PROFESSOR K. PEARSON AND DR. A. LEE ON 
90 
It will at first appear, therefore, that the fraternal regression with the size of 
families actually occurring will vary from - 35 to '4. 
To some extent these values would be modified by assortative mating, which 
actually exists in the case of eye-colour. The correlations between parent and 
offspring and between brothers would both be slightly increased. Thus if p be the 
coefficient of assortative mating, Ry the fraternal correlation with and iy without 
assortative mating, and r the coefficient for parent and offspring, # 
T) _ r f + 2 pr~ 
n f ~ 1 - 2 pr 2 ‘ 
If we put r f — - 36, r — '5, p = T, we find 
U f = -39. 
Thus we see that the regression or correlation for fraternal inheritance in the case 
of exclusive inheritance could not, with the average size of families, be very far from 
'4 of blended inheritance. 
A further source which can modify immensely, however, the fraternal correlation is 
the prepotency of one or other parent, not universally, but within the individual family. 
In the extreme case all the offspring might be alike in each individual family. Thus 
fraternal correlation might be perfect although parental correlation were no greater 
than \5. Hence, where for small families we get a fraternal correlation greater than 
‘4 to '5, it is highly probable that there exists either a sex prepotency (in this case, 
one of the parental correlations will be considerably greater than the other) or an 
individual prepotency (in which case the parental correlations based on the average 
may be equal). We shall see that fraternal correlations occur greater than ’5 in our 
present investigations. I have dealt with these points in my Memoir on the ‘ Law of 
Reversion,’f and also in the second edition of the ‘ Grammar of Science. 
Another point also deserves notice, namely, that with the series ’5, ‘25, T25, &c., 
for the ancestral coefficients in the direct line, the theorems proved in my Memoir 011 
Regression, Heredity, and Panmixia§ for the series of coefficients r, r 2 , r 3 . . . exactly 
hold. That is to say, if we have absolutely exclusive inheritance, the partial regres¬ 
sion coefficients for direct ancestry are all zero except in the case of the parents. 
This it will be observed is not in agreement with Mr. Galton’s views as expressed in 
Chapter VIII. of the ‘ Natural Inheritance.’ But I do not see how it is possible to 
treat exclusive inheritance on the hypothesis that the parental regression is about *3.|| 
Actual investigation shows that for this very character, i.e., eye-colour, it is nearer ‘5. 
If we take Table XIX. of Mr. Galton’s appendix, and make the following groups, both 
* This is shown in a paper not yet published on the influence of selection on correlation. 
t ‘Roy. Soc. Proc.,’ vol. 66, pp. 140 d seq.. 
I “ On Prepotency,” p. 459; “ On Exclusive Inheritance,” p. 486. 
§ ‘Phil. Trans.,’ A, vol. 187, p. 602, etc. 
|| Mr. Galton takes A throughout his arithmetic. 
