608 



THE AMERICAN NATURALIST 



rVoL. XLVII 



Aa X Aa=< 



a A 



Now since the above expressions give not only the 

 probable distributions of the characters in the whole 

 progeny population, but also the probable distribution of 

 these characters within any single family, it necessarily 

 implies that the constitution of the sister of any male is 

 equally likely to be any one of the four possible combina- 

 tions. Or, in other words, 



The constitution of any particular sister of any partic- 

 ular AA <$ is equally likely to be either AA, or Aa, or 

 aA, or aa. 



The constitution of any particular sister of any partic- 

 ular A a (or a A) $ is equally likely to be either A A, 

 or Aa, or aA, or aa. 



The constitution of any particular sister of any partic- 

 ular aa $ is equally likely to be either AA, or Aa, or 

 aA, or aa. 



This clearly means that the progeny of the next gen- 

 eration produced, by hypothesis, from the mating of 

 brothers X sisters of this generation will be gametically 

 such a progeny as it produced by mating at random a 

 male population of the constitution 

 AA + 2Aa + aa 

 with a female population of the same constitution, namely 



AA + 2Aa + aa. 

 But, as Pearson 18 first showed, this results in a progeny 



16AA + 32Aa + 16aa. 

 There is no increase in the proportion of homozygotes, 

 which was the point to be proved. Of course the same 

 reasoning obtains in regard to the next and any number 

 of other generations. In other words the proof is general 

 and complete that no increase of the proportion of homo- 

 zygotes in the population follows inbreeding save under 

 one or the other of two special conditions, viz. 



