288 On Theories of Association 
theory of the correlation of sub-universes has been dealt with by Pearson* and the 
formulae obtained in 1901 were known shortly afterwards to be perfectly general, 
although the proof of this generality was only recently published j\ 
Let s x be the standard deviation of the sub-universe selected, o-j its standard 
deviation before selection. Let the three characters be represented by the sub- 
scripts 1, 2, 3 ; then if we write 
Pvi = r 12 Vl - s-f/af , p is = r ri Vl - s-fja* , 
we have for the correlation within the sub-universe 
ir 23 = 
^23 pnPis + 
Vl -p 12 - Vl 
pn V 1 - p ls * 
and ,7V ~ " ruVls 
Vl — r 12 2 Vl — r v3 2 
for the partial correlation coefficient. 
In the case of normal correlation to which Mr Yule is referring, Si is the 
standard deviation of the truncated portion of a normaL curve. For his special 
cases when that portion is one-half the frequency curve 
and x= * / — o"i. Hence 
2 /2 
1*23 
2 
^23 ?']2?*]:S 
7T 
i 2 / 2 2 
1 - - rj a/ 1 - - r n 2 
7T V IT 
But it is equally feasible to get almost in a line the value of ^ for any 
truncated portion of the normal curve other than one-half, and tables for 
determining the values of x ^ 2 and the moments of the tail about the severance 
ordinate at x, giving 
*i 2 = OY (a/*a - xPa'), 
were calculated by Dr Alice Lee and published in 1908§. These functions were 
termed the incomplete normal moment functions. Had Mr Yule paid attention to 
any of this work, he would hardly have published his special illustration and 
remarked "At present I have not been able to carry the matter further" (loc. cit. 
p. 628). The general formula had been given eleven years ago, and tables from 
which it was quite easy to calculate special cases were published four years ago ! 
* Phil. Trans. Vol. 200 A, pp. 1 — G6. 
t Biometrika, Vol. vm. p. 437. 
t Phil. Trans, loc. cit. Eqn. (lvi), p. 23. 
§ Biometrika, Vol. vi. p. 00. 
