106 ANALYSIS OF THE FOUR PRINCIPLES. 



Method o) Testing Table A. 



f+ 



Let M- 10 ; m ~-= 7; c- -; then ~ f- according to the table. 



4 F 2 



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

 breeds 3^ times that number. 



*j 



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



s~* 



nih generation, we find that -=- '- is correct for all subsequent gene- 



s~* 



=- 



rations; 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 4 i)th generation, the pure-breeds will be 



2 x (MMc) l 2 I io- - ! = 15. In the (n + 2)th generation 



I 4 J 



the pure-breeds == 2 x (M -- Me) 2 = ; 2 ' (7.5)2 = 112.5. 



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

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

 breeds of the previous generation == 7 2 4 . In the (n4- 2)th gen- 

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

 generation, plus 7 >< one-quarter of the pure-breeds of the (n. 4 i)th 



generation == 73+ L.+ El x 7 = 393.75. 





