1911.] Stable Simple Mendelian population. 



27 



As before we assume an equi-partition of the sexes, that is there are to be 

 32 st males and 32 st females in each group. Not only are males and 

 females to be equally numerous in the group as a whole, but also in each 

 family is the number of protogenic males to be equal to the number of 

 protogenie females, and so on for each type of constitution. 



If we number the groups in Mr. Snow's table from 1 to 36, from left to 

 right taking the rows in turn, we have 15 different types of second sibling 

 family. Then, mating male and female, and putting the fertility of each 

 mating as 4m, the offspring of inbreeding second siblings is 



. (AA)[8p s + 49p 7 q + 54 : pY + 2opY + ±pY + " 2 PY + 1 ^P 5 <f 

 + 64pY + 9 2> Y + 72 pY + oipY + 8pY +lY + QpY] 

 + (Aa) [14p 7 2 4- 36jj 6 q 2 + 30 p h q z + 8pY + 48p e q 2 + 



. +128^Y + 30pVH144pV+180^V + ^ 8 i 5 ¥ + 14 i 9 2 7 + 36 Z'Y] 

 4- (eta) [the expression for the coefficient of (A A) with p and q interchanged]. 



When the terms are collected, this reduces to 



(p + of [(8p + q) V (AA) + Upq (Aa) + (p + 8q) q (aa)]. (2) 



This . last expression, multiplied by 16 stm, gives the distribution of 

 children of all second siblings. Now we have mated together at random 

 all the second siblings in each group of families, so that both first cousin 

 marriages and brother-sister marriages have been included. To obtain the 

 offspring of first cousin marriages only we must subtract from the above 

 result the offspring of brother -sister marriages in the second generation. 

 These can be found from the expression for the children of brother-sister 

 marriages in the first generation by changing 16 s into 4£, t into m 

 multiplying by 16 s (p + q)*, and dividing by 16 s. 



The first three changes are clear. The final division by 16 s is necessary, 

 because, without it, we should obtain the offspring of second siblings, on 

 the supposition that all these siblings contracted brother-sister marriages ; 

 whereas, if, as has been done above, the second siblings mate at random 

 each in their own group of families, there will be 16 s first cousin marriages 

 to one brother-sister marriage, there being 16 s families in each group. 



Hence the children of brother-sister marriages in the second generation are 

 given by the expression (1) multiplied by 2 rut (p + qf. Subtracting this 

 from the expression for the offspring of all second siblings, we get, omitting 

 the factor 2 mt (p + q) 15 , 



8 S [(8p + q)p (A A) + 14^ (Act) + (p+8q)q (aa)-] 



- [(4p + q)p (AA) + 6 pq (Aa) + (p + 4q)q (aa)] 

 = [(64.s-4)p + (8s-l)^]^(AA) + [112s-6]^(Aa) 



+ [(8s-l)p + (64s-4)q]q(aa). (3) 



