Rainard B. Robbins 201 



equations so as to get the results for random mating and for self- 

 fertilization. If we set cr = 0, we have, for random mating, 



r„=(2ro + 6-„)V4; «„ = (2ro + s„) (2^o + 5o)/2 ; tn = {2t, + s„y/'^. 



Setting a = 1 gives for self-fertilization 



rn = r„-|-6-o(l-l/2")/2; Sn = So/2^; tn^t, + s,(l - l/2'')/2. 



The results for brother and sister mating, starting with a family 

 VqAA + SoAa + toCia, have been published by the present author (2). 

 They are 



r„ = />/2-Z„/4«+', (13) 



s„=2X„/4»+\ (14) 



^„ = (2-p)/2-X„/4"+S (15) 



in which Ln = ^[^3(1 + V5)«+i -K,{1- V5)«+^] , 



^^ (H-V5) (1-V5), . ,, 

 and Ks = ^ ^ So + ^ — ^ — - (so - 4ro«o) , 



From these results we readily calculate that as n increases indefinitely, 

 rn, Sn, tn approach respectively the values 2ro-fSo, 0, 2tQ + s^ In words, 

 the heterozygous type tends to disappear in brother and sister mating and 

 the homozygous types approach a proportion equal to that of their respective 

 gametes in the original population. 



But equation (8)) shows that in our combination of self-fertilization and 

 random mating, the heterozygous type can never disappear, if a is different 

 from unity. In fact equation (8) shows that 



lim*« = p(l-cr)(2-p)/(2-o-). 



n=oo 



Thus it is clear that no such combination of random mating and self- 

 fertilization can represent brother and sister mating. In every case of 

 inbreeding which the present writer has examined, the proportion of 

 heterozygotes approaches zero as the number of generations increases. 

 Equation (8) shows that so long as a- =^1, i.e. so long as we have a fixed 

 proportion of each generation mating at random while the others are 

 self-fertilized, the proportion of homozygotes cannot vanish. It would 



