1911.] 



Stable Simple Mendelian Population. 



25 



numbers, mate them in pairs, aud multiply the offspring by the frequency 

 ■of the particular type of family. 



The total number of children in each first sibling family is taken as 16 s, 

 -where s does not vary from family to family. In practice, of course, 

 as every family does not consist of a fixed number of children with the 

 brothers and sisters equally numerous, this assumption would appear to 

 be rather a large one. It will be necessary to consider later how far the 

 -constancy of s, and its equal partition into male and female, are 

 hypotheses likely to give reasonable approximations. 



Certainly the complexity of the analysis would be very greatly increased 

 if every possible size of family, with every possible distribution of the 

 sexes, were considered, and probably no very great modification of the 

 result is to be anticipated. 



Let the offspring of each brother-sister marriage be 4=t, where t is also 

 •constant, but may^differ from 4s. 



In the first family there are 8 s brothers and 8 s sisters. There will be 

 -8 s matings of brother and sister, and the progeny will be 8st 4(AA). 



In the second type of family there are 4s brothers (AA), and 4s brothers 

 •(Act), and a like distribution of sisters. The progeny will be 



2st {9 (AA) + 6 (Aa) + (aa) }. 



In the third type there are 8s (Aa) brothers, and the progeny is 



8st{ (A A) + 2 (Aa) + (aa) }. 



In the fourth type there are 2s (AA) + 4s (Aa) + 2s (aa) brothers, and 

 -the progeny is 8 st { (AA) + 2 (Aa) + (aa) }. 



In the fifth type there are 4s (Aa) + 4s (aa) brothers, and the progeny is 



2st { (A A) + 6 (Aa) + 9 (aa) }. 



In the sixth type there are 8s (aa) brothers, and the progeny is Sst4:(aa). 



The frequency of these types is p*, 4p 3 q, 2p 2 q 2 , Ap 2 q 2 , 4pq 3 , 



Multiplying the offspring by these frequencies and adding, we find the total 



offspring of all types of brother-sister marriages is (omitting the factor 



-2 st) :— 



16p 4 (AA) 

 + S6p s q (AA) + 24p 3 2 (Aa) + 4p 3 q (aa) 

 + 8])Y ( AA) + 16 p 2 q~ ( Aa) + 8 p 2 q 2 (aa) 

 + 16pV ( AA) + 32p 2 q 2 ( Aa) + 16^V (aa) 

 + fyq 3 ( AA) + 2±pq 3 (Aa) + 3Qpq 3 (aa) 

 + 16 q i (aa). 



