CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 
263 
are sought of the fertility of mothers in mankind, the women will appear under their 
husbands’ names, and the labour of ascertaining whether two sisters have been 
included is enormous, when large numbers are dealt with. But if two or more sisters 
have been included, their mother has been weighted with her fertility, and when we 
seek the correlation between mother and daughter, it will be between mothers and 
daughters when weighted with fertility. Bat a still more serious difficulty arises 
from the fact that all records are themselves weighted records ; the same number are 
not married from each family, hence we are more likely to find a member of a large 
family included than a member of a small. The large families, when we seek a record of 
two generations, are more likely to appear than small families. Precisely the same diffi¬ 
culty occurs when we are dealing with thoroughbred horses ; a mare with large 
fertilit}' is less likely to have all her offspring colts, or all her progeny sold abroad, 
some one or more will probably ultimately come to the stud, and thus mares of large 
fertility are, d i^riori, more likely to contribute to our fecundity correlation cards. 
We do not get over this difficulty by taking the mother and only one of her offspring. 
The record is still weighted with fertility. The practical verification of this lies 
in the experience that the fertility of mothers will always be found to be greater 
than that of daughters, although the fertility of the community may really be 
increasing ; the weighting, of course, excludes sterility in the generation of mothers, 
but the mere exclusion of the sterile is far from accounting for the whole difference. 
What we actually find from our records are M'l and cr'i, but what we want for the 
problem of heredity are Mi and Equations (i.) and (ii.) do not suffice to determine 
these, because we cannot evaluate the third moment S {(a: — Mi)^ 2 :]. We can hardly, 
even for a first approximation, assume it zero, for the standard-deviation, and there¬ 
fore the individual variation is large as compared with the mean in the case of 
fertility, i.e., the distribution is markedly skew. 
Turning to offspring of the same sex as the parents, say : let Mg be the mean 
fertility of offspring taking one only to one parent for the number Ni of parents, 
supposing the parents not weighted with their fertility ; let Mb be the mean in the 
same case when the parents are weighted with their fertility; and let M'b be the 
mean of all recorded offspring of the second generation. Let cto, ct'^, o-'b be the 
standard deviations in the fertility of the offspring for the same three cases, and 
r, r, r' be the corresponding coefficients of correlation between fertility in parent 
and in offspring. It seems to me that r is the coefficient which actually measures 
the real inheritance of fertility, but that in any correlation table that we can form 
we shall get r or r”. 
Let y be the fertility of any individual among the offspring, and x the fertility of 
the corresponding parent; let \.x as before be the weighting of the parent, and \'x 
the number of offspring included in the record, X' being supposed a constant.* 
* I have been unable so far to find any sensible correlation between size of family and number 
married in man, but the point is worth a more elaborate investigation. 
