2.54 Proceedings of the Royal Society of Edinburgh. [Sess. 
Arranging these as before, we find for blended inheritance the following- 
groups : — 
£c 4 , 4x s y, Qx 2 y 2 , 4 xy 3 , y% 
and for mixed dominant inheritance, 
x 4 + 2 x 3 y, 2 x 6 y + Qx 2 y 2 + 2 xy 3 , 2xy 3 + y 4 . 
If y = 2x the latter reduce to 
5, 44, 32, 
so that the ultimate population is given by 
(5 + 44 + 32) p 
where p is the number of double pairs originally present in either parent. 
This case is obviously asymmetric, but with p large quickly approaches 
symmetry. It is not, however, necessary to assume that two races mix. 
In each race diverse pairs must necessarily exist. It is possible that these 
may appear in proportions sensibly obeying the normal law. This, however, 
makes no difference in the preceding theory. Variation of individuals will 
but tend to smooth the curve (see note). Asymmetry also arises if the 
number of dominant elements from each race is unequal. 
If the hypotheses above discussed are granted, two points emerge 
demanding consideration : — 
(1) How far does this grouping accord with the observed statistics ? 
(2) How far does the grouping permit of the correlation coefficients 
found by observation ? 
(1) Every large series of statistics of a population give groups to which 
Type IV. corresponds better than the normal curve. Those of Pearson,* 
Powys,f etc., may be taken as types. In the case of the diagrams given by 
the former, it is very noticeable that at the apex the statistics give a 
value in excess of that shown by the normal curve fitted to the statistics, 
and at the limits there is a defect on the part of the statistics resembling 
that just found theoretically. With the curves of the Scottish insane given 
by Tocher J the same is also found. 
(2) With regard to the values of the correlation coefficients we are on 
firm ground. As we have seen, the hypothesis of blended inheritance leads 
to correlation coefficients as follows : — 
Parent and offspring ..... *5 
,, „ grand-offspring . . . . '25 
,, ,, great-grand-offspring . . . T25 
,, „ great-great-grand-offspring . . '0625 
* Pearson, “ On the Laws of Inheritance in Men,” Biometrika , vol. ii. p. 357. 
t Powys, “ Data for the Problem of Evolution in Man,” Ibid., vol. i. p. 30. 
X Tocher, “ Anthropometry of the Scottish Insane,” Ibid., vol. v. p. 298. 
