334 THE AMERICAN NATURALIST [Vol.LVI 



and offspring, respectively. The coefficient for the path, 

 dam to offspring, is similar. 



In the case of sire's sire (G) and individual, we have 

 po.g = po.s ^«.i; = iV(l + + /o), and for any ances- 



tor (A) we have for the co efficient pertaining to a given 

 line of descent po.a= (i)''V (1 + /«)/(! + A), where n is 

 the number of generations between them in this line. 



The correlation between two individuals (rsc) is ob- 

 tained by a summation of the coefficients for all connect- 

 ing paths. 



Thus 



where n and n' are the number of generations in the 

 paths from A to B and from A to C, respectively. 



The formula for the correlation between uniting 

 gametes, which is also the required coefficient of inbreed- 

 ing, is 



/o = i/;.A/(l+70(l+>), • 



where tsa is the correlation between sire and dam and /« 

 and fd are coefficients of inbreeding of sire and dam. 

 Substituting the value of rsd we obtain 



If the ancestor {A) is not inbred, the component for 

 the given path is simply (^)«^"'"^ where n and n' are the 

 number of generations from sire and dam respectively 

 to the ancestor in question. If the common ancestor is 

 inbred himself, his coefficient of inbreeding (/a) must 

 be worked out from his pedigree. 



This formula gives the departure from the amount of 

 homozygosis under random mating toward complete 

 homozygosis. The percentage of homozygosis (assum- 

 ing 50 per cent, under random mating) is 1(1 fo) 



xioo. 



