PROBABILITY THEORY AND TELEPHONE ENGINEERING 57 



2<ir/ = ag( ± Z2), (23) 



then the "mean" of the product UV vanishes, that is, 



UV = 0, (24) 



and the mean of U^ and the mean of V^ have the values expressed by 

 the equations 



IIP = JZpi |z^|, (25) 



2F2 = IZI^T iZ^I, (26) 



and these values are extremum values in the sense that one is a maxi- 

 mum and the other a minimum when <^a has either of the values 

 ^.4' given by (23). Regarding the double signs in equations (23), (25), 

 (26), it is hardly necessary to remark that the upper signs go together 

 as one set, and the lower signs as another set. However, the presence 

 of the double signs is a triviality; for the UAV-axes (Fig. 11) with 

 respect to which equations (23), (24), (25), (26) are fulfilled are unique 

 except merely as to their designations (U versus V, with signs), the 

 values of '^a' differing only by a multiple of t/2. (In numerical appli- 

 cations it will usually be convenient to choose the upper set of signs, 

 so that U~ will be the maximum quantity and T'^ the minimum.) 



The particular Z7^ F-axes (Fig. 11) for which equation (24) is ful- 

 filled and for which ^a therefore has a value ^Z given by equation 

 (23) are calledjthe "principal axes" through A; and the corresponding 

 mean squares IP and V^ given by (25) and (26) are called the "prin- 

 cipal mean squares." It will therefore be natural, and will be found 

 convenient, to call ^a', f/^ V^ the "principal paramete rs" p ertaining 

 to the point A; they are seen to depend only on Z^ and \Z\^. 



More generally, when the point A in Fig. 11 is not restricted to 

 being the origin of the scatter-diagram of the given complex chance- 

 variable but is any point in that scatter-diagram and when the XA Y- 

 axes and the UA F-axes are any two pairs of rectangular axes through 

 ^, it is readily seen that the formulas (23), (24), (25), (26) remain 

 unchanged, although of course Z no longer represents the given chance- 

 variable but now represents merely the position of any point T with 

 respect to the XA F-axes, while W represents the position of T with 

 respect to the UA F-axes. The quantities ^^Z, TP, V^ given by equa- 

 tions (23), (25), (26) w^ill naturally continue to be called the "principal 

 parameters" relating to the point A, which is now any point. Thus 

 the "principal parameters" are more general than the last three 

 (^c\ Su, Sv) of the "leading distribution-parameters," to which the 

 " principal parameters " reduce when A coincides with the "center" c. 



