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THE AMERICAN NATURALIST [Vol. XLVIII 



The proportionate number of each of these types of families 

 will change in successive generations according to the following 

 system. 



Let Q n _ t denote the number of A A families in the n — 1th 

 generation, and 

 Pn_i denote the number of AA-\-Aa families in the 



n — 1th generation, and 

 g„_i denote the number of Aa families in the n — 1th 



generation, and 

 r n _! denote the number of AA and 2Aa and aa families 



in the n — 1th generation, and 

 u n -i denote the number of Aa + aa families, and 

 denote the number of aa families. 

 It will be possible to write down u and v in any case without 

 calculation because of the symmetrical relations of a Mendelian 

 population, since always under normal conditions such as are 

 assumed in the general treatment, we have 



It is necessary, therefore, to consider only the coefficients for 

 the first four types of family. In the wth generation the consti- 

 tution of the population in respect of families (not individuals) 

 will be as follows : 



Families in nth generation 



=s(On- t + l/4p«_ 1 + l/16r fl _ l )AA families 

 + s(l/2p„_ x + l/±r n _ x )AA-\-Aa families 

 + s(l/8r„_ 1 )4a families 

 + s(l/2pn_x + + l/Arn.. t )AA 



+ 2Aa + aa families 

 -{-s(un)Aa + aa families 

 + s(>„)aa. families. 

 Or, taking coefficients alone we have 



^ = 0^ + 1/4^ + 1/16^, 

 ^ = 1/22^ + 1/4^, 



^ = 1/2^ + ^ + 1/4^, ; 



tt.=l/2u w . 1 .+ l/4r n ^=p„, 



v n = Vn-t + 1/4^-x + l/16r n _ x = o n . 



