R. A. Fisher 
517 
and these we shall compare with the form 
III, 
r= (a + cot a — a cot^ a), 
p 
•1000 
•2000 
•3000 
•4000 
•5000 
•6000 
•7000 
•8000 
•9000 
•9500 
I 
•0853 
•1710 
•2578 
•3463 
•4377 
■5333 
•6347 
•7443 
■8649 
•9304 
II 
•0847 
•1697 
•2555 
•3419 
4310 
•5241 
•6236 
•7330 
•8566 
•9254 
III 
•0850 
•1704 
•2570 
•3451 
•4360 
•5301 
•6290 
•7357 
•8540 
•9209 
It will be observed that the approximations lie on either side of the exact 
value over the greater part of the range, and that the error of the first 
approximation increases up to the value when p = •9. The second formula 
gives the correct value somewhere between S and ^9, and is thereafter too 
large. 
For the particular case p — ^6608, 
I find (formula III) r = "5897, nearly the maximum difference from p, 
Mr Soper gives (p. 109) the value ^5933 
and the experimental data •5609. 
The two theoretical values are much nearer to each other than either is to 
the experimental value. On the whole, it is obvious that even in this unfavour- 
able case Mr Soper's formulae possess remarkable accuracy. 
10. The use of the correlation coefficient r as independent variable of these 
frequency curves is in some respects highly unsatisfactory. For high values of r 
the curve becomes extremely distorted and cramped, and although this very 
cramping forces the mean f to approach p, the difference compared with 1 — p 
becomes inordinately great. Even for high values of n, the distortion in this 
region becomes extreme, and since at the same time the curve rapidly changes 
its shape, the values of the mean and standard deviation cease to have any very 
useful meaning. It would appear essential in order to draw just conclusions from 
an observed high value of the correlation coefficient, say •99, that the frequency 
curves should be reasonably constant in form. 
The previous paragraphs suggest that more natural variables for the treatment 
of our formulae are afforded by the transformations 
t = tan /3 
r 
Vl - jO^ 
The expression for the frequency curve (II) 
T = tan a = 
