58u 



THE AMERICAN NATURALIST [Vol. XLVII 



mating of relatives is accomplished, or whatever the de- 

 gree of relationship of the individuals mated together, 

 the case in last analysis comes back to the above state- 

 ment, namely that there are actually in the pedigree of 

 the inbred individual fewer different ancestors in some 

 particular generation or generations than the maximum 

 possible number. 3 



The idea suggested in the foregoing paragraph may be 

 expressed symbolically as follows. If there is absolutely 

 no collateral relationship between any of the individuals 

 in a pedigree, the number of different individuals in suc- 

 ceeding ancestral generations will be given by the series 



(1)2** (2)4** (3)8** (4)16** (5)32 < > (»)2*, (i) 



where the numbers in parenthesis denote the numbers of 

 the ancestral generations (1 = parents, 2 = grandparents, 

 3= great-grandparents and so on), and the free figures 

 denote the maximum possible number of different ances- 

 tors in the indicated generation. If in any generation in 

 the series relatives are bred together the same individual 

 will appear more than once in the ancestral series, and 

 the number of different individual ancestors in the higher 

 terms will be accordingly diminished below the maximum 

 number as given in (i). The series will then become 



*** (1)2** (2)4-^(3)8-^ m 

 «*(4)16—y g «*(5)32— 

 where y lf y 2 , y 3 , • — may, in the nth generation have any 

 value not greater than 2 n — 2, in the case of organisms in 

 which two individuals must cooperate in the process of 

 reproduction. The final limiting case is, of course, self- 



in inbreeding (heterozygosis in the sense of East and Hayes, etc.), will be 

 undertaken in a later section of the paper (p. 605). 



'This generalized concept of inbreeding seems to me to be in essential 

 (though not entirely in verbal) agreement with that of O. F. Cook, whose 

 interesting general discussions of this and related problems are summarized 

 in a recent paper ("The Superiority of Line Breeding over Narrow Breed- 

 ing," U. S. Dept. Agr., Bur. Plant Ind. Bui. 146, 1909). I use "inbreed- 

 ing" as a generic term, while Cook regards it as a species of "line breed- 

 ing." This seems to me to be a purely terminological difference, and not 



