1911.] 



Stable Simple Mendelian Population. 



33 



The ratio of the two quantities of allogenic constituent is 



which agrees for s = oo with the result already obtained for brother-sister 

 and first cousin marriages. 



6. We will now consider In more detail the offspring of intermarrying 

 second cousins, introducing the necessary fertility factors and making the 

 deductions for the inclusion of brother-sister and first cousin marriages in the 

 result already obtained. 



The distribution of all the offspring of second cousins, first cousins in the 

 second generation, and of brother-sister marriages in the third generation is 

 found by putting n = 2 in (9), and we have 



The absolute frequency of each type must now be considered. 



We started with (p + q) 2 individuals who mated in every possible way 

 with (p + q) 2 other individuals, and had 16 s offspring to each mating. 

 Hence the first siblings number 16s {p + qf- 



The next fertility being 4£, the second siblings number Q-ist (p + qf. 



The next fertility being 4m, the third siblings number 2oQstm {p + q) w . 



Let the fertility of third siblings inbreeding with each other be 4«. Then 

 the children of third siblings will number 1024stmn (p + q)™- 



Xow the number of individuals in (10) is l§{p + qf. Hence (10) must 

 be multiplied by QAstmn (p + q) u to give the absolute frequencies. 



Now in the expression (2) multiplied by 16stm we have the offspring of 

 all first cousin and brother-sister marriages in the second generation. We 

 must transfer this to obtain the offspring of first cousins in the second 

 generation and • of brother-sister marriages in the third generation, and 

 subtract the result from the offspring of third siblings generally. 



The result is obtained by changing 16s into 4t, t into m, m into a 

 and multiplying by 16s (p + qf, and finally dividing by 16s, since the 

 chance of a brother-sister and first cousin marriage in the random marriages 



of third siblings generally is —J— of the whole number of mating, 

 lbs 



Hence the offspring of first cousin marriages in the second generation and 

 of brother-sister marriages in the third generation is Q-istrun ( p + qf . -^L 

 multiplied by the expression (2). That is, we must subtract 



from the expression for the offspring of inbreeding third siblings generally. 



p (16p + q) (AA) + dOpq (Aa) + 2 + 16 2 ) (aa) 



(10) 



Umn (p + 2 )" [(8^ + q)p (AA) + Upq (Aa) + & + 8q)q (aa)] 



VOL. LXXXIV. — B. 



D 



