34 



Mr. S. M. Jacob. Inbreeding in a 



[Mar. 18, 



Omitting the common factor 4 tmn (p + q) u , the result is 

 Ws[p (Up + q) (A A) + 30pq (Act) + q(p + 16q) (act)] 



- [P (8 j? + q) ( AA) + 14^ 2 (Act) + q(p + 8q) (act)]. 

 Thus the offspring of second cousin marriages is given by 

 4 [8p (32s- 1) + (1 6 s- 1) q] (A A) + 2pq (240 s - 7) ( A«) 



+ 2 [(16.5-1)^ + 8^(32.5-1)] («a). (11) 



The proportion of allogenic element is 



(16*-l)p/g 



(1+iV?) 2 8(32s-l)(l+p/ 2 ) 2 ' 

 and the ratio of this to the amount of allogenic element in the non-inbreeding 

 population is 



16s- 



8^32s-l) q 



For s = oo this becomes l+-i -, and for 16 s = 4 it is 1 + ^-, the 

 16 <i 06 q 



corresponding values for first cousin marriage being 1 + and 1+ — 'i-. 



8 q 12 q 



1, We shall now proceed to consider the results obtained with particular- 

 reference to the marriage of first cousins. In the first place it will be 

 necessary to examine how far the numerical proportions of pure dominant, 

 hybrid, and pure recessive differ in the offspring of first cousins from the 

 proportions in a non-inbreeding population.* For this purpose the ratio of 

 p to q has been selected to give in the non-inbreeding population 25, 10, 5, 

 4, 3, 2, and 1 per cent, and the fractions 1/1,000, 1/10,000, 1/20,000, and 

 1/1,000,000 of pure recessive. 



Table I gives the percentages of each type of constitution in the offspring 

 of non-consanguineous and of consanguineous first cousin marriages. 



This table illustrates the absolute decrease in the percentage of pure 

 recessive in the offspring of consanguineous marriages, while it is seen that 

 the ratio of the percentages of allogenic constitution steadily rises as the 

 frequency of occurrence of the allogenic element diminishes. 



If we have an evil which is a Mendelian recessive and is of common type, 

 such as tuberculosis,! then first cousin marriage will not be much more 

 likely to produce a defective offspring than any other kind of marriage. 



* By a " non-inbreeding population " in this paper is meant one in which inbreeding is 

 not universal, i.e., it corresponds to the general population resulting from random mating. 



t It is not intended to imply that tuberculosis has been proved to depend on a 

 simple Mendelian recessive factor. 



