Karl Pearson 
261 
offspring mean exactly follows the shift of the parental mean, or the mean of the 
offspring is the offspring of the mean parent. This gives at once the secular 
change due to weighting the parentages with their fertility. 
Further we must have 
(l-ii^)Sr J' 
(1 I-P'-WJ 
R 1 
(I - R') I- p'a^a,' 
From these How readilv 
R= ^ --^- -^ (ix), 
V^i + d-r)^ 
S.- , (X), 
^ (xi). 
2., = a, I 
1 + 
These results show us at once that differential fertility: (i) reduces the variability 
(Sg) of the offspring, (ii) reduces the variability of the eff"ective parentages, (iii) re- 
duces the apparent correlation between parentage and offspring. It should be 
noted that the non-diff"erential fertility is obtained by putting ctq = oo ; thus 
any diff'erential fertility up to cro= 0, which denotes breeding from a single grade, 
will give lowered values for R, 2i and 
While the variabilities and correlation are thus changed, it is worth noting 
that the regression of offspring on parents, i.e. Rl.Jlj^ pa^/a^ and is therefore 
unchanged. Further the equation to the regression line is 
z — m., = p ~{x — m^), 
or since m2 = m-^p<T2l'^x, it follows that 
z = p — X 
or is identical with the regression line for non-diff'erential fertility. This is, of 
course, only a case of the general principle found long ago* that selection of a 
character A does not alter the regression of a second character B on A, but it is 
interesting to note that the effect of differential fertility is as it were a pushing of 
the population down its regression line. 
Biometrika vii 
* Phil. Trans. Vol. 197, pp. 20, 21. 
34 
