Mathematical Contributions to the Theory of Evolution. 387 



The confidence I put in the truth of the law is not measured by 

 Mr. Galton's researches on stature or on colour in Basset hounds, 

 however strong evidence these may provide, but rather on the fact 

 that the theory gives a priori the correlation between parents and 

 offspring, and that this correlation is practically identical with the 

 value I have myself determined from these and other observations. 



With reservations as to how "mid-parent" shall be denned, I 

 would state the law of ancestral heredity as follows : — 



If k s be the deviation of the sth mid-parent # from the mean of 

 the sth ancestral generation, and Jc be the probable deviation from 

 the mean of the offspring of any individual, o s the standard deviation 

 of the sth mid-parental generation, <r of the generation of the off- 

 spring, then 



h = \ 12*,+ pi.+p h+ + 



°1 °2 °3 °4 



This is the somewhat generalised form of the law, which Mr. 

 •Galton sums up as " each parent contributes on an average one- 

 quarter, or (0*5) 2 , each grandparent one-sixteenth, or (0'5) 4 , and so 

 on, and that generally the occupier of each ancestral place in the nth. 

 degree, whatever be the value of n, contributes (05) 2n of the 

 heritage, "f 



The generalised form above allows for a secular modification of 

 the means and variabilities of the successive generations. 



(2) Let ri, r 2 , r 3 , r 4 , . . . . , be the coefficients of correlation between 

 offspring and parent, grandparent, great-grandparent, &c, respec- 

 tively. Then, if correlation remains constant during the successive 

 generations, r Hn , t would be the correlation between the parent of the 

 nth generation and the parent of the sth generation if they be in the 

 direct line of ascent, for one of these is the (n^-s)th parent of the 

 other. It must be remembered that if r q be the correlation between 

 an individual qth parent and his offspring, r q may theoretically 

 have a great variety of values according to the proportion and order 

 of the sexes in the line of descent. If all these r q s be unequal, then 

 rq shall be taken to represent their mean value. It will be necessary 

 for our investigations to find the correlation between the nth and 5th 

 mid-parents in terms of these rs, which give the correlation between 

 individuals. Let p ns be the correlation between the nth and sth mid- 

 parents. Let qh s be the deviation of any organ of the qth male sth 

 parent from the mean of that organ for the sth generation of male 

 parents, q h\ that of his female mate ; let m be any constant not yet 



* A father is a first parent, a grandfather a second parent, a great-grandfather a 

 third parent, and so on, in the notation here adopted. The mid sth parent or the 

 .sth mid-parent is derived from all 2* individual sth parents. 



f 'Roy. Soc. Proc.,' vol. 61, p. 402. 



. 2 f 2 



