506 TRANSMISSION 



binomial B + R expanded to the fourth power according to the 

 binomial theorem. 



Succeeding populations follow the law of the binomial theorem 

 except as interrupted by selection or differences in fertility. In 

 the breeding of this third generation inter se we find the numbers 

 becoming rapidly complicated, but from the fact that it always 

 follows the binomial theorem we can write the normal distribu- 

 tion for the fourth generation of descendants of any pair of 

 characters as follows : l 



(B + R)* = B* + 8 B~R + 28 B^R* + 56 B*R* 



+ 70 B*R* + 56 B*R* + 28 B*R* + 8 BR 1 + X*. 



Analyzing this "fourth-generation population," we find : 



1. That no less than nine color combinations are represented, 

 ranging all the way from pure black to pure red. 



2. That the frequency numbers representing the various com- 

 binations, i, 8, 28, 56, 70, 56, 28, 8, i, form a symmetrical fre- 

 quency distribution whose total is 256, only two individuals of 

 which are pure. 



3. That future breedings would become rapidly complicated, 

 but that we should always have one pure black and one pure 

 red, with all possible combinations betiveen the two. 



4. That the actual color combination of the individual cannot 

 in most cases be inferred from appearances. For example, there 

 is but one real black and but one real red in the whole popula- 

 tion. However, the 28 iPR* will look like blacks, because there 

 have been six infusions of black to but two of red, while the 

 reverse is true of another equal number, 28 B 2 J^. The 70 B*R*, 

 which is the largest number of all, constituting nearly one third 

 of the entire population, is equally balanced as to color tenden- 

 cies, but will appear to be of the color that is most noticeable, 

 in this case probably a dark red. 



Combinations of three characters. Though the numbers become 

 more rapidly complicated, the same principles apply when 'deal- 

 ing with three or more characters. For example, suppose we 

 introduce a third color, white. We shall then have as the result 

 of the first mating the following : 



1 The student can easily verify these figures by the plan already outlined. 



