32 Mr. S. M. Jacob. Inbreeding in a [Mar. 18, 



Now let us evaluate 



(2«-2-r)lrl + (2«-l-r)l(r-l)I 

 _ (2w-2)!{2t<;(2?/;-r-l) + r} 



(2«-r-l)Irl 

 = (2vg-l)?(2w-r) = (2w-r) 2 Q 

 (2io-r-l)lrl 2w 2w r ~ 



Thus 



r = o 



= ' ~2 2w[2wx 2a ,_ 2 C r + 2w _ 2 C r -i]^V 



= 2w(2^ + 2)^"|" 2 L_2C^ 2w -'-V/- 1 

 = 2^(2^+2)^(^+^-2. 

 Take next the coefficient of (Aa). It is 



= 2 r f ,- w 



1= 2 (2kQ (2w-l) (2^-2)! 



~,to (2«-r)!r! P 5 



= 2 (4tv>-2)pg~ 2 ~ 2 l- 2 CrP 2w - r - 2 'i r 

 = 2w(4:W-2)pq(p + qY>»-2. 

 The coefficient of (aa) is clearly found by interchanging p and q in the 

 coefficient of (AA), and it is 



2lo(p + 2wq)q(p + q) 2 ^ 2 . 

 Thus the distribution of the constituents in the offspring of inbreeding 

 nth siblings is (after dividing by 2vj (p + q) 2w ~ 2 ), 



(2up + q)p ( AA) + (4 w - 2) pq ( Aa) + (p + 2 ioff) j (aa). 

 Hence the offspring of nth cousins is 



(2* +2 2 > + q)p (AA) + (2» +3 -2) M (Aa) + (p + 2 n+2 q) <L (««)■ (9) 

 This, of course, includes the offspring of brother-sister, first cousin, second 

 cousin, up to (n — l)th cousin marriages. 



Thus the proportion of the allogenic element determined by (9) is 



, + — so . as against — — ^— — - in the non-inbreeding popu- 



(1+p/qf 2» + Xl+p/q) 2 * (l+p/rj) 2 ° F 1 



lation. 



