THE PROBLEM OF INBREEDING 151 



always under normal conditions such as are as- 

 sumed in the general treatment, we have 



u n -i = Pn-i, 



V n -i = O n _i, 

 Un = Pn, 

 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 n -i -f ^ r_i) AA families 

 + s (? p n -i + \ r n -i) AA -\- Aa families 

 + s ( r n -i) A a families 

 + s Q p n ^i + q n -i + | r n _i) AA 



+ % Aa + aa families 

 + s (u n ) Aa + aa families 

 + s (v n ) aa families. 



Or, taking coefficients alone, we have 

 o n = o n _i + | p n .i + T V r n _i, 



Pn = \ Pn-l + | r n _i, 



q n = I r B _i, 



fn = \ Pn-l + Q'n-l + 4 ^n-1, 



w = I tt_i + i r n _i = p n , 



^n = -! + I U n .i + T V r n _i = O n . 



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



