150 



Prof. Karl rearson. 



It is quite possible that eye-colour in man and coat-colour in hounds 

 are exclusive and not blended inheritances, so that (i) would cover 

 these cases. On the other hand, I have found parental correlations as 

 high as 0*5 for a new and large series of stature data in man, without 

 fraternal correlation approaching unity. Here (i) can hardly apply, 

 a,lthough (ii) may, for the coefficient of assortative mating in this case 

 is remarkably high, nearly 0-3. But I think that, even if (i) or 

 (ii) might help us over our difficulties in certain cases, we ought 

 to carefully reconsider the assumption referred to in (iii). It would 

 surely only be justifiable in the case of an absolutely stable popula- 

 tion, each generation of which has existed under an identical en- 

 vironment. In itself it seems to exclude any secular change due to 

 natural selection, or to improved physical or organic environ- 

 ment. In fact, we must proceed with caution when applying the 

 statement that the average of all the offspring of an absolutely 

 same system of midparents would be like those midparents ; for a 

 portion of such offspring have very probably been removed by 

 selection, and our average is not really that of all the offspring, but of 

 the fitter. In the like manner, we must treat with some caution the 

 principle on which Equation (i) of the present paper is based. It 

 assumes that all the ancestral contributions are to be found in the 

 present progeny ; but what if the contributions of certain ancestors by 

 selection, artificial or natural, have been eliminated before reaching 

 the existing generation 1 What if the coat-colours of certain ancestors 

 were unfashionable, and only their unlike descendants have been put to 

 the stud 1 Our theory may be quite correct, but it may appear erroneous 

 when tested by facts observed in the case of horse or dog breeding. 



Let us investigate whether independent y and fS in our expressions 

 for parental and fraternal correlations would enable us in the case 

 of blended inheritance to reach a value of the former as high as 0'5 

 without the latter becoming perfect. I find if ri be the parental corre- 

 lation, = col/ J'2, from Equation (xiii) of my memoir on the ancestral 

 law (p. 394), and if r be the fraternal correlation obtained from 

 Equation (xviii) of the same memoir : 



' V2 1-/5^1+27) 



l-^2(l + 2y) 



(xxix). 



(xxx). 



Whence, eliminating y/5, we have 



,0 _ 2(r-4ri2) i 

 r{2-r) 



(xxxi). 



and 



