334 BELL S YSTEM TECH NIC A L JOURNA L 



together with all points in the straight line segment extending downward 

 from the point at 0.9376 to the point at 0.7979 [= 2/ \/2^ = P'{R\\) for 

 R = 0+]. The curve of P(0; 1), because it has no area under it, is the 

 'non-effective' part of the curve of P{R\\). 



Starting at the origin of coordinates, where i? = 0, the complete curve 

 of P{R\\) consists of the curve of P(0;1), described in the preceding para- 

 graph, in sequence with the curve of P'(R;\), given by (3.21). Thus the 

 complete curve of P(R;\) is the locus of a tracing point moving as follows: 

 Starting at the origin of coordinates, the tracing point first ascends in the 

 P{R; b) axis to a height 0.9376 [= MaxP(i?;l)]; second, descends from 

 0.9376 to 0.7979 [= 2/ V2^ = P'iR'A) for R = 0-\-]; and, third, moves 

 rightward along the graph of P'(R;\) [b = l] toward i? = -f co . The locus 

 of all of the points thus traversed by the tracing point is the complete 

 curve'' of P{R;l). 



In addition to being the principal part ('effective' part) of the curve of 

 P{R;\), the curve of P'(R;\), whose formula is (3.21), has a further impor- 

 tant significance. For the right side of (3.21), except for the factor 2, will 

 be recognized as being the expression for the well-known 1 -dimensional 

 'normal' law; the presence of the factor 2 is accounted for by the fact that 

 the variable i? = | i? | can have only posiive values and yet the area under 

 the curve must be equal to unity. This case, b = 1, is that degenerate 

 particular case in which the equiprobability curves, instead of being ellipses, 

 are superposed straight line segments, so that the resulting 'probability 

 density' is not constant but varies in accordance with the 1-dimensional 

 'normal' law (for real variates), as noted in my 1933 paper, at the top of p. 45 

 thereof (p. 11 of reprint). 



All of the curves of P{R;b), where O^b^l, pass through the origin, 

 the curve of PiR;\.) [b = 1] being no exception, since the part P(0;1) passes 

 through the origin. 



Formula (3.12), supplemented by (3.15), shows that P(R; b) = at 

 i? = 00 ; and this is in accord with the consideration that the total area 

 under the curve of P{R;b) must be finite (equal to unity). 



Since P{R;b) — slI R — and a.t R — co, every curve of P{R;b) must 

 have a maximum value situated somewhere between R ~ Q and R — oo — 

 as confirmed by Figs. 3.1 and 3.2. These figures show that when b increases 

 from to 1 the maximum value increases throughout but the value of R 

 where it is located decreases throughout. 



The maxima of the function P{R;b) and of its curves (Figs. 3.1 and 3.2) 

 are of considerable theoretical interest and of some practical importance. 



''^ The presence, in the curve of F{R; 1), of the vertical projection, or spur, situated in 

 the P{K; b) axis and extending from 0.7979 to 0.9376 therein, is somewhat remindful 

 (qualitatively) of the'Gibbs phenomenon' in the representation of discontinuous periodic 

 functions by Fourier series. 



