R. A. Fisher 
521 
which leads by a process of simplification to the equation 
dx 
0 (cosh X — pry 
{r — p cosh x) = 0. 
Since cosh a; is always greater than pr, the factor in the numerator, r—p cosh, x, 
must change sign in the range of integration. We therefore see that r is greater 
than p. Further an approximate solution may be obtained for large values of n. 
The integrand is negligible save when x is very small, and we may write 
CO 
1 + for cosh X 
and (1 - pry " for (cosh x - prj'. 
Then rj^^ / = p/J (l + f ) / ^ dx, 
and in consequence, as a first approximation. 
The corresponding relation between t and t is evidently 
1 
i = T 1 + 
\ In 
It is now apparent that the most likely value of the correlation will in general 
be less than that observed, but the difference will be only half of that suggested 
by the mean, t. 
It might plausibly be urged that in the choice of an independent variable we 
should aim at making the relation between the mean and the true value approach 
the above equation, or rather that to which the above is an approximation, or 
that we should aim at reducing the asymmetry of the curves, or at approximate 
constancy of the standard deviation. In these respects the function 
log tan -J- (^a + that is, tanh~^ p 
is not a little attractive, but so far as I have examined it, it does not tend to 
simplify the analysis, and approaches relative constancy at the expense of the 
constancy proportionate to the variable, which the expressions in t exhibit*. 
* [It may be worth noting that Mr Fisher's t is the 0-square root mean square contingency — of the 
more usual notation, and is tlie expression used in determining the probabiHty that correlated material 
has been obtained by random sampling from uncorrelated material. Ed.] 
