98 GENETICS IN RELATION TO AGRICULTURE 
TaBLeE XV.—PuENoTYPIC Ratios OBTAINED BY EXPANSION OF THE BINOMIAL 
(3 + 1)". 
Pai f . . . Number of 
faces (38 + 1) | Phenotypic ratio Ca binations 
| 
1 3+11)/3 +1 4 
2 (3 +1)? |324+23+4+1 | 16 
3 (3+ 1)? | 334 3.37+3.3+41 | 64 
4 (3 + 1)4 | 34 + 4.33 + 6.32 4+ 4341 256 
5 (3 + 1)5 | 35 + 5.34 + 10.33 + 10.37 + 5.3 +1 1,024 
- - -—2 
n | + |3* +7391 +s mot nea ars +. 4 an 
For three pairs of factors, therefore, we interpret this table to mean 
that the distribution with respect to phenotypes is as follows: 
297A BC:9ABc:9AbC: 9aBC:3Abc:3aBc:3abC: labe 
If it is desired now to write down the numbers of each particular geno- 
type in a given phenotype, the procedure according to the method of pro- 
gression is very simple. Let us select the class 27ABC the genotypes of 
which are as follows: 
1AABBCC 4A ABbCE 
2AABBCe 4AaBBCce 
2A A BbCC 4AaBbCC 
2AaBBCC 8AaBbCc 
It may be noted that there is one phenotype in each class homozygous 
for all its factors. In this class starting with this phenotype, we double 
the number of individuals each time an additional pair of factors becomes 
heterozygous. Thus there are three genotypes possible with only one 
heterozygous factor, and there will be two individuals of each of these, 
there will be three different genotypes having two heterozygous factors, 
and each of these will be represented by four individuals, and finally 
there is only one genotype with three heterozygous factors and it is 
represented by eight individuals. The method of progression is based 
upon the symmetrical relations which exist in the phenotypic ratios and 
in the ratios of genotypes within a phenotype and is a very convenient 
method for general use. 
For illustrative purposes when it is desired to bring out relations 
graphically the checkerboard method of Punnett is much used. This 
method has already been employed in this book and needs no extended 
discussion here. The accompanying general checkerboard for three 
pairs of factors will illustrate the relations obtaining when this method is 
employed consistently. As shown in Fig. 48 the gametic series is written 
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