]t)2 Shull. 



and recessive individuals in the F2. In the dihybrid this ratio is 15 : 1, 

 in the trihybrid. 63 : 1, in the tetrahybrid, 255 : 1, Ac, or grenerally, 

 4" — 1 : ], in which n is the number of the dui)licate determiners which 

 were present in the germ-cells entering into the cross. The occurrence 

 iif (inc of these ratios in tlie F^ is the first evidence for the existence 

 of duiilicate determiners, and is the only evidence yet available in some 

 of the crosses for which the duplication of identical genes has been 

 adopted as an explanation. As the same ratios may be produced by 

 means uthei- than the repetition of Jlendelian determiners, the evidence 

 supj)lied by the F^ ratio is never conclusive. This fact has been strik- 

 ingly illustrated by the studies of Kajanus (1912, 1913) with beets 

 and turnii)S. 



The F3 gives a much better criterion and if an adequate number 

 of sufficiently large F3 families is grown, the results will be fairly 

 decisive. This is due to the fact that, while all Fa families from a given 

 cross are the same, F3 families produced by selfing the dominant indi- 

 viduals of the Fa are of n + 1 different kinds, i.e., there will be, besides 

 a gi'oup of families which contain only the dominant type, as many 

 different ratios in the different ¥3 families as there were duplicate deter- 

 miners involved in the original cross. The ratios placed in the several 

 s(iuares of Fig. 4 represent the ratios expected in the Fs of a dihybrid, 

 and the reason for the expectation in each case will become clear on 

 consideration of the genotypic formula included in the same square. The 

 ratio 1 : <i. — the pure breeding of the dominant type, — occurs when 

 any one or more of the rejteated determiners are homozygous. When 

 all of the independent genes for the given character are present, but 

 heterozygous, the Fa ratio will be the same as the ¥■>. When one or 

 more of these genes are absent and the remainder heterozygous, one of 

 the Mendelian ratios lowei' than that of the F2 will be produced. 

 Thus, in a tetrahybrid which produces in the Fa only families present- 

 ing a ratio of 255 : 1, there will be some families in the Fa which 

 will breed true, others which will I'epeat the 255 : 1 of the F2, and still 

 other families which will consist of the dominant and recessive types 

 in the ratios 63 : 1, 15 : 1, and 3 : 1. Not only are all of these five 

 kinds of families due to appear in the F:i of a tetrahybrid, but if the 

 number of Fa families is large enough, the relative number of each of 

 these ratios is definitely fixt in the proportion 175 : 16 : 32 : 24 : 8. In 

 the F.1 of a trihybrid some families will breed true, and others will 

 give ratios 63 : 1, 15 : 1 and .{ : 1. in the proportion 37 : 8 : 12 : 6. In 

 thr dihybi-id the F3 families fdrmed by selling the dominant individuals 



