No. 548] SHOETER ARTICLES AND DISCUSSION 



489 



each of two or more things separately, the probability that all 

 of them shall happen is the product of the separate probabil- 

 ities for each. Now, we know that the probability x for the 

 homozygotic condition with respect to one character is 

 2"- l 



For two characters it is then 



For three characters it is of course ( oT' ) ' and in £ eneral > 

 for any number m of characters, the probability x for pure ho- 

 mozygotes (or the proportional number of pure homozygotes) is 



■- W 



By similar reasoning, the proportion of all the organisms that 

 will be heterozygotic with respect to all the m characters is 

 ¥=(%)"■. 



With two or more characters, there will be of course a consid- 

 erable number of the organisms that are homozygotic with re- 

 spect to some characters, heterozygotic with respect to others. 

 If we call the proportion of these z, then 

 z = l-(x + y). 



And if we let v be the total proportion that contains any 

 heterozygotic characters (so that v = y + z), then 

 /2--1V" 2"-(2"-l)- 

 v = 1 ~{^2^) ~ 2- * 



These formula? may readily be deduced algebraically, or veri- 

 fied, by a detailed analysis of a case of two or more characters. 

 It may be worth while to indicate the method followed, by taking 

 U P the simpler case of two pairs of characters. Call these j Q 

 and {b ' The g am etes formed are AB, Ab, aB, and ab. When 

 these combine in all possible ways (as indicated in the diagrams 

 given in Bateson's Mendelism), these give the following results: 

 \ABAB + lAbAb + laBaB -f \abab + 2ABab + 



2AbaB + 2ABAb -f 2ABaB + 2Abab + 2aBab = 16. 



It will be observed that of the entire 16, the first four are pure 

 homozygotes, the second four are pure heterozygotes (heterozy- 



