538 TRANSMISSION 



to be mediocre on the one hand or something more than 

 mediocre on the other. 



Assuming mediocrity beyond some definite point in the back 

 ancestry. Galton's form of the law of ancestral heredity may 

 be written 



to infinity, in which h has the meaning denned on page 533. 



If the variabilities of successive generations be taken as 

 equal, the fuller statement of this law as given on page 533 

 reduces at once to this simple form. 



As the simpler form of statement gives a good approximate 

 value, we shall, for the sake of simplicity and elegance of results, 

 be content here to investigate what grows out of this law in the 

 way of establishing a character for which selection and breeding 

 are being carried on. 



i. If we assume mediocrity back of the immediate parents, 

 we must make 



Then h = 0.5 H ; 



that is, one half the desired character is present in the offspring. 

 2. If we assume mediocrity back of the grandparents, we 

 must make 



Then h = 0.5 H^ + 0.25 H*. 



If we have a fixed standard of selection, H^ = H^ , and 

 ^ = 0.75^. 



3. If we assume mediocrity back of the great-grandparents, 

 we must make 



H, = HI = o. 



Then h = 0.5 H^ + 0.25 H^ -f- o.i 25 H z ; 



and with a fixed standard of selection 



H\ = J-J% = HZ ; 

 from which h 0.875 H^ 



