478 THE GRAMMAR OF SCIENCE 



generation is or is not different. It allows, therefore, for 

 secular changes. 



Except, however, in the case of pedigree stock few 

 records will be available of ancestry, and we shall generally 

 have no means of determining the types and variabilities 

 of each past generation. 



If we suppose the race stable in type and variability 

 then the law of ancestral heredity may be written : — 



/i = y{aU^ + am.-, + am.,+ . . . +alOHj(,+ 



Now if an individual had mid-parents all of the same 

 deviation from the racial type right away back, ie. if 

 H^ = H^ = Hg = ... = H^Q = ... = H, we should reason- 

 ably expect him also to have a deviation H, but if 

 /i = H, then : — 



H=y{a + a3 + a3+ ... + a^^ + . .. ] H, 



or as is shown in algebra for geometrical series : — 



i=ya/(i-a). 

 Thus, 0^=1/(^1+.^) 



We can now write our law of ancestral heredity : — 



"=-/ { ^,;^H, + ^^H,+ . . . , _L_ ^iLH„+ . . . I . 



Clearly then all hereditary influence depends upon this 

 one quantity 7, the constant of heredity. Mr, Galton 

 puts 7=1. If 7 be not unity, we may be said "to tax 

 the bequests of each generation," for each generation then 

 contributes to the offspring not once, but 7 times the 

 quantity a"H„ peculiar to it. Thus a may be looked upon 

 as the taxing factor for each portion of the heritage for each 

 grade of distance the ancestor stands from the heritor ; 

 while 7 is the taxing factor on the total heritage so 

 reduced that comes to the heritor.^ 



The reader must pardon the amount of symbols used 

 in the previous discussion on account of the extreme im- 



^ We must also consider the possibility of y greater than unity, or an 

 accumulative interest on the heritage. 



