PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 353 



The fact that the curve for 6 = 1 is the straight Hne Qid;l) = 1/4 = 0.25 

 corresponds to the fact that for 6 = 1 the scatter diagram has degenerated 

 to be merely a straight Hne coinciding with the real axis, so that no point 

 outside of this line makes any contribution to Q{d;\). 



The fact that, at ^ = 90°, Qi9;b) = Q{90°;b) has for all b the value 1/4 = 

 0.25 corresponds to the fact that the area of a quadrant of the scatter 

 diagram is one-fourth the area of the entire scatter diagram. Hence 

 Q(360°;b) = 4Q{90°;b) = 1, which is evidently correct. 



Acknowledgment 



The computations and curve-plotting for this paper were done by Miss 

 M. Darville; those for the 1933 paper, by Miss D. T. Angell. 



APPENDIX A 



Derivation of Formula (2.15) for P{R,d) 



(2.15) will here be derived from (2.11) by utiHzing the fact that the 'areal 

 probabiUty density', G, at any fixed point in the scatter diagram must be 

 independent of the system of coordinates; for G dA gives the probability 

 of faUing in any differential element of area dA, and this probabiUty must 

 evidently be independent of the shape of dA (assuming that all linear dimen- 

 sions of dA are differential, of course). Thus, indicating the element of 

 area by an underline, we have, in rectangular coordinates, 



G dUdV = P{U,V)dUdV, (Al) whence G = PiU,V). (A2) 



In polar coordinates, 



GRdddR - P(R,d)dRde, (A3) whence G = P{R,d)/R. (A4) 



Comparing these two expressions for G shows that 



P{R,e) = RP(U,V). (A5) 



Thus, a formula for P(R,6) can be obtained from (2.11) by merelv multiply- 

 ing both sides of that formula by R. However, in the resulting formula it 

 will remain to express U and F in terms of R and 6, by means of the relations 



U ^ R cos d, (A6) V = R sin d. (A7) 



The final result, after a simple reduction, is (2.15), which is thus proved. 



APPENDIX B 



Formulas of the Curves in Fig. 3.3 



As in equation (3.22), Re will here denote the critical value of R, that is, 

 the value of R at which P{R) = P{R',b) has its maximum value; and Tc 



'2 Formula (A5) can be easily verified by the entirely different method which utilizes 

 (1.23). 



