MATHEMATICAL CONSIDERATIONS 91 



resents the number of character pairs involved. The 

 exponent of the first term gives the number of hetero- 

 zygous and the exponent of the second term the number 

 of homozygous characters. As an example, suppose we 

 desire to know the probable character of the fifth segre- 

 gating generation (F^) when inbred, if three character 

 pairs are concerned. Expanded we get 



13 + 3 [12(31)] +3 [1 (31)2] + (31)3 



Reducing, we have a probable fifth-generation population 

 consisting of 1 heterozygous for three pairs; 93 hetero- 

 zygous for two pairs; 2883 heterozygous for one pair; 

 28,791 homozygous in all three character combinations. ' ' 

 Of the 32,768 total number of individuals in this genera- 

 tion, 2977, or 9.09 per cent., are heterozygous in respect to 

 some characters. Of the 98,304 total number of allelo- 

 morphic pairs involved in all the individuals of this gen- 

 eration, 3072, or 3.125 per cent., are heterozygous. This 

 is the percentage which is obtained by halving 100 per 

 cent, five times. It is the per cent, of heterozygous allelo- 

 morphic pairs in all the individuals making up the popu- 

 lation as a whole that follows curve 1 in Fig. 24. The 

 per cent, of individuals heterozygous in any factors in any 

 generation inbred by self-fertilization depends upon the 

 number of heterozygous elements concerned at the start. 

 The curves where 1, 5, 10 and 15 heterozygous allelo- 

 morphs are present in the beginning are given in Fig. 24. 

 These are calculated from the formula given and illus- 

 trated above. The curve for the reduction in hetero- 

 zygous individuals where one factor only is concerned at 

 the start, is identical with the curve showing the reduction 

 in the number of heterozygous factors in an inbred popu- 



