MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
81 
could determine a quantitative scale, would give a distribution obeying—at any rate 
to a first approximation—the normal law of frequency. 
The whole of the theoretical investigations are given in a separate memoir, in 
which the method applied is illustrated by numerical examples taken from inheri¬ 
tance of eye-colour in man, of coat-colour in horses and dogs, and from other fields. 
We shall not therefore in this paper consider the processes involved, hut we may 
make one or two remarks on the justification for their use. If we take a problem 
like that of coat-colour in horses, it is by no means difficult to construct an order of 
intensity of shade. The variable on which it depends may be the amount of a 
certain pigment in the hair, or the relative amounts of two pigments. Much the 
same applies to eye-colour. In both cases we may fail to obtain a true quantitative 
scale, but we may reasonably argue that, if we could find the quantity of pigment, 
we should be able to form a continuous curve of frequency. We make the assump¬ 
tion that this curve—to at any rate a first approximation—is a normal curve. Now 
if we take any line parallel to the axis of frequency and dividing the curve, we 
divide the total frequency into two classes, which, so long as there is a quantitative 
order of tint or colour, will have their relative frequency unchanged, however we, in 
our ignorance of the fundamental variable, distort its scale. For example, if we 
classify horses into bay and darker, chestnut and lighter, we have a division which is 
quite independent of the quantitative range we may give to black, brown, bay, 
chestnut, roan, grey, &c. 
Precisely the same thing occurs with eye-colour; we classify into brown and darker, 
hazel and lighter, and the numbers in these classes will not change with the 
quantitative scale ultimately given to the various eye-tints. Our problem thus 
reduces to the following one : Given two classes of one variable, and two classes of a 
second variable correlated with it, deduce the value of the correlation. Classify sire 
and foal into bay and darker, chestnut and lighter ; mother and daughter into brown 
and darker, hazel and lighter, and then find the correlation due to inheritance 
between the coat-colour or eye-colour of these pairs of relations. The method of 
doing this is given in Memoir VII. of this series. Its legitimacy depends on the 
assumptions (i.) and (ii.) made above, which may I think be looked upon as 
justifiable approximations to the truth. 
Of course the probable error of the method is larger than we find it to be when cor¬ 
relation is determined from the product-moment. Its value varies with the inequality 
of the frequency in the two classes given by the arbitrary division. It will be 
least when we make that frequency as nearly equal as possible—a result which can 
often be approximately reached by a proper classification. In our present data the 
probable errors vary from about ‘02 to "04, values which by no means hinder us from 
drawing general conclusions, and which allow of quite satisfactory general results. 
(2.) So far we have only spoken of the two classes, which are necessary if we 
merely want to determine the correlation. But if we wish to deal with relative 
VOL. CXCV,-A, 
M 
