1903.] of Alternative Inheritance. 507 



results (a) to (d) involve what I have termed the law of ancestral 

 heredity, which is thus seen to hold for a general population obeying 

 the most general theory of the pure gamete hitherto developed 

 mathematically. 



(e). The distribution of the population itself follows a skew binomial 

 law, which approaches closer and closer to the Gaussian distribution 

 as we increase the number of couplets in the constitution of the zygote. 

 All these results are in broad agreement with what biometric investiga- 

 tions have shown us to hold for the inheritance and distribution of 

 frequency in populations. 



(3). We now turn to points of divergence : 

 (/). The parental correlation is ^. 



(g). The ratio of diminution of ancestral correlation at each stage is J. 

 (h). The correlation between brothers is sensibly 0'4. 

 One-third is the value originally given by Francis Galton for parental 

 correlation ; is the value of the ratio of ancestral diminution which 

 really follows from his geometrical series J, J, T \-, etc., for ancestral 

 contributions, and 0*4 is the corresponding value of fraternal correla- 

 tion.* 



The generalised Mendelian theory thus fixes absolutely inheritance 

 for all characters in all races, i.e., inheritance is no function of the 

 number of couplets determining the constitution. These results, while 

 in accordance with Gallon's original views, do not seem in accordance 

 with more recent statistical observations on populations. 



The value for parental correlation is not constant either for character 

 or species ; it varies sensibly, and it clusters about a value 0'45 to 0'5, 

 sensibly higher than the value i above given. The value for fraternal 

 correlation is, indeed, somewhat larger than that for parental corre- 

 lation, but it clusters about a value 05 to 0-6, and is sensibly higher 

 than the value 0'4 obtained from this Mendelian theory. The values 

 i i i found for grandparental, great grandparental and great great 

 grandparental correlations from this theory are all very sensibly less 

 than the values actually obtained from observation, 

 diminution is more nearly f than J. 



Hence, this Mendelian generalisation fails when we test it by act! 



numbers. , 



(4). Points of theoretical divergence, not depending upon n 



measures, are the following : 



(i). If p be the number of allogenic couplets in the fathe!, 2 m 

 mother, the number m,,, to be expected on the average in the offspring 



0-4 and not 2/3 a, in " Natural Inheritance "is ,he value f or fraternal c^rela- 

 tion which flows from Gallon's hypothesis, see < Koy. Soc. Proe., vol. 6-, p. 



