H. E. SoPER, A. W. YounCt, B. M. Cave, A. Lee, K. Pearson 329 
formulae he gives for r and of the distribution of the correlation r in samples 
of n from a population of correlation p , we have found in practice the most exact 
are* 
^ = P[^- 2^, (1 - P") - 8^2 (1 - P') (1 + (1) 
and ^ v / j ^ r ^^^^ 

w = /• (r) = ^'-p^ ^ a _ ,.2r2 '-f"-> ] (iv) 
Soper also by assuming a Pearson curve of limited range + 1 to — 1 of type 
* = "»('-0""(^ + 
deduces the modal value f of r as approximately 
{1 - (a,,2 + r')} 
so that r would be determined from a knowledge of r and a/ + r^. 
The next step was taken by R. A. Fisher who gave in 1915 1 the actual frequency 
distribution of r, namely the curve 
II - 1 
(l-p2) 2~^^ ^Jl-* dn-^ / cos-^ i-pr ) 
TV 
Except for very low values of w this expression for i/„ does not provide a formula 
from which the ordinates of the frequency curve for /• can be readily determined, 
and as the problem was left by Fisher there were no rapid means of numerically 
determining either f or r or again a/. 
Clearly in order to determine the approach to Soper's approximations, and 
ultimately to the normal curve as n increases we require expressions for the moment 
coefficients of (iv), and further for practical purposes we require to table the 
ordinates of (iv) in the region for which n is too small for Soper's formulae to 
provide adequate approximations. These are the aims of the present paper. It 
is only fair to state that the arithmetic involved has been of the most strenuous 
kind and has needed months of hard work on the part of the computers engaged J. 
On the other hand the algebra has often been of a most interesting and suggestive 
character. 
(2) On Properties of the Function U = cos~^ (— x)jVT 
We have - ^ ^xcos'H-x) 
dx 1 — a?^ 
(1 
or {i^x^)^=l + xU. 
* See loc. cit. pp. 105 and 107. f Bimmtrilca, Vol. x. p. 507 e< seq. 
X Besides those whose names are given under the title, we have to thank I. Horwitz for some 
calculating aid, Ethel M. Elderton and D. Heron for occasional assistance, especially in the experimental 
part of the work, and lastly but very far from least we have to acknowledge the untiring work of 
H. Gertrude Jones and Adelaide G. Davin in the construction of the models the beauty and accuracy 
of which are not more than suggested in the plates. 
Biometrika xi 22 
