The Theory of Polymery 345 



If six genes of this type were interacting to determine height, 

 the r2 would fall into more classes, but the same principles 

 would hold true. Let us suppose that the smaller plant is again 

 30 cm tall and the taller is 54 cm, but let us suppose that the 

 difference is caused by six interacting genes each of which con- 

 tributes 4 cm to the height of the plant. The two parents would 

 be TiTi ^2^2 ^3^3 XX (54 cm) and U^t.t.UhXX (30 cm), 

 and the Fi would be T^t^ ^2^2 TshXX (42 cm). The contribu- 

 tions of the various Fi genes could be indicated as 4 + -j- 4 + 

 + 4 + + 30, which equals 42 cm. The F2 from any Fi plant 

 would fall into various classes, as in Fig. 966. As in the ex- 

 ample involving four contributing genes, the intermediates are 

 more frequent in the F2 than the members of any other class, 

 parental types are recoverable, and no plants are to be expected 

 more extreme than either parent. Since F2 plants which are 

 phenotypically alike do not necessarily have the same genotypes, 

 the F3 families may differ from one another both in average size 

 and in variability, even though they have come from F2 plants 

 which are indistinguishable. 



In both these examples, the number of interacting genes is 

 relatively small. Conceivably, any number might be interacting 

 in various situations. If we admit this possibility, we can de- 

 termine easily the number of expected classes in the F2 and the 

 frequency of each class for any given number of pairs of dupli- 

 cate, cumulative, nondominant genes merely by applying the 

 binomial theorem. If n equals the number of interacting genes 

 (and n/2 will therefore equal the number of loci), the F2 ratio 

 can be determined from the expansion of (a + b)'\ The coeffi- 

 cient of a given term of the expansion indicates the frequency of 

 the corresponding class of the F2, and the exponent of a indicates 

 the number of contributing genes and therefore the strength of 

 the expression of the character of that class. The Fi is always 

 intermediate. The number of classes in the Fo is one greater 

 than the number of contributing genes, and the greater the num- 

 ber of contributing genes, the smaller the relative frequency of 

 the class of intermediate size. The greater the number of con- 

 tributing genes, the smaller the relative frequency of either 

 parental type and, as they are in all cases the classes of least 

 frequency, the smaller the chance of recovering either parental 

 type. If two parents are of the same size in two or more crosses 



