PALITIAL SELF-FERTILIZATION CONTIIASTED 

 WITH BROTHER AND SISTER MATING. 



By RAINARD B. ROBBINS, 



Instructor in Mathematics, University of Michigan, Ana Arbor, Mich. 



In a recent issue of this Journal, A. B. Bruce(l) has published an 

 article entitled "Inbreeding," the purpose of which was "to express the 

 genetic consequences if in each generation, selfing and mating at random 

 take place in the fixed ratio *■:(!— s)." Bruce gives as one of the reasons 

 for the importance of such a problem, that the results of brother and 

 sister mating may be assumed to be given by a proper choice of s. The 

 excuse for such an assumption is suggestive parallelism in particular 

 cases. The fact is that the assumption is erroneous, as the results of 

 this paper will show. 



The theory will now be developed for combined selfing and random 

 mating. The results for the general case of brother and sister mating 

 have already appeared(2). It is by comparing these results that our 

 conclusions are drawn. Only one pair of simple Mendelian factors will 

 be considered. 



Suppose we start with a pcjpulation which can be represented by 



r^jAA + s^Aa + t^aa. 



This means that the pure dominants, the heterozygotes and the recessives 

 occur in numbers proportional to ?•,,, ,v„, <„. For convenience we choose 

 ?•„, So, t„ such that 



r„ + «„ + <„ = !. 



Suppose that self-fertilization occurs in a part of each generation 

 indicated by the proper fraction a and that mating is at random in the 

 remainder of each genei-ation. Of course we assume that the individuals 

 which are self-fertilized are chosen at random in each generation. 



