Rainard B. Robbins 201 



equations sci as tu get tlie results for random mating and for self- 

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



)■„ =(2r, + *■„)■-/•* ; '^n = { 2''o + ■%) ( 2<„ + s„)/2 ; t,, = (2f„ + 6-„)V4. 



Setting a = 1 gives for self-fertilization 



/■„ = /■„ + *„(!- l/2»)/2 ; s„, = s„/2" ; t-a = t, + *•„ ( 1 - l/2")/2. 



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

 r„AA + s„Aa + t„aa, have been published by the present author (2). 

 They are 



r„ = p/2-Z„/4"+", (13) 



6v=2X,J4"+^ (14) 



i„ = (2-p)/2-AJ4"+', (15) 



in which L„ = -^ [K, (1 + ^5)"+' - K, (1 - \/5)"+'], 



and Ki = ' ^ s„ + — ^ — - (s„ - 4r„io) , 



„ 1 - Vo (H- \/5) , , , ^ . 

 Ki = — g— So -I- ^ (.V - ir„t„). 



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

 rn, Sn, tn approach respectively the values 2r\ + So, 0, 2<Q-f s„. In words, 

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

 the homozygous types ap)proach 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 cr is different 

 front unity. In fact equation (8) shows that 



lims„ = p(l-^)(2-p)/(2-cr). 



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

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

 inbreeding which the present Avriter has examined, the proportion _oiL 

 heterozygotes approaches zero as the number of generations increases. 

 Equation (8) shows that so long as a =f\, 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 



