INHERITANCE, WITH SPECIAL INFERENCE TO MENDEL'S LAWS. 85 



frequency ; there is no room for the appearance of " mutations," although certain 

 variations with very small frequency would he extremely rare in a limited population. 

 A mutation a variation not hitherto observed would only appear in the offspring 

 of the hybrids between two pure races ; after this with random mating the mixed 

 race woiild remain perfectly stable until disturbed by sexual or natural selection. 

 These are the only mutations which arise on the generalised theory of the pure 

 gamete, i.e., two pure races form one mixed race, breeding true to itself; it is difficult 

 under these circumstances to account for the origin of the two pure races by a 

 mutation-theory of the differentiation of species ! 



(ii.) Between any two relations ^if we measure the character by the number of 

 allogenic or protogenic couplets in the zygote of the individual we have a linear 

 regression. The frequency distribution of any character is skew, approaching closely 

 the normal distribution as the number of couplets which determines the constitution 

 of the zygote is increased. 



(iii.) The correlations between pairs of blood relations take definite numerical 

 values absolutely independent of the number of couplets, and the same for all 

 characters and races. 



(iv.) The ancestral correlations form a geometrical series of common ratio one-half. 



(v.) Fraternal correlation is fixed between narrow limits depending on the number 

 of brothers per familv dealt with, and is verv slightly larger than parental 

 correlation. 



(vi.) The theory of the midparent for a considerable number of couplets approaches 

 closely that originally given by FRANCIS GALTOX, except for extreme values of the 

 character, when the regression becomes rapidly smaller and ultimately vanishes. 



We thus see that a generalised theory of the pure gamete would be of very great 

 advantage if it could be accepted. It would lead to a system of inheritance' in 

 randomly mating populations with non-differential fertility, which in its broad 

 features would be essentially the same as that which has been biometrically 

 developed not from theoretical hypotheses, but from the statistical description of 

 observed facts in populations. 



Unfortunately, however, when we come to the actual numerical values for the 

 coefficients of heredity deducible from such a theory of the pure gamete, they do not 

 accord with observation. They diverge in two ways : First, they give a rigid value 

 for these, coefficients for all races and characters a result not in reasonable 

 accordance with observation. Secondly, they give values distinctly too small, as 

 compared with the average values, or with the modal values of large series of 

 population observations. 



We thus reach the point we have so often had to insist upon : that the biometric 

 or statistical theory of heredity does not involve a denial of any physiological theory 

 of heredity, but it serves in itself to confirm or refute such a theory. Mendelian 

 formulae analytically developed for randomly mating populations are either consistent 



