PROBABILITY THEORY AND TELEPHONE ENGINEERING 59 



and \Z — Zp, though each of these depends on Z, which plays the 

 part of a reference value. 



The foregoing formulas, by aid of (88) and (89) in Subsection 4.2, 

 can be written also in the forms: 



2^Pc' = ag( ^iZ'- Z ]), (34) 



-2 — -2 



2Su' = |Z|2 - |zr± IZ^-Z I, (35) 



25,2 = -|2p _ \z\' =F |Z^ - Z !, (36) 



which are completely determined by the three mean values Z, Z^, 



Su and Sv are termed the "principal standard deviations," obviously 

 because they relate to the "principal central axes," namely the par- 

 ticular ucv-axes corresponding to V'c = ^c' (Fig. 10). They are special 

 values of the "standard deviations" Sx and Sy, which latter relate to 

 any specified central axes, xcy, and are defined by the equations 

 Sx^ = x^ and SJ^ = y^. 



By aid of the pairs of equations (29), (30) and (32), {33) and (35), 

 (36), the parameters b and S defined by equations (13) and (14) can 

 now be written in the following more explicit forms: 



\¥\ _ \{Z -ZY\ [Z' - z\ 



R"2 ■\z-z\' TzV'-\z\'' ^^^^ 



S^ = |2|2 = \Z - Z|2 = |Z|2 - \Z\\ (38) 



Returning now to the general case in which point A in Fig. 11 is any 

 point in the scatter-diagram of the given complex chance-variable, it 

 will be recalled that formulas (23), (25), (26) give the values of the 

 "principal parameters" relating to the point A. Let it now be re- 

 quired to formulate the principal parameters relating to any other 

 point, a, in terms of quantities relating to the point A. With this 

 purpose, consider Fig. 12. Here the XA F-axes are any rectangular 

 axes through A; but the UA F-axes are the principal axes through A, 

 as implied by the symbol ^^' for their orientation-angle. The xay- 

 axes are merely a pair of auxiliary axes through a drawn parallel to 

 the XAY-axes', and the way-axes are the principal axes through a. 

 Z, W, 2, w represent the position of any point T with respect to the 

 axes XA Y, UA V, xay, uav respectively; and Za represents the position 

 of point a with respect to the XA F-axes. Then, corresponding to 



