MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 411 
lotteries. But the introduction of these skew curves leads us to two important 
conclusions :— 
(i.) Ifa material be heterogeneous we have no right to suppose it must be made up 
of groups of homogeneous material each obeying the normal law of distribution. Each 
homogeneous group may follow its own skew distribution. 
(u.) If material obeys a law of skew distribution, the theory of correlation as 
developed by Gatton and Dickson requires very considerable modification, 
We may note two points bearing on these two conclusions, which do not seem 
without interest for the general problem of evolution. Fever mortality curves are 
skew curves. The general mortality curve—frequency of death at different ages—- 
is a compound of many diseases, but with sufficient approximation, it can be resolved 
into five components; three of these components are markedly skew, the other two 
less so. Selection, according to age, is thus distributed with different degrees of 
skewness about five stages in life; this at least suggests that selection according to 
the size or weight of an organ may be compound, if we take a considerable range of 
size, and that the components may have varying degrees of skewness. 
The correlation of the ages of husband and wife at marriage is a subject with 
regard to which we have a very fair amount of material. For a given age of the 
husband, the frequency of marriage with the age of the wife fits very closely a curve 
of Type IV., and with sufficient exactness very often a curve of Type III.* The 
sections of the surface of frequency are oval curves differing entirely from the ellipses 
of the Gatron-Dickson theory, but resembling in general the “oval” polygons 
obtained by taking horizontal sections of the frequency polyhedron for the correlation 
of cards of the same suit in two players’ hands at whist. Plate 9, fig. 19, shows how 
widely these differ from ellipses. There seems therefore to be considerable danger 
in assuming in vital statistics, whether in man or the lower animals, that the “ con- 
tributory” causes are independent. All the statistics for sizes of organs in animals, 
which I have yet analysed, if they are not compound, seem to agree in following a curve 
of Type IV., and suggest this kind of inter-dependence of the “ contributory ” causes. 
Their correlation surfaces of frequency will thus have for lines of level skew ovals— 
what for want of a better name may be termed “whist ovals” as distinguished 
from the ellipses which flow from the normal frequency surface. The remarks from 
quite a different standpoint of RANKE on skull measurements seem to lead to the 
same conclusion. I propose on another occasion to illustrate the resolution of 
compound curves into skew components, and further to deal with the main features 
of correlation in cases of a skew frequency distribution. 
* I have fitted some of Purozzo’s marriage statistics with skew curves, but reserve their discussion 
for the present, as they belong properly to the theory of skew correlation. 
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