272 PROF. K PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
To obtain Table II., tables of double entry* were formed for the class enumerated 
in the first column, e.g., height of husband and height of wife as the variables 
x and y, and frequency of each pair of heights as From this table S ( xy ) was 
calculated by very laborious but straightforward arithmetic. This product moment 
was reduced to parallel axes (x , y') through the centroid of the system and r 
determined from the formula r = S (affi/^/Aoqoq (see p. 265). The p.e. of r was then 
found from the formula on p. 266. 
The coefficients of regression, in Table III., have the value roq/<x 2 , given on p. 267, 
where, if oq be the standard deviation of A, and oq of B, roq/oq is the regression of 
an A array on a, B type, and roq/oq, the regression of a B array on an A type. 
In Table IV., the array is first stated and then the type ; e.g., in the first line the 
type is the husband of given height, the array the distribution of all waves of 
husbands of this height. The first S.D. is that of the array obtained from the 
formula S.D. = oq\/l — r 3 , of p. 267, oq being the second S.D of Table IV., or the 
S.D. of the whole group from which the array has been extracted by selecting a 
particular value of the correlated group. 
Table V. gives the ratio for corresponding groups of the two sexes of the constants 
given in Table I. 
Now, a consideration of the probable errors recorded in Tables I. and II. shows us 
that, in several cases, definite conclusions may be drawm, and in certain other cases 
very probable conclusions. In particular, the probable errors of the correlation 
coefficients of inheritance are sufficiently small to show that these coefficients give 
the chief features of heredity in the group and for the characteristic v 7 e are dealing 
with. We may note one or two special features. 
(i.) Natural Selection .—We are dealing with two adult populations, and, therefore, 
should only expect to find traces of secular natural selection. The data, however, 
are not suited, either by their nature or number, to illustrate this point. There 
is a slight increase in height of sons over height of husbands, and a larger increase in 
height of daughters over height of mothers. Neither can be definitely asserted to be 
significant. Even if they w r ere significant they might be accounted for by (a) 
shrinkage due to old age,t and (b) increased physical activity and exercise in the 
middle classes of the younger generation, especially daughters. If we turn from 
means to S.D.’s we see again an insignificant change in the range of variation of 
husbands and sons, the sons being slightly less variable than fathers. This result, 
were it necessary to account for it, would be more likely due to our having taken 
sons from a less general population than husbands—a point to be borne in mind 
* It did not seem necessary to publish these tables, but the corresponding tables will be published 
when the fuller data for heredity in man, which I am at present collecting, are complete. 
f In my own collection of data, several parents state that they are now shorter than they used to be. 
The shrinkage in the case of fathei's of sons cannot be great in Mr. Ualton’s statistics, to judge by the 
means, unless we suppose a sensible regression in sons’ stature. 
