1895.] Inheritance in the Case of Two Parents. 241 



Let T! = co-efficient of correlation (Galton's function*) for the two 

 organs (or same organ) of the parent population, i.e., TI is a measure 

 of the amount of sexnal selection between the parents of the popula- 

 tion with regard to these organs. 



Let r 3 = co-efficient of correlation between fathers and offspring 

 for the organ (or two organs) under consideration, i.e., r 2 is a measure 

 of the paternal inheritance. 



Let r 3 = co-efficient of correlation between mothers and offspring 

 for the organ (or two organs) under consideration, i.e.,r 3 is a measure 

 of the maternal inheritance. 



The value of HI is given by 



% 

 i n *2 i 



"31 



and the standard deviation 2 of the fraternity due to parents H 2 and 

 H 3 is given by 



2 = 



ri z -r a 2 -r 3 z + 2 r,r 2 r 3 ) ' 



Thus the distribution of fraternities is the same for all parentages ; 

 it depends, however, upon the strength of sexual selection, and on 

 the paternal and maternal inheritances for the community at large 

 with regard to the organs under consideration. 



The portion of regression due to either parent alone is not depen- 

 dent solely on maternal or paternal inheritance ; it is influenced not 

 only indirectly by sexual selection but directly by the inheritance 

 from the other parent owing to the presence of the terms r,r 3 and 

 rir a . Further, the greater the variability of one sex (i.e., the greater 

 <r 2 or <r 3 ) the less, other things being the same, the parent of that sex 

 contributes to the inheritance of the offspring. The above two formulaa 

 seem to embrace the chief laws of heredity in populations. The 

 whole of the constants involved can be found by comparatively simple 

 measurements, and, indeed, have been, to some extent, found in the 

 case of man by Mr. Galton.f 



* The probable error of a determination of Galton's function 



= 0-674506 -- P^- 

 Vn(\+rS) 



where n is the total number of correlated pairs. Mr. Galton having kindly placed 

 at my disposal his Family Faculty Eecords,' I find that r, for height is, as he 

 supposed, small, = 0'093. But the probable error of the determination (n == 198 

 only) is 0-047. Hence the balance of probability is in favour of a certain small 

 amount of sexual selection as to height in human marriage. I hope shortly to have 

 sufficient data to confirm this result. 



t They do not seem, however, to fully justify his theory of the midparent. I 

 hope at a later date to discuss its special limitations, e.g., r 2 anc * ff a differ consider- 

 ably in several series of skull measurements with which I have had to deal. 



S 2 



