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BELL SYSTEM TECHNICAL JOURNAL 



XA F-axes by Z = X -\- iY, whence z = Z — Zc. Any pair of axes, 

 such as ucv, through the center c are called "central axes"; \}/c denotes 

 their orientation-angle with respect to the xcj'-axes, and hence with 

 respect to the XA F-axes. When ypc has such a value \pc' that m; = 0, 

 the central axes ucv are called "principal central axes"; the corre- 

 sponding values of ii- and v"^ are denoted by Su and 5/ respectively, 

 and Su and Sv are called the "principal standard deviations " pertaining 

 to the chance-variable w = u -\- iv. 



Conformably to the implicit definition in the first paragraph of this 

 Section, we may now state that the "leading distribution parameters" 

 of any complex chance-variable Z = X -{- iY are the four quantities 

 Zc, 4^c', Su, Sv defined and named in the preceding paragraph; it will 

 be recognized that these four quantities would be sufficient for fixing 

 the distribution if it were " normal." 



(Still referring to Fig. 10, it may be noted that an alternative set 

 of four parameters fixing the distribution of any " normal " complex 



chance-variable consists of Zc, Uxv, S-r, Sy, where Ux 



xy, Sx 



Sy" = y"^. The set Zc, \p/, Su, Sv was chosen as being much prefer- 

 able for this paper.) 



With a view to formulating precise definitions of the various addi- 

 tional technical terms needed, and to establishing general formulas 

 from which to deduce the desired formulas for the last three of the 

 "leading distribution parameters" Zc, ^Z, Su, S^, consider Fig. 11, 



Fig. 11 



which is a partial reproduction of Fig. 10, with the addition of the 

 axes UA V, which are any pair of rectangular axes through A, so that 

 W = U + iV represents the position of any point T with respect to 

 the UA F-axes, the position of T with respect to the XA F-axes being 

 represented by Z = X + *F, of course. Then it can be shown (Sub- 

 section 2.1) that when the orientation-angle ^a of the UAV-axes 

 (Fig. 11) with respect to the XA F-axes has either of the values '^a' 

 given by the equation ^ 



^ In this paper, if Z denotes any complex quantity, then agZ denotes its angle, 

 \Z\ its absolute value, Z its conjugate, ReZ its real part, and Im Z its imaginary 

 part (that is, the cofactor of i when Z is written in rectangular form). 



