Correlation and Application of Statistics to Problems of Heredity 23 



understands, for having determined the midparental contribution to be £ from 



either series, he now writes* of the values of - and — : 



n m 



"These values differ but slightly from J, and their mean is closely J, so that we may fairly 

 accept that result. Hence the influence, pure and simple, of the midparent may be taken as £, 

 of the nridgrandparent T , of the midgreatgrandparent |- and so on. That of the individual parent 

 would be T , of the individual grandparent y 1 ^, of an individual in the next generation F * T and 

 so on." 



Thus Galton reaches his Separate Contribution of each Ancestor to the 

 Heritage of the Child, a principle which is often spoken of as his Law of 



Ancestral Heredity. In reaching it he apparently drops his - series altogether 



and follows his — series with its geometrical system of taxation. This is 



distinctly more in keeping with the expression for the generant deviate U 

 above, which runs in a geometrical series. If we assume all the ancestors to 



have the same deviation h, we have U= r> h, and, if the offspring value 



might in such a uniform breed be also taken as h, it follows that y = 1 — /S. 

 Hence if we take the first midparents' contribution to be ^, i.e. y = ij, with 

 Galton, it follows that /3 = <^, and our series is Galton's geometrical series with 

 his radix value, a half. But I venture to think it was inspiration rather than 

 correct reasoning which led him to a geometrical series for U. 



On the other hand his multiple regression coefficients ^, \, |-, ... suffice to 

 determine what the correlations between an individual ancestor in any 

 generation and the offspring ought to be. They take the values for parents *3, 



for grandparents ^ x '3, for great-grandparents — x '3 and so on. Galton found 



Li 



his midparental regression § and took his parental to be \\. This is not so 

 far from "3, that we could say it confutes Galton's Ancestral Law. But we 

 find Galton taking the grandparental regression and therefore the correlation 

 \, the great-grandparental ^y and so on. These values form a series a, a 2 , 

 a', ... for the individual ancestral correlations and lead to y= 1, fi = 0, or to 

 the generant U being solely determined by the parents, the higher ancestry 

 contributing nothing to the generant J. Hence it follows that Galton's 

 Ancestral Law is not in keeping with the values he has taken for his 

 individual ancestral correlations. The reasoning by which he has reached 

 one or the other is defective. As I have said Galton's guess at a geometrical 

 relation for the coefficients of £7 was an inspiration, but his idea that a grand- 

 son is the son of a son and so his regression (and with a stable population 

 his correlation) must be | x ^ = \ is fallacious. Regression coefficients cannot 

 be obtained from each other in this manner. 



* Roy. Soc. Proc. Vol. lxii, p. 62. 



f This will be equal to the correlation, for the variabilities of both variates are taken to be 

 the same. 



t See Phil. Trans. Vol. 187, A, p. 306, 1896. 



