MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
89 
type of inheritance is possible. We may have one progenitor, prepotent over all others 
and absorbing all their shares, who hands down to the offspring not a proportion of 
his character, but the whole of it without blend. If this progenitor is a parent we 
have exclusive inheritance, if a higher ancestor a case of reversion. I have dealt at 
some length with this type of inheritance under the title of the Law of Reversion 
in another paper. # We must consider in outline the main features of such inheritance, 
for the cases of eye-colour in man and coat-colour in the horse approximate more closely 
to the numerical values required by it, than to those indicated by the law of ancestral 
heredity. The chief feature of exclusive inheritance is the absolute prepotency of 
one parent with regard to some organ or character. It need not always be the 
parent of the same sex, or the same parent throughout the same family. Some 
offspring may take absolutely after one, others after another parent for this or that 
organ or character only. I believe Mr. Galton first drew attention, in his ‘ Natural 
Inheritance’ (p. 139), to this exclusive or, as he terms it, 'alternative heritage 
in eye-colour. In going through his data again I have been extremely impressed by 
it; even those cases in which children might be described as a blend, rare as they are, 
are quite possibly the result of reversion rather than blending. If we sujipose exclu¬ 
sive inheritance to be absolute, and there to be no blending or reversion, it is not hard 
to determine the laws of inheritance. Supposing the population stable, one-half the 
offspring of parentages with one parent of given eye-colour would be identical with 
that parent in eye-colour, the other half would regress to the general population 
mean, i.e., the mean eye-colour of all parents. Hence, taken as a whole, the regression 
of children on the parent would be "5. In the case of the grandparent the regression 
would be '25 ; of a great grandparent ‘125, and so on. With an uncle a quarter of 
the offspring of his brother will be identical in eye-colour with him, the other three- 
quarters will regress to the population mean, thus the regression will be '25. If we 
have n brethren in a family, and take all possible pairs of fraternal relations out of it, 
there will be \n(n — 1) such pairs; \n brothers will have the same eye-colour that of 
one parent, the other \n brother that of the other parent. Hence selecting any one 
brother, \n — 1 would have his eye-colour, and on the average \n woidd have 
regressed to the mean of the general population. In other words, the coefficient of 
regression would be (\n — 1 )/{\n — I + \ri) — n — 1 )j{n — 1). 
Accordingly 
n = 3 
Regression = 
•25 
n — 4 
99 
'3333 
1! 
99 
*3649 
■*<> 
<>> 
II 
cn 
99 
•375 
n = 5’3 
99 
•3833 
n — 6 
99 
•4 
n — oo 
99 
•5 
* ‘Roy. Soc. Proc.,’ vol. 66, pp. 140 d seq. 
VOL. CXCV.—A 
N 
