Correlation and Application of Statistics to Problems of Heredity 21 



If U for any character be the deviate of the generant from the mean of 

 the race, then the individual endowed with such U's for all characters would 

 represent the stirp of any family. Unfortunately Galton does not give us 

 any method for determining the U of the generant. I think, however, if 

 we take the character U of the generant to be that linear function of the 

 characters of all the ancestry which gives the highest correlation R with the 

 character in the offspring, it throws Tight on Galton's idea. In this case U is 

 simply proportional to the multiple regression expression. If we make the 

 following hypotheses, which have considerable experimental evidence in their 

 favour, namely : 



(a) that the individual correlations of offspring with male and with 

 female ancestors are equal, 



(b) that such correlations with individual ancestors die out in a geometri- 

 cal ratio, i.e. the correlations of the offspring with individual parents (father 

 or mother), with individual grandparent (male or female), with individual 

 great-grandparent, etc. form a series r„ i\a, r,a 2 , etc., where a is less than 

 unity, then it can be demonstrated that the deviate U will be given by 

 the formula* 



where h lt h 2 , h 3 , etc. are the deviates of the midparental characters in the 

 successive grades of ancestry and y, /3 are constants, which can be found in 

 terms of r, and a. Further, the fraternity of which U defines the stirp will 

 vary round U with variability crJl—R*, where R (the "coefficient of 

 multiple correlation") is known in terms of r, and a, or of y and /S. 



The expression for IT, or the deviate of the generant which defines the 

 stirp, has been termed the Law of Ancestral Inheritance^. It is not a 

 biological hypothesis, but the mathematical expression of statistical variates, 

 which obey, as many measurable characters in man, certain forms of frequency 

 distribution, these being maintained in successive generations. It can be 

 applied with special values of y and /8 to many biological hypotheses. We 

 are, however, not concerned to discuss these matters here, but merely to 

 point out that in the papers we are now dealing with Galton was feeling his 

 way upwards towards this Law of Ancestral Inheritance, though I venture 

 to think by a faulty stairway. The somewhat complicated mathematics of 

 multiple correlation with its repeated appeals to the geometrical notions of 

 hyperspace remained a closed chamber to him, necessary as multiple correla- 

 tion now is for many practical problems of modern statistics. As I have said 

 there is a true generant, i.e. one in which we insert the true values of the 

 different ancestral midparental deviates, namely h lt A 2 , h 3 , ... as above, and a 

 probable generant for which we only know h x and put in probable values 



* Biometrika, Vol. vin, pp. 239-243. 



t Roy. Soc. Proc. Vol. lxii, p. 386. For the fuller mathematical treatment see Biometrika, 

 Vul. vin, pp. 239-240 and Vol. xvn, pp. 129 et seq. 



