^Mathematical Contributions to the Theory of Evolution. 411 



further, by Galton's law, r i2 = r 3i = p, for both are coefficients of 

 direct heredity. 

 Hence 



2 



P ~~ vf + v*— 2v 1 v i R > 



where R is the organic correlation between the two organs in the 

 oame individual. Thus it follows at once that 



Or the mean of the two coefficients of cross heredity is the product of 

 the coefficient of direct heredity into the correlation of the two organs 

 in the same individual. Now in all cases of interchangeable relation- 

 ship, i.e., brother and brother, or cousin and cousin, r 14 = r 23 , and 

 it is highly probable that this is also true where the relationship 

 is not interchangeable, e.g., parent and offspring.* Thus we reach 

 the exceedingly simple rule for cross heredity. Multiply the coefficient 

 of direct heredity by the coefficient of organic correlation, and vje have the 

 coefficient of cross heredity. 



For example, the organic correlation between femur and humerus 

 is about 085 for Aino or French males. Hence we should expect to 

 find the cross heredity between femur of parent and humerus of 

 offspring to be about 03 X 085 = 0*25. Thus Galton's law, even if 

 it be not absolutely correct, will still serve as a useful standard to 

 test the problems of cross heredity. 



(13) Conclusion. — The above illustrations of Galton's law will 

 suffice to prove the wide extent of its applications. If either that 

 law, or its suggested modification, be substantially correct, they 

 embrace the whole theory of heredity. They bring into one simple 

 statement an immense range of facts, thus fulfilling the fundamental 

 purpose of a great law of nature. It is true that there are difficulties 

 which will have to be met, among which I would note two in par- 

 ticular : 



(i) Galton's law makes the amount of inheritance an absolute 

 constant for each pair of relatives. It would thus appear not to be a 

 character of race or species, or one capable of modification by 

 natural selection. This seems to me a priori to be improbable. I 

 should imagine that greater or less inheritance of ancestral qualities 

 might be a distinct advantage or disadvantage, and we should expect 

 inheritance to be subject to the principle of evolution. This diffi- 



* For example, the correlation between the arm length of one brother and the 

 stature of a second, must be equal to the correlation between the arm length of the 

 second and the stature of the first. It is probable, but requires statistical confir- 

 mation, that the correlation between stature of parent and arm length of offspring 

 is equal to the correlation between arm length of parent and stature of offspring. 



