326 BELL S YSTEM TECH NIC A L JOURNA L 



With the purpose of reducing the number of parameters by 1 and of 

 dealing with variables that are dimensionless, we shall henceforth deal 

 with the 'reduced' modified variate W = U ■\- iV defined by the equation 



W ^ w/S = u/S + iv/S = U + iV. (2.9) 



Thus we shall be directly concerned with the scatter diagram of W = 

 U + iV instead of with that oi w = u -\- iv. 



The distribution function P(L'*,T') for the rectangular components U 

 and 1' of any complex variate W — U -\- iV is defined by (1.3) on setting 

 p = i' and cr = T; thus, 



p{u,v)dudv = p{U<u'<u-\-du,v<r'<vi-dV). (2.10) 



When the given variate z — x -\- iy is normal, so that the modified variate 

 11) — u -{■ iv is normal, as represented by (2.1), then, since S is a mere con- 

 stant, the reduced modified variate W — U -{- i]' defined by (2.9) will 

 evidently be normal also, though of course with a different distribution 

 parameter. Its distribution function P(t',l ) is found to have the formula 



1 r t/2 F2 ■ 



where P{1) and P{V) are the component distribution functions: 



t/2 



= F{U)P{V), (2.11) 



^(^) = vOT)^-r 



P(V) = ./..; . ^exp[-^4 



(2.12) 

 (2.13) 



\/ir(l - b) 



These three distribution functions each contain only one distribution 

 parameter, namely b; moreover, the variables U = u/S and 1' = v/S are 

 dimensionless. 



' The distribution function P{R,6) for the polar components R and 6 of 

 any complex variate W = R{cos 6 -\- i sin 6) is defined by (1.3) on setting 

 p = R and a — 6; thus 



P{R,e)dRd9 = p{R<R'<R^dR. d<d' <d-\-de). (2.14) 



For the case where 11' is 'normal,' it is shown in Appendix A that 



R [ -R' 



VT 



^'(^'^) = -Wr^-T. exp -^-fi:2 (1 - & cos 2d) 



(2.15) 

 exp[-L(l - 6 cos 20)], (2.16) 



"This formula can be obtained from (2.1) by means of (2.7), (2.8), (2.9) and (1.17) 

 after specializing (1.17) by the substitutions p = u,a = v and h = k = 1/5. It is (16) 

 of my 1933 paper, but was given there without proof. 



