1911.] 



Stable Simple Mendelian Population. 



31 



We shall now determine the frequencies. From (7) we see that the 

 frequency of the general type of family group in the nth sibling generation 

 is given by the product 



v$i i p w ~ li q l1 x w Gi. 2 p w ~'*q 1 *, 

 where h and l 2 can take any value from to w. 



Now, whenever h + h = r, we get an operator of the form (2w— r) A + ra 

 Hence the frequency of this type of operator is 



2 wCi,T w ~ V 1 x v$h'P w ~ (where h + l 2 = r). 



Now take the identity 



(l+xY>(l+x)» = (1 + xf», 

 and equate the coefficients of x r on both sides. 



We have at once 'f, ,„C (l . u ,C r _j 1 = 2w C r . 



?! =0 



Hence the frequency of the operator of type (2 w—r) A + ra is 2w G T p 2w ~ r q r 

 This is the operator which operating on itself produces the (« + l)th sibling 

 families. Hence the (?i.+ l)th generation is given symbolically by 



L?*A?* B "Y {(2w-r)A+w}]J 

 = [ 2 [^Cr^-Y {(2" +1 -r) A+ra}] . ( 8 ) 



r = 



Thus it is shown that if the result (7) be true for the ?ith sibling generation, 

 it is true for the (%+l)th. But it has been verified for the first and second 

 sibling generation. Hence it is universally true. 



5. To obtain the offspring when the nth. generation of siblings inbreeds, 

 we take each of the families in (7) and mate each type with itself. The 

 result will be given by taking in (8) only the square terms, and omitting the 

 cross-products, except that the frequencies are not to be squared, as we 

 divide each group into an equal number of males and females, as has been 

 done throughout. 



We have then, neglecting absolute frequency, the offspring of inbreeding 

 ?ith sibling groups given by 



' l" [{(2w-r) 2 (AA) + 2r(2w-r)(Aa) + r 2 (aa)} x ^C^-y]- 



Consider first the coefficient of (AA). It is 



' ~% W (2w-r) 2 2w G r p 2 ' v -Y- 



