44 BELL SYSTEM TECHNICAL JOURNAL 



diagram with respect to the «,y-axes. P„ and Su have the values 

 already defined by equations (1) and (2) respectively, and P^ and 5^ 

 are defined by those same two equations after changing utov through- 

 out; Su and Sv are distribution-parameters called the "standard devia- 

 tions" of u and v respectively. 



It will be recognized that the u,v-2i\es are the "central principal 

 axes," namely that pair of rectangular axes which have their origin 

 at the "center" c of the scatter-diagram of z, and hence oiw = u -{- iv, 

 and are so oriented in the scatter-diagram that uv = 0. By the "cen- 

 ter" c of the scatter-diagram of any complex chance-variable s is 

 meant that point Zc with respect to which as origin the "mean value" 

 (Section 4) of the chance-variable is zero, that is, such that z — Zc = 0; 

 thus, Zc = z. In the case of the chance-variable w = u -}- iv, whose 

 origin is the center of the scatter-diagram, so that zUc = 0, it is thus 

 seen that w = 0; the fact that the M,t;-axes have their origin at c may 

 conveniently be indicated by designating them as the ucv-axes. 



Instead of taking S^^ and S^ as the distribution-parameters it will be 

 found preferable to take b and S, defined by the equations * 



, Ojj 'Ju" l \'-^v/ 'Jul M "^^ 



It is convenient, and fairly natural, to call S the "resultant standard 

 deviation" of ^ u and v. More explicit formulas for b and 5" are (37) 

 and (38) established in Section 2. 



Equation (12) shows that the equiprobability curves of the complex 

 chance-variable iv = u -{- iv are a set of similar ellipses centered at the 

 center c of the scatter-diagram; and that the axes of these ellipses 

 coincide with the principal axes of the scatter-diagram and have 

 lengths proportional to Su and S^, and hence proportional to Vl + b 

 and Vl — 6 respectively, since, from (13) and (14), 



2SJ = (1 + b)S\ 25,2 = (1 _ 6)52. 



Thus, when S^ = Su and hence when b = 0, the ellipses degenerate 

 to circles. When ^^ = or kS„ = and hence when & = -f 1 or 



^ A parameter which itself is simpler than b is a = SJSu; but if a were used in- 

 stead of b most of the formulas in the unpublished Appendix A, mentioned in foot- 

 note 3, would be rendered consid erably longer and more complicated. 



« It is to be noted that |m) I'' is not equal to .Sf,^,, if, as is natural, this is defined 

 by the equation 



5f^i = (M -WD' = kP - H . 



