]\rATHE:\rATICAL CONTETBUTIOXS TO THE THEORY OF EVOLUTION. 289 
of each family, but, of course, the difficulty of a,voiding this is increased when a 
pedigree must be traced through three instead of two genei’ations. 
If d denotes granddaughter, g grandmother, the following results were obtained 
(Table YIII.) 
4-411, M, = 5-657, 
cr^ = 2-897, cr„ = 3'056, 
n.,= -ii23. 
The coefficient of regression of daughter s fertility on grandmother’s fertility = -1065, 
The probable error of — -0211. 
According to the law of ancestral heredity""' we should expect the grandparental 
correlation and regression to be half the parental and equcl to -15. Comparing the 
present result with Case (iv.), we see that -1123 and "1065 have to be compared with 
|•(-2130) and |-(•2233), or with -1065 and "1116. These are tlifterences well within 
the probable error of our results, or we may conclude that the correlation of a woman 
with her paternal grandmother is exactly what from Case (iv.) of Section (9) we 
shoidd expect to find for her correlation with her maternal grandmother. The reduc¬ 
tion from -15 to ‘1123 is just what we might have predicted after the maternal 
reduction from '3 to "2130. We, therefore, conclude that the fertility of woman is 
inherited through the male line with the same intensity as through the female, and 
this intensity is most probably that which would be indicated by the law of ancestral 
heredity. 
(12.) We do not stay to consider many jroints which flow from our tables, such, for 
examjale, as the amount of restraint indicated by the hump at the start of our various 
frequency distributions for size of families, partly because such consideration would 
lead us beyond our present scope, the inheritance of fertility, and partly because this 
point has been already dealt with by one of us in a paper on ‘Reproductive Selection.’ 
We consider that we have shown fertility in mankind to be an inherited character in 
both lines, and probably obeying the law of ancestral heredity.! By aid of our 
theoretical investigations it is clear that the average size of a family (Md, as deduced 
from our record data (M'l or M'd): is about 3-5 children, if the marriage lasts till the 
death of one partner, or at least till 15 years; it is about 3-9 to 4 children if the 
duration of the marriage is at least 15 years. Reproductive selection would increase 
tliis average by about "5 child per generation were its influence not counteracted 
* “ Mathematical Contributions to the Theory of Heredity, on the Law of Ancestral Heredity,’’ 
‘ Roy. Soc. Proc.,’ vol. 62. p. 397. 
t In the paper on “ The Law of Ancestral Heredity ” (‘ Roy. Soc. Proc.,’ vol. 62, p. 412} it is .stated 
that fertility is probably inherited, but the amount falls below that which would be indicated hj' the 
law of ancestral heredity. At that time only Case (i.) of Section (9) and Case (i ) of Section (10) had 
been worked out in detail. It is the rise of correlation with more stringent limitation of opposing 
influences, which suggests that after all that law^ is true for fertility as for other characters. 
VOL. CXC’TI.—A. 
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