MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 283 
that fertility is certainly inherited in the female line. By selecting fairly homogeneous 
material with a more definite and complete record than exists for the heterogeneous 
material of the previous case, we have carried up the correlation to five times its 
previous value, and within a reasonable distance of the value *3 which would he 
required by the law of ancestral heredity. The homogeneity of our material is 
evidenced by the reduction in both standard deviations ; the greater completeness of 
the record by the rise in the fertility of daughters ; and the non-weighting of the 
fertility of mothers by tlie fall in their mean fertility. 
If the reader will turn back to the theory of the influence of heterogeneity on 
correlation in section (7), he will notice that the expression in (xxxiv.) will be negative, 
and therefore the apparent correlation less than the real, if we form a mixture of two 
groups in which and m ^ Now the entries of women in the Landed 
Gentry and other records are very often entries of “ heiresses,” while the entries of 
women in the Peerage are entries because of class. An “ heiress” naturally has fewer 
brothers and sisters than another woman on an average, or we may expect > m\,. 
On the other hand an “ heiress ” need not have fewer children than other women, 
unless her heritage is the result of her coming from an infertile stock, and is not a 
result of the incompleteness of her parents’ marriage. If she belongs to a somewhat 
lower social grade, she may possibly be more fertile than the average of a higher 
social grade. In this case will be > and when we come to mix records of the 
Peerage with those of the Landed Gentry and Family Histories, we need not be 
surprised to find the correlation of fertility much weakened, as it undoubtedly is (as 
shown by (i.) and (ii.) above) by the mixture. 
Let us next apply our theory to the above results. We are now dealing with 
M’l, Mb, cr'i, cr'o. Assuming that there is no secular change cti = a,, and accordingly 
since a'l = cr'., sensibly, formula (xi.) shows us that both = cti. 
Further, if Mj = M.,, formula (v.) is a quadratic equation to find ; substituting 
for M'l, aj, and r, w'e have, on solving and taking the only admissible root. Mi =: 3’4625. 
'fhen, applying formula (i.) to find Mb, wc have : 
Mb = 5-GGO. 
This is not cpiite as high as the observed value 5'85G, but it suflices to show that 
our theory expresses the main facts. In all probability we have not entirely freed 
our results from weighting with fertility ; because, although eve)y endeavour was 
made to take only one from each family, it is possible that pairs of sisters have occa¬ 
sionally crept into the record. 
(iii.) Table III. gives the result of 1000 cases taken from tlie Landed Gentry. As 
we have already noted, the women recorded are largely “heiresses,” and we believed 
this might be one of the chief sources of the heterogeneity of the material in Table I. 
The conditions of selection were made somewhat mure stringent, and were as follows :—■ 
Only one daughter was taken from each family, and her marriage must have lasted at 
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