestricted 

 equation (52) shows that (24) will be fulfilled for all values of ^x. 

 This remark serves to prove the statement made in the paragraph 

 containing equation (27). 



2.2 Outline of a Purely Analytical Treatment of the Leading 

 Distribution-Parameters 



This Subsection is supplied, in accordance with the second para- 

 graph of Section 2, in order to show that the leading distribution- 

 parameters can be equivalently defined and formulated in a purely 

 analytical manner, that is, without the aid of the " scatter-diagram " 

 concept. 



With Z = X -{- iY denoting the given chance-variable, let Zc de note 

 that particular value of Z determined by the equation Z — Zc = 0, 

 so that Zc = Z, the superbar connoting the "mean value" ("expected 

 value") of Z, as defined just after equation (2). On account of the 

 restriction of the present Subsection to pure analysis, Z^ cannot here 

 be consistently called the "center of the scatter-diagram"; instead it 

 will be called the "central value" of Z. 



Next let z = X -\- iy and w = u -\- iv be the auxiliary chance- 

 variables defined by the equations 



2 = Z - Zc, (57) w = zexpi- i^c), (58) 



