•28 



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



The proportion of the allogenic constituent in the progeny of first cousin 

 marriages is 



1+ 8s ~ 1 1 



[(8a-l)p + 4(16g-l) g ]g = 4(36s-l)g a) 



The excess of the recessive constituent over the amount produced by the 

 non-inbreeding population is 



(5) 



4(16.s-l)(l + ^) 



The ratio of the two rates of production of allogenic element by first cousin 

 and non-consanguineous marriage respectively, is 



1 + 8s ~ 1 P. (6) 

 + 4(16s-l) 2 V ; 



The question now arises as to what value should be given to s. Theoretically, 

 «ven in man, the possible gametes of both male and female are very large in 

 number, and the resulting zygotes would also be very numerous. It seems 

 not unreasonable then to take the actual population, in which each family is 

 limited by a variety of causes, of which no account can possibly be taken, as a 

 random sample of the population which would arise, if every possible zygote 

 were formed and developed. In this case we must put s practically infinite. In 

 doing this we do not make each individual family infinite, but we assume that 

 existing families are a random sample of what would occur, if every possible 

 gamete of both parents survived to form a zygote. This, of course, would not 

 be legitimate if applied to the individual family, but it seems reasonable as a 

 means of predicting the constitution of the whole population. To assume that 

 each family has a variable number — as a rule too few to give the possible 

 Mendelian variations — would lead to an immense algebraical extension of the 

 work, and it is hard to see that it could lead to results differing from the 

 supposition that the actual population is a random sample of the theoretically 

 possible population, as an indication of the variation produced in 'the results 

 by limiting fertility. I have put 16 s equal also to the average size of the 

 family met with in practice. The chief objection to this is that it obviously 

 gives fractional frequencies, within the individual family, to some of the 

 Mendelian possibilities. 



In the present case, to make s infinite means that the expression (2) 

 alone gives the offspring of first cousin marriages, whereas it actually includes 



