H. E. Soper 
93 
Since a,? = (1 — r 2 )/\, and must grow very small as the number in the sample 
grows large, i.e. A, grows large, we see that r and r rapidly become equal as the 
sample increases or the distribution becomes symmetrical. 
The value of y 0 can be found from (Phil. Trans., loc. cit. p. 369) 
2/0-2 (m 1 + m 2 )™i+^ r(m 1 + l)r(m 2 +l) {Xi) ' 
The problem of the distribution of r would thus be completely solved, if we 
knew : 
(a) r in terms of p, 
(6) o> 2 in terms of p. 
Using Stirling's Theorem we can reduce the expression of y 0 to 
t . L( 1 1 1 \ 
-A (nij + m 2 + 1 ) v (mi + m 2 ) 12 
2/o = — 7= , e 
2V27T vm-.m, 
/ a/ 'i :• -)U + -)/a/U + V 1 + 
V27TO-,. V V mi/ V m 2 // V \ m^ + mj \ m^ Wa+l 
— (_i — - — 1 
A T (, 1 / 5 5 29 \1 
!+To - + -- ^ ( x11 )- 
V27TCT,. I 12 Vwij m 2 n^ + nh 
This approaches rapidly to the Gaussian value i\ r /(v / 27rcr ) .), if 07 be at all small 
and therefore A, and m 1 and m 2 large. For most purposes it is sufficient to take 
where the relation between r and r is given by (x) and it only remains to 
determine r and a r in terms of p. 
(2) Now the product moment value of the coefficient of correlation, p, between 
two measured characters in any population is defined by 
_ Pn ~PioPoi / • n. 
Pv>, P20 being the first and second moments in respect of the first character, 
Pan P02 those in respect of the second character, and p u being the first product 
moment, all derived from measures of individuals taken from some arbitrary 
origins of measurement in the two characters. 
If samples of number n are selected at random the moments will have 
different values p 10 ', p 20 ' etc., and in consequence the coefficient of correlation a 
different value, r, in any sample, and 
P\\ p\a pia 
V(iV -i>io' 2 ) x V(po 2 ' - Po/ 2 ) 
.(xv). 
