io6 



ANALYSIS OF THE -FOUR PRINCIPLES. 



Method of Testing Table A . 



Let M = 10; w 7; c~ -; then-^=Z according to the table. 



4 f 2 



We now place under pure-breeds any number, and under cross- 

 breeds 3^ times that number. 



Starting with the fraction -s given in Table A, as correct for the 



C 7 

 wth generation, we find that -R= - is correct for all subsequent gene- 



j-ations ; and this proves the formula to be correct. 







If the denominator of the fraction representing the value of = is 



o, or less than o, the disproportion increases with each generation ; 

 that is, cross-breeds become the overwhelming element. 



In this case by which we are testing the correctness of Table A, 

 suppose the pure-breeds to be 2 and the cross-breeds to be 7 in the 

 generation with which we commence. In the next generation, which 

 we designate as the (n + i)th generation, the pure-breeds will be 



2 x (M Me) 1 2 I 10 - I = 15. In the (n -f 2)th generation 



L 4 J 



the pure-breeds ^ 2 X (M Me) 2 = 2 x (7.5)2 = 112.5. 



The cross-breeds in the (n + i)th generation = 7 x the cross- 

 breeds of the previous generation, plus 7 X one-quarter of the pure- 

 breeds of the previous generation = 7 2 + -. In the (n + 2)th gen- 

 eration the cross-breeds wiljrbe 7 x the cross-breeds of the (n x i)th 

 generation, plus 7 x one-quarter of the pure-breeds of the (n + i)th 



2 f ^\ 



generation = 7 3 + + - x 7 



