MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
115 
eye-colour extend. It is noteworthy that while the- two highest correlations are 
reached for nephew with paternal and with maternal uncles, nearly the two lowest 
are found for niece with paternal and with maternal aunts. Without laying 
special stress on each small difference, it must be admitted that the avuncular 
correlations vary in a remarkable manner with sex, and differ very widely from 
the practical equality of resemblance which we might a priori have expected to 
exist in this relationship. 
(cl.) Direct Heredity. First Degree .—Here we have a mean value of the paternal 
correlation = ’4947. This is in excellent agreement with the ‘5 to be expected by 
our theory of exclusive inheritance ; it is thus in practical agreement with the value 
of the parental correlation obtained for the inheritance of coat-colour in horses. It 
would not be inconsistent with a high value for y in the theory of blended inheri¬ 
tance, but such a value of y is rendered impossible by the values we have obtained 
for collateral heredity (see ‘Roy. Soc. Proc.,’ vol. 66, p. 140 et seep). 
We may draw the following special conclusions :—(i.) The son inherits more 
strongly from his parents than the daughter, the mean correlations are as ’5160 
to ‘4733; (ii.) The son inherits more strongly from his father than his mother, 
and the daughter more strongly from her mother than her father. 
This is part of the general principle which we have seen to hold, namely : that 
change of sex weakens the intensity of heredity. 
The correlation of father and daughter appears to be abnormally below the other 
three, but something of the same kind has been noted in certain stature data ; as it 
is, the high correlation of father and son renders the mean paternal correlation with 
offspring (‘4936) sensibly equal to the mean maternal correlation (‘4956). 
(e.) Direct Heredity. Second Degree.-—ii we take the mean value of the eight 
grandparental correlations, we find it equals ‘3164, while the mean value of the 
regression of offspring on their grandparents is ‘3136. These results are absolutely 
incompatible with the ‘15 required by Mr. Galton’s unmodified theory, and they in 
fact put the theory of blended inheritance entirely out of court. At the same time, 
unlike the cases of parental, avuncular, and fraternal inheritance, they cannot be said 
to be in good agreement with the value '25 required by the theory of exclusive 
inheritance. We have to admit that our grandparental data are shorter series than 
in the other cases, and that guesses as to grandparents’ eye-colour, based on memory, 
miniatures, &c., were more likely to be made. Further, such guesses might easily 
be biased by a knowledge of the eye-colour of more recent members of the family. 
Still a reduction from '32 to '25 is a very large reduction, and we have to remember 
that for long series in the case of the thoroughbred horses, with no such guessing at 
colour as may occur with ancestors’ eyes, we found '3353 for the maternal grand- 
sires, a result in excellent agreement with the '3343 found for the maternal grand¬ 
fathers in the present case. Thus while the theory of exclusive inheritance without 
reversion suffices to describe the quantitative values we have found for the parental, 
y 2 
