482 THE GRAMMAR OF SCIENCE 



certain number of the ancestors, we may be able to go 

 back two, three, or even in some special cases up to ten 

 generations. But beyond this what were the ancestry ? 

 Are we (i.) to suppose the earlier mid-parents identical 

 with the general population type, or (ii.) to suppose that 

 they already diverged from that type ? It is very 

 needful to be clear on this point. We may want to 

 know the probable type of the offspring of certain 

 parents with known ancestry for n generations, but in 

 the case of pedigree stock it is very unlikely that the 

 exceptional character of the ancestry stops at the known 

 n generations. On the other hand, if we start with any 

 wild race, stable in its environment, and change that 

 environment or artificially select for n generations, we 

 shall be quite safe in supposing the mid-parents some 

 little way back to be sensibly identical with the then type. 

 In the latter case we know Hj, Hg, Hg . . . H^^^ as a 

 result of our selection, and we take H^^^^, H,j^2> H!%+3 • • • 

 all zero. Our estimate must therefore be based on the 

 law of ancestral heredity as given on p. 477. It is 

 accordingly : — 



// = y{a-H.+a-^-H,+ . . . + «« -H„ I • 



If we suppose that the race without selection would 

 exhibit no bathmic influence (p. 375) tending to modify 

 its variability, this may be written : — 



where ^S^, ^., . , . ^^ ^^e, if we adopt Mr. Galton's 

 hypothesis as fairly near the truth, -g-, 4 • • • ^ respec- 

 tively. 



Further, the variability % of the selected group, in terms 

 of the racial variability, may be shown by an extension of 

 the theorem on p. 469 to be given by 



v.2^^2/ , _ to _ Jfe^ _ ~ _I^ - etc. I 

 where r^, ro, r^ . . r^^^ etc., are the correlation coeffi- 



