Duplicate genes. for capsule-form in Bursa bursa-pastoris. 103 



of the Fo will be distributed as follows: Seven-fifteenths or 46'7 per 

 cent will contain only the dominant type; the remaining 53'3 per cent 

 will split into dominants and recessives, half of the families presenting 

 a ratio 15 : 1 and half having the ratio 3:1; in other words, the three 

 kinds of families will appear in the proportion 7:4:4. 



It is obvious that, as the number of the repeated determiners 

 increases, both the number and size of the families which will be re- 

 quired to adequately test the inheritance-ratios, will be rapidly increased 

 beyond the numbers practicable to the experimental breeder. Thus, 

 with no more than four independent genes for the same character, only 

 one individual of the recessive type can be expected in 256 F 2 offspring, 

 and by chance a much larger number might readily fail to include a 

 recessive individual, in which case the recessive type would appear to 

 have been completely lost or "swamped". If Fs families were grown 

 from such an F 2 , 68'7 per cent of those families, or more than two- 

 thirds, would continue to breed true, and only one-fourth would give 

 ratios sufficiently low that families of no more than a hundred or two 

 would be adequate for their discovery. For this reason conclusive results 

 may not be secured in certain cases, without the aid of the F4 or even 

 later generations. 



The Fi gives an additional criterion of the duplication of deter- 

 miners, for while every sufficiently large series of Fs families must be 

 distributed among n -j- 1 types with respect to the ratios of dominants 

 and recessives (n being as before the number of repeated genes involved 

 in the original cross), the F 4 families will exhibit all of these ratios 

 only when they are derived by selfing dominant individuals in an F$ 

 family which repeated the F 2 ratio 4 n 1:1. F families which are 

 formed by selfing dominant individuals in those F 3 families which 

 exhibited the lower inheritance-ratios, can present no instance of the 

 high ratio which characterized all the F 2 families and some of the F 3 

 families; and, in general, no hybrid family produced by self-fertilization 

 can contain both dominants and recessives in a ratio higher than that 

 which existed among the sibs of its parent, because the ratio is dependent 

 upon the number of heterozygous genes present, and this number can 

 be decreased but not increased by segregation. The formation of negative 

 homozygotes with respect to one after another of the duplicate deter- 

 miners reduces the tetrahybrid to the trihybrid, the dihybrid, the mono- 

 hybrid, and finally, to the recessive, and the units thus omitted could 

 be regained only by a process of positive mutation, or by some sort of 

 rearrangement of determiners such as is discussed later in this paper. 



