oSO 



THE AMERICAN NATURALIST [Vol. XL VII 



Let us now proceed to the calculation of the coefficients 

 of inbreeding Z , Z lf Z 2 and Z,. For Z we have 



Z Q = = 0. ' 



In the same way 



Zi= wl=* = so. 



4 



_ _ 100(8 - 2) _ „ 



Z 3 ■■ 



100(11', - 2) 



These results may be expressed verbally in the follow- 

 ing- way: In the last two ancestral generations x is 50 per 

 cent, inbred ; in the last three generations it is 75 per cent, 

 inbred ; and in the last four generations it is 87.5 per cent, 

 inbred. 



This pedigree table and the constants will repay 

 further consideration, since the case is a limiting one. 

 With the table at hand it is possible to grasp a little more 

 clearly the precise meaning of the coefficients of inbreed- 

 ing. Thus it is seen that what the value of Z x = 50 really 

 signifies is that because the individuals a and b were 

 brother and sister the number of different ancestors 

 which x can possibly have in any ancestral generation can 

 not be more than 50 per cent, of the total number theo- 

 retically possible for the generation. That is, x's sire 

 and dam having been brother and sister means that x can 

 not have had more than 2,048 different great-great-great- 

 great-great-great-great-great-great-grandparents, instead 

 of the possible 4,096. He may have had fewer than 2,048, 

 but Z 1= =50 tells us that he could not have had more. 

 Similarly Z 2 = 75 indicates that since c and d, the grand- 

 sire and grand-dam of x were brother and sister, x can 

 not have in any ancestral generation more than 25 per 

 cent, of the theoretically possible number of ancestors for 

 that generation. And so on for the other values of Z . 



