490 



THE AMERICAN NATURALIST [Vol. XL VI 



gotic with respect to both characters) ; while the last 8 are mixed 

 (homozygotic with respect to one character, heterozy gotic with 

 respect to the other). Letting x = pure homozygotes, i/==pure 

 heterozygotes, z — mixed,. we find thus that 



*=y4>y=*y4,*=y2, of an. 



Now, by an analysis of the sort already given, it will be found 

 that at the next self-fertilization, x remains r\ y breaks up, 14 of 

 these 'becoming x, y 2 becoming z, and 14 remaining y • z breaks 

 up, y 2 of these becoming x, y 2 remaining z. 



Now, when we recall that before the second fertilization x was 

 V-i ; V, V4, and z, y 2 of all, we see from the above that after the 

 second fertilization 



+ (y 2 x y 2 ) + (y 4 x y 4 ) =%«= (^~— Y. 

 y=(y4X%)=M«==(y 2 ) 2 », 

 ^= (y 2 x y 2 ) + (y 2 x y 4 )=%= (y 2 )»+ (y 2 )« +i . 



These are the same formula? for x and y that were obtained 

 by the other method (since here n and m are each 2). This 

 method however gives in addition a direct formula for z. 



It is easy to verify the formula? for three pairs of characters, 

 though of course the conditions become here somewhat more 

 complex. 



We may now summarize our formulae, and show the results 

 they give in certain examples. 



Let # = the proportional number of organisms that are pure 

 homozygotes (with respect to all the characters con- 

 sidered), 



t/ = the proportion that are heterozygotic with respect to 



all the characters considered, 

 z = the proportion that are mixed, 



v = the proportion that have any heterozygotic characters. 

 Then, if w = the number of successive self fertilizations and 

 m = the number of pairs of characters, 



■<%)-, ( 2 ) 

 -(• + », < 3 > 



Examples.— (1) Suppose that there have been eight self-fer- 

 lizations, and that we are dealing with 10 pairs of characters, 

 ifhat proportion x of the organisms will be homozygotic with 



