268 
PROFESSOR KARL PEARSON, MATHEMATICAL 
(b.) If no reproductive selection exists, i.e., if fertility be not inherited, then r = 0, 
and 
Cr''^ = <j' 2 — Cft, M 2 = M^2 = M^'2» 
or, however we form a record of offspring, the mean value and variability of their 
fertility ought not to be changed. We shall see later that this is very far from the 
truth, and that these values are in whole or part sensibly affected by the manner in 
which the record is formed. 
(c.) Although there be no reproductive selection, Mj, M'l, and M"i will not all be 
equal, it is impossible that they should be. Further, cr,, a-\ and a-”i need not be 
equal; their degree of sensible divergence will depend on the nature of the primitive 
frequency distribution for parents. 
(c/.) If fertility be inherited, or reproductive selection be an actual factor of 
evolution, then we see, by comparing (v.) with (i.) and (vii.) with (viii.), that the 
mean fertility of mothers will always be apparently greater than the mean fertility 
of daughters. This follows, since r is always less than unity, and if the race be not 
subjected to secular evolution, other than that due to reproductive selection, cr, 
cannot differ very widely from cti."" 
(c.) An argument from means, as to whether fertility is inherited or not, is very 
likely to be misleading. We may choose two groups from the record for comparison, 
neglecting the fact that their frequency in the record is not necessarily that of their 
fre(|uency in the general population. Thus, if one person, say, in four were married, 
a marriage record of the community might exhibit the proper frequency of families 
of four, but it would not do so of families of one. The sort of fallacious arguments 
we have to be prepared for are, for example : 
(i.) That the fertility of the community is diminishing, because Mb is less 
than Mb- 
(ii.) That the fertility of the community is increasing, because Mb might be > M, 
or M'b be > Mb. 
(iii.) That fertility is not inherited, because, owing to natural selection, or other 
factor of evolution, one or other of these means for offspring is sensibly equal 
to one or other of these means for parents. 
Owing to tl)e extreme difficulty of insuring that the method of extracting the 
record really gives us definitely Mb, say, and not M'b (or M'b in part), I have 
discarded all use of the mean values in attempting to ascertain whether fertility is 
inherited. The following result, however, is tenqffing, and might possibly be made 
" A difference between and <73 '^ould mark natural selection, sexual selection, or some other factor 
of secular evolution at work; of secular, not periodic, evolution, as parents and offspring must have 
reached the same adult stage to have had their fertility measured, 
