350 BELL SYSTEM TECHNICAL JOURNAL 



In the case b — \, the curves in Fig. 6.1 suggest, by Hmiting considera- 

 tions, that P(0;1) is zero for all 6 except d = 0°, and that P{d;\) is infinite 

 for 6 = 0°. These conclusions are verified by formula (6.3), as this formula 

 shows that: 



P{d;\) = for ()°<d<mr; P{d;\) = --c for 6 = 0°, 180°. 



The curves in Fig. 6.1, though having the advantage of directly rep- 

 resenting P{d;b) as function of 6 with b as parameter, are somewhat trouble- 

 some to use because of their numerous crossings of each other. This 

 difficulty is not present in Fig. 6.2, which gives curves of P{d;b)/Ma,x 

 P(6;b), obtained by dividing the ordinates P{6;b) of the curves in Fig. 6.1 

 by the respective maximum ordinates of those curves, as given by (6.4), 

 so that the equation of the curves in Fig. 6.2 is formula (6.7). 



7. The Cumulative Distribution Function for the Angle 



The cumulative distribution function Q{<6,R]2) = Q{6) for the angle 6 

 of any complex variate TF ^ R{cos 6 + / sin 6) is defined by equation 

 (1.11) on setting p = d, a ^ R, pi = di =^ 0, ai = Ri = and 02 = R2 = »= ; 

 thus 



Q{d) = p{0<d'<d, 0<R'<oo). (7.1) 



A 'double integral' for Q{d), in the form of two 'repeated integrals,' can 

 be written down directly by inspection of the p( ) expression in (7.1) 

 or by specialization of (1.8); thus 



Q(d) = f \ [ P(R, d)dR dd ^ I f P(R, e) dd dR. (7.2) 

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Evidently these can be written formally as two 'single integrals,' 



Q{d) = f P(9) dd = \ P{R\< d) dR, (7.3) 



by means of the distribution functions P{d) = P(e\ R12) and P{R\ <d) 

 given by the formulas 



P(d) - [ P{R, 6) dR, (7.4) P(R \ <d) = f P(R, 6) dd. (7.5) 



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(7.4) is the same as (6.2). (7.5) is a special case of (1.6), and the left side 

 of (7.5) is a special case of P{p \ <a) defined by (1.13). 



The rest of this section deals with the case where W = R{cos d -\- i sin 6) 

 is 'normal.' Since this case depends on b as a parameter, Q{d) is here an 

 abbreviation for Q{6;b). 



A natural and convenient way for deriving formulas for Q(d) is afforded 



