THE PROBLEM OF INBREEDING 151 



always under normal conditions such as are as- 

 sumed in the general treatment, we have 



U„-i = p„-i, 



V n -l = O n _l, 

 U n /-'nj 



V n = O n . 



It is necessary, therefore, to consider only the 

 coefficients for the first four types of family. In 

 the nth generation the constitution of the popula- 

 tion in respect of families (not individuals) will 

 be as follows : 



Families in the nth generation 



= s(o n -i + \ p»_i + is r n -i) AA families 

 + s (| p„_i + \ r»_i) AA + Aa families 

 + 5 (| r„_i) Aa families 

 -|- s (| p n -i + q n -i + \ r n -i) AA 



+ 2 Aa -\- aa families 

 + 5 (u„) Aa + act families 

 + s (v n ) aa families. 



Or, taking coefficients alone, we have 



O n = 0„_l + \ p n -l + 16 ^n-l, 



p n = \ p n -i + \ r n -u 



On ~ 8 n— 1» 



r n = \ Pn-i + o n -\ + \ r n -i, 



v n = v n -i + i w n _i + T V T n -l = o n . 

 Let us see how this formula works out in a con- 



