GROWTH AND MORPHOLOGICAL CHARACTERS. 



49 



But, on the other hand, if the two factors Ai and A2 in this same cross 

 were incompletely dominant, the Fi generation would be intermediate 

 and the F2 generation would present a normal curve. Bearing in mind 

 the cumulative effect of each added dose of a dominant factor, according 

 to this hypothesis, and that the visible effect of a factor in the hetero- 

 zygous condition is half that of a homozygous condition, there should 

 be a graded series in the F2 generation ranging from individuals with 

 the cumulative effect of four doses of the dominant factors Ai or A2 to 

 individuals without either Ai or A2. For example, if one dose of either 

 Ai or A2 was responsible for increments of height to the extent of 1 inch, 

 the dominant was 12 inches tall, and the recessive parent was 8 inches 

 tall, then the Fi class would be intermediate or 10 inches tall; for, 

 there would be two increments of an inch each added to the recessive 

 type due to the presence of Ai and A2 in single dose. The F2 classes, 

 however, would consist of an array due to segregation and recombi- 

 nation of factors, this array being like a symmetrical curve. There 

 would be : 



1 = 12 inches, effect of 4 incompletely dominant factors. 

 4 = 11 " " 3 " " " 



6 = 10 " " 2 " " " 



^ _ g " "1 " *' " 



1 ^ S " ''0 " " " 



The formula for such an F2 distribution is obviously not 4° — 1 domi- 

 nants : 1 recessive, as in the previous case involving complete domi- 

 nance. With incomplete dominance, the numerical distribution of F2 

 classes followed the expansion of the binomial (1 + 1)*, i. e., (1 + 1)^°, 

 where n equals the number of allelomorphic pairs. The smallest 

 total number of individuals necessary for a complete representation of 

 all F2 combinations in their proper proportions is 16, or 4°, the sum 

 of the series (1 + 1)^". In dealing with crosses, such as the one just 

 used, in which the size-characters are theoretically due to multiple 

 factors without complete dominance, we should obtain numerical dis- 

 tributions of classes in the F2 generation, together with the number of 

 times an incompletely dominant factor (D) is represented, as follows: 



Allelo- 

 morphic 

 pairs. 



Distribution of F2 classes. 



l(2D)+2(D)+l(d) 



l(4D)+4(3D)+6(2D)+4(D)+l(d) 



l(6D)+6(5D)+15(4D)+20(3D)+15(2D)+6(D)+l(d). 

 1 (8D) +8(7D) +28(6D) +56(5D) +70(4D) +56(3D) + 



28(2D)+8(D)+l(d) 



(1+1)^° 



Total 

 of indi- 

 viduals. 



4 

 16 

 64 



256 



4° 



