288 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(8.) Double Regression and Biparental Inheritance. 
(a.) General Formula and Comparison with the Theory of the Midparent. —If we 
apply the results of Section (7) to the problem of inheritance, we obtain some interesting 
results. Let i\ — coefficient of correlation for the same or different organs in two 
parents, he., be the measure of assortative mating; r 3 = coefficient of correlation of 
organs of offspring and male parent, he., be the measure of paternal inheritance ; 
r 2 = coefficient of correlation of organs of offspring and female parent, i.e., be the 
measure of maternal inheritance ; then the above formulae express the chief charac¬ 
teristic of biparental inheritance as modified by assortative mating. If rq, as 
probably is frequently the case, be small, then we see that the effect of assortative 
mating is to reduce the deviation of the offspring. Suppose there were no assortative 
mating, then the mean deviation of the offspring of selected parents would be 
h = ^ h 2 + r 8 J l h , 
and the actual value rq, being small, is clearly less than this. Again, even admitting the 
insignificance of the assortative mating in some cases, we see that, unless r 2 = r 3 , 
and further special relations hold between the variations of parents and offspring, this 
formula is not reducible to a mid-parent formula. 
For example, in the case of stature, consider the male offspring of two pahs 
of parents. In the first case, let the father be 4"'and the mother '923" above the 
average; in the second, let the father be 1" and the mother 3"'692 above the 
average. In both cases the height of the mid-parent is 2"’5 above the average, 
and the average male offspring will, on the mid-parent theory, exceed the mean by 
l"'67. But in the first case, the bi-parental formula gives l" 1 95, and in the second, 
1"'52. In the case of the female offspring of the same pairs, the mid-parental formula 
gives 1 " 54 for both pairs, and the bi-parental formula 1"'41 and 1"'25 respectively. 
These differences are due to the prepotency of paternal inheritance, and to the 
inequality of the variation in different male and female groups. 
These results have, of course, no greater validity than the statistics upon which they 
are based—a validity which Mr. Galton has been very careful to weigh (‘Natural 
Inheritance/ pp. 73, 131), but, I think, they suffice to show that the mid-parent theory 
must be looked upon as only an approximation of a rough kind. 
It must further be borne in mind, that the variability of a fraternity with given 
mid-parent is, if assortative mating be neglected, = oq ffl — ?y — r 3 2 ; or if r 2 be 
= r 3 , it is equal to <r ls /1 — 2 r 3 3 , and not oq ffl — r 3 . 
(b.) Effect of Assortative Mating on Cross Heredity. —Our formula of course 
applies to the problems I have classed as those of cross heredity. Unfortunately, I 
have no statistics at present to give any illustration of the intensity of cross-heredity. 
