50 GENETIC STUDIES ON A CAVY SPECIES CROSS. 



To cover the general case involving any number of such multiple 

 factors with incomplete dominance, we may say that with "n" allelo- 

 morphic pairs w^e theoretically obtain in a total of 4" individuals a 

 series of classes with coefficients derived from the expansion of the 

 binomial (1 + 1)^°, i. e., the series: 



l(2n D)+2n[(2n-l)D] + g^^^[(2n-2)D]+ 2n(2n-l)(2n-2) ^^^^,3^^^ 

 + . . . . +2n[{2n-(2n-l)}D]+ l(2n-2n)D. 



The character of each class is shown by the number of times it 

 contains a dominant factor, D. This is in the form of an arithmetical 

 progression, the first term being 2nD and each succeeding term being 

 smaller by D, until the last term becomes 2nD — 2nD, meaning that 

 the ultimate recessive contains no dominant factors whatsoever. In 

 other words, the progression is the same as the exponents of the first 

 term of our expanded binomial (1 + 1)^°. 



^Vhen the F2 generation is produced from the Fi, not by mating Fi 

 individuals inter se, as above, but by mating these Fi individuals to 

 either the larger or smaller parent, then the formula (1 + 1)'", does not 

 fit the distribution of classes in the F2. The expression (1 + 1)° is used 

 instead. This can be easily seen by taking a hypothetical case, in 

 which a larger race homozygous in three dominant size factors and 

 having a zygotic formula AABBCC is crossed with a smaller race 

 lacking these, and having a formula aabbcc. The heterozygous Fi 

 generation would have a formula AaBbCc. In crossing these Fi 

 hybrids back to the smaller parent, we get a distribution of classes as 

 follows : 



ABC + AbC + ABc + Abe + aBC + aBc + abC + abc . . . Fi gametes. 



abc + abc smaller parent gametes. 



AaBbCc + AaBbcc + Aabbcc + aabbcc j 



AabbCc aaBbcc > F2 zygotes. 



aaBbCc aabbCc J 



1(3D) + 3(2D) + 3(D) + 1(d) F2 distribution of classes. 



It is apparent that with three allelomorphic pairs the coefficients of 

 the classes are derived from the expansion of (1 + 1)^, and that each 

 class has the dominant-size factor represented one less time than the 

 preceding class, the first class having it three times. The total number 

 of individuals is 2^ or 8. Hence, for "n" allelomorphic pairs we would 

 theoretically expect a series as follows : 



1 (nD)+n[ (n- 1)D] + ^-^ [(n-2)D] + "^"Y^j^""-^ [(n-3)D] 

 + .... + n[{n-(n-l)}D] +l(n-n)D. 



This means that the coefficients for the classes are derived from the 

 expansion of (1 + 1)" and the dominant factors are represented in the 



