58 BELL SYSTEM TECHNICAL JOURNAL 



Continuing to regard A in Fig. 11 as any point in the scatter- 

 diagram, it can be shown that in the degenerate case characterized by 

 Z^ = all pairs of rectangular axes through A are "principal axes"; 

 for when Z^ = 0, equation (24) is fulfilled for all values of ^a (as will 

 be shown in the last paragraph of Subsection 2.1). Furthermore the 

 mean squares with respect to all pairs of rectangular axes through A 

 are then equal, as is shown by the fact that equations (25) and (26) 

 reduce to 



2t/2 = 2F2 = |Z|2 = X2 + P. (27) 



Since A in Figs. 10 and 11 can be any point, the desired formulas 

 for the last three of the "leading distribution-parameters" Zc, \pc' , 

 Su, Sv, relating to the point c in Fig. 10, are now seen to be immediately 

 obtainable from formulas (23), (25), (26) for the "principal param- 

 eters" relating to the point A, by merely letting A coincide with c, 

 the XA F-axes with the xc3'-axes and the UA F-axes with the WCT-axes; 

 for then ^x', U, V, Z become ypc , u, v, z respectively; whence, after 

 writing Su^ and Sv^ for u- and v^, the desired formulas are seen to be 



2^/^ag{±¥), (28) 



25,2 =1^2^ |72|_ (29) 



25.2 =T¥p:p li^l, (30) 



where, as will be recalled, z = Z — Zc = Z — Z represents (Fig. 10) 

 the position of any point T of the scatter-diagram of Z with respect 

 to the axes xcy through the center c parallel to the XA F-axes, which 

 latter are there the axes of Z; thus 2 = 0, though of course Z ^ in 

 general. In accordance with (28), (29), (30) the la_st three of the 

 leading distribution-parameters of Z = s -f Zc = s + Z, which are the 

 same as the last three of the leading distribution-para mete rs of z, are 

 completely determined by the two mean values z^ and | z | -. 



In order to represent explicitly the last three of the leading distri- 

 bution-parameters of Z as depending on Z — Z, it seems worth while 

 to rewrite (28), (29), (30) in the following equivalent forms: 



2rP/ = ag{ ± [Z - ZP), (31) 



25„2 = |Z -Z|2± |[Z - Z]2|, (32) 



25.2 = IZ-Z|2t |[Z-Z]2|, (33) 



which are completely determined by the two mean values [Z — ZY 



