1911.] 



Stable Simple Mendelian Population. 



29 



the offspring of the brother-sister marriages in the second sibling generation.* 

 The inclusion of these marriages has the effect of increasing the proportion 

 of protogenic and allogenic element at the expense of the hybrid constituent. 



Seeing, then, that the proportion of allogenic element in the offspring of 

 first cousins increases with the fertility of the first generation, where the 

 fertility does not vary from family to family, it would appear not unlikely 

 that where the fertility is made variable from mating to mating, the pro- 

 portion of allogenic elements will not be greater than the value determined 

 by putting the fertility constant and equal to the maximum value recorded 

 in the observations. Further, it would appear that the true value must be- 

 greater than that given by the least recorded fertility. 



It is true that the gametic correlations of Mendelian collaterals, which, 

 by putting s infinite, agree closely with statistical results for somatic 

 characters, become too small when we put 16 s = 4, say. "We have, in fact,- 

 the values -1 ^, for the fraternal, avuncular, and first cousin gametic 

 correlations respectively. However, until Mendelism is more firmly estab- 

 lished, this ex-post-facto justification, though of some weight, cannot be 

 regarded as decisive. 



In the numerical calculations that follow, the results have been given both 

 for s infinite and for 16 s = 4. 



Before considering, however, the numerical consequences of the results 

 obtained for first cousin marriages, the general expression for the progeny 

 arising from the intermarriage of any grade of cousin will be investigated. 



4. If we take two populations of the forms u.i(AA) + {3 1 (Aa) + y 1 (aa) and 

 « 2 ( AA) + /3 2 ( Are) + 72 where «i + /Si + 7i = a 2 -H/82 + 72, and mate them 

 at random in every possible way, the offspring of each mating numbering 

 4m, the result is easily shown to be 



[(2 ai + /3 1 )A + (fa + 2 71 ) re] [(2 « 2 + &) A + (ft + 2 72 ) re], 

 where, after algebraical multiplication, (AA) is written for A 2 , (Are) for Aa, 

 and (««) for a 2 . Hence the rule for finding the offspring of any two 

 populations is easily seen, We simply write down for each population 

 the number of each kind of allelomorph and add. We thus obtain two 

 factors which, algebraically multiplied, give the required resultant offspring, 

 provided A 2 , Aa, a 2 , are interpreted as noted above. Thus, in the first of the 

 two populations given above, the numbers of dominant and recessive 

 allelomorphs are respectively 2ai + /?i and (81 + 271, and these, multiplied 

 by A and a respectively, give the first of the required factors. 



* This of course indicates that brother-sister marriages, when all matings are random,, 

 would only form an indefinitely small fraction of cousin marriages. 



