VII NATURAL CROSSING 115 



subsequent years. Thus, the years would yield : — 

 A, per cent, B, 10 per cent., C, 19 per cent., D, 27 '1 per 

 cent., &c. 



Again, if we start with Fi plants, all being identical, 

 allow no further crossing, and consider one allelomorphic 

 pair only, we shall obtain the following series, P denoting 

 homozygotes, and if denoting heterozygotes. A, 100 per 

 cent. H: B, 50 per cent. H and 50 per cent. P. C, 25 

 per cent. H and 75 per cent. P., &c. In other words, 

 assuming the productivity of H and P to be equal, the 

 hybrid form will decrease to infinitely small proportions. 

 When two pairs of characters are involved, the rate of 

 decrease of H will be slower. Instead of a 1 : 2 : 1 ratio 

 in the year B, or 2 H -.2 P, we shall have the ratio of 

 1 : 1 : 2 : 2 : 4 : 2 : 2 : 1 : 1, or 12 £Z" : 4 P, being 25 per cent, 

 instead of 50 per cent, of homozygote forms. 



Combining these two antagonistic processes, crossing 

 and segregation, we come to the following general 

 algebraic statement. 



For y pairs of simple allelomorphs involved in a cross 

 we obtain in F2 : — 



2y homozygotes (P) from iy individuals. 



Since crossing is renewed every year we can consider 

 this as a general value for purification. 



In each generation let xP become H, by crossing, and 

 yH become P, by segregation. 



Then the composition of the crop will be : 



1st Year : P only. 



2nd „ : {1-x)P + xH:. 



3rd „ : {{l-xY + xy)P + {x{2-x-y}}H. 



4th „ : {{l-xy + xy{3-2x-y)}P 



+ x{3-S{x + y) + {x + yy\H. 

 Hence 



nth. year : H= -^ ^^ , ^ — '- 



•' x + y 



I 2 



