488 



THE AMEBIC AN NATURALIST [Vol. XL VI 



A, and a, and when these combine in all possible ways, they give 

 zygotes AA -j- Aa + aA -f aa; that is, two homozygotes and two 

 heterozygotes. Thus, the self-fertilization of such an organism 

 gives y 2 the progeny homozygotic (with respect to this charac- 

 teristic) ; y 2 heterozygotic. If we let rr = the proportion of 

 homozygotes, y the proportion of heterozygotes (with respect to 

 one character), then after the first self-fertilization 



y = y 2 of all. 



Now, after the next self-fertilization, of course the homozy- 

 gotes x remain pure, so that half of all the progeny are still ho- 

 mozygotes on this account. The heterozygotes y of course again 

 break up, in the way already set forth, one half into x, the other 

 half remaining y. Since y included half of all, this will give y 2 

 of y 2 (=% of all) as x, y 2 of y 2 (=1/4 of all) as y. 



So the total proportion for the homozygotes x becomes after 

 the second fertilization 



z=y 2 -f(y 2 ) 2 =%, 



while 



2/=(y 2 ) 2 =y4. 



This process is repeated after each fertilization, so that if 

 there are n fertilizations in succession, the total number of homo- 

 zygotes x, becomes 



*-y 2 + (y 2 ) 2 +(y 2 ) 3 . . . * P to (y 2 )». 



This expression reduces to x = -- ^ n \ where n is the number 

 of fertilizations. , 



For the heterozygotes, y, on the other hand the formula is 

 simply 



2/=(y 2 )». 



These then are the formula in case we deal with but one pair 

 of characters. They express (1) the proportion of all the organ- 

 isms that will be homozygotic (or heterozygotic as the case may 

 be), after a given number n of fertilizations; (2) also they of 

 course express the relative probability for a given case, as to 

 whether it shall be homozygotic or heterozygotic. 



2. When we are to deal with two or more pairs of characters, 

 the problem may be attacked in two ways. One is by the gen- 

 eral principles of probabilities; the other is by analyzing the 

 ease of two or more characters in the way exemplified above. 

 The two methods give the same results. 



The first method is far the simpler. It is merely an appli- 

 cation of the principle that when we know the probability for 



