EVOLUTION 493 



Let us look at this a little more closely from the 



numerical standpoint. In a given group of N brethren, 



let n xN follow directly the father, // x N the mother, 



supposing both sexes equipotent ; let ;/i x N follow 



directly each of the grandparental types. Thus 41/1 x N 



are really the number of the offspring who directly revert 



to the grandparents, but some of these will be like the 



parents, just as some like the parents will be like the 



grandparents, for a certain percentage of parents are like 



grandparents. Now in the case of a grandparent, in x N 



of the offspring will directly follow his type ; and if 



PiX N be the total number of offspring like a parent, 



p^x n xN of the 7/ x N offspring who directly follow a 



parent will be also like a grandparent. Hence, if p^ x N 



be the total number of grandchildren like a grandparent, 



then .. ^T , XT 



p.2 X N = /Ji X N + p.^ X n X N, 



p., = 7/i + p^x /r . . . . (1.). 



Again, of the total offspring like a parent, or p^ x N, there 

 will be ;/ X N who follow the parent directly, and 2 x p^ x 

 in X N who will be like the parent, because they revert to 

 one or other of the two grandparents, who in the fraction 

 p^ of cases are like the parent. Hence 



p^ X N = /7 X N + ip^ X 111 X N, 



or ,.. . 



Pj = ;z + 2w/)j .... (11.). 



Finally, 2/^ x N + 4.V1 x N must be the total number of 

 offspring or N ; thus 



2;/ + 4?;^= I . . . . . (iii.). 



Taking p^ = -4947, as given by the observations on eye- 

 colour in man, we find from (i.), (ii.), and (iii.) 



/3^ = .2474. 



This cannot be said to be in good agreement with the 

 observed value .3166 (see p. 491) for grandparental 

 correlation in eye-colour. Nor, as can be shown, do we 

 get more satisfactory results by extending the reversions 



