324 



BELL SYSTEM TECHNICAL JOURNAL 



where / stands generically for p and for a, and g is any positive real constant, 

 standing generically for h and for k; (1.20) and (1.21) are easily seen to be 

 true by imagining every variate in the universe of the /-variates to be 

 multiplied by g, thereby obtaining a universe of (g/)-variates. A second 

 way of deriving each of the three formulas (1.17), (1.18), (1.19) depends on 

 general integral relations of the sort 



( f{t) di = ^^ r fit) d{gt) ^u" f (-) d\. (1.22) 



•'« g ^ga g Jga \g/ 



A third way, which is distantly related to the second way, depends on the 

 use of the Jacobian for changing the variables in any double integral; thus, 



P(p,<r) 



(1.23) 



the first equality in (1.23) depending on the fact that the two sets of vari- 

 ables and of differentials have corresponding values and hence are so re- 

 lated that 



p(p<p'<p-\-dp, a<y<(T-\-da) = p(\<y<X-\-d\ m<m'<M+^/)u), (1-24) 



whence 



P(p,a) 1 dpd<j I = Pi\,fi) I dXdfjL |. 



2. The Normal Complex Variate and Its Chief Probability Functions 



The 'normal' complex variate may be defined in various equivalent ways- 

 Here, a given complex variate z = x -\- iy will be defined as being 'normal' 

 if it is possible to choose in the plane of the scatter diagram of s a pair of 

 rectangular axes, u and r, such that the distribution function P{u,v) 

 for the given complex variate with respect to these axes can be written in 

 the form^ 



P{u,v) 



1 



ZTTOuOv 



exp 



2Sl 



41 

 2Sl\ 



P(u)Piv). 



(2.1) 



We shall call w = u -\- iv the 'modified' complex variate, as it represents 

 the value of the given complex variate g — .t -f iy when the latter is referred 

 to the w,r-axes; P(u) and P{v) are respectively the individual distribution 1 

 functions for the u and r components of the modified complex variate ; and 



■^ Defined by equation (L3) on setting p = it and a = v. 



"This ecjuation is (12) of my 1933 paper. It can he easily verified tliat the (double) 

 integral of (2.1) taken over the entire n, ii-plane is equal to unity. 



