516 Distribution of the Correlation Coefficients of Samples 
The absolute frequency dj, with which r falls in the range dr, is therefore 
91-1 
A 2 
n - 4 
-3 
sin ^B^y sin 6 
dr. 
8. I do not see how to integrate the other expressions of the type 
^+11-^ cot ^ 7-Pdr 
sin^ <^ _ ^2 ' 
although a form could probably be obtained when p is even. The general 
expression for the second moment may, however, be deduced by means of a 
reduction formula. 
By a process of integration by parts it appears that, if we write 
n-i 
then = 7,1+2.0 + '«/„. 0 2, 
1 ■ T a /tan^a 
and since 1^ .2= Z7r[ - 
we may obtain successively 
V 2 
tan a + a 
■r ^, /tan^a tan^a \ 
16.2 = 2477 I — ^ + tana-al, 
/tan' a tai\^ a tan-' a 
-\ tan a + a , 
V 6 .5 ■ 3 
and so on, yielding, when n is even, the expression 
, = — TT |?i — 2 j tan'*""^ xdx, 
a form which may well hold when n is odd. 
The above expressions are useful in tabulating the numerical values of the 
second moment, + cr^ of the unit curve, which may easily be calculated in 
succession for different values of n when tan- a is taken to have some simple 
value. 
9. Before leaving this aspect of the subject it is worth while to give a more 
detailed examination of the mean of the frequency curves of r when n = 4. 
Two formulae are arrived at by Mr Soper, which are equivalent approximations 
of the second degree 
I. r = p 
II. r = p 
In [ 4/1 
(1 + Sp') 
1 - 
l + ^(l+3p^) 
1 
2(n-l)[ 4(?^-l) 
1-p"- 
i-j2(i-V) 
