1909.] Correlations of a Mendelian Population. 



227 



(4) These distributions correspond to the cases of 2, 1 and A elements 

 in the gametic constitution of the nth parent. And we have at once the 

 following result : — 



Number of protogenic elements Average number of same 



in ?ith parent. elements in array of offspring. 



2 2n+1 P + 2 V =f 2 , 



(P"-l)p + q 



2»(p + q) 9U 



2-Q> + 2 ) ~ ya - 



Accordingly, the average number of protogenic elements in the array of 

 offspring decreases uniformly with the decrease in number of the like 

 elements in the nth parent, i.e. 



Thus the regression between the nth parent and the offspring is linear, 

 and the correlation coefficients form a geometrical series of ratio \, and 

 first term \. Further, the exact constitution of the population, as far as 

 the number of protogenic, allogenic or heterogenic individuals is concerned, 

 is of no influence on the result at all. For all mixtures following the simple 

 Mendelian rule : (AA) x (act) = 4 (Aa), the ancestral correlations for gametic 

 constitution are : 



Parental correlation O500 



Grandparental correlation O250 



Great grandparental correlation 0125 and so on. 



It will be seen at once that these correlations are of the type p, p 2 , p s , etc., 

 for which, in my memoir of 1896, I worked out the multiple regression 

 formula, and showed that the ancestors were quite indifferent. " A know- 

 ledge of the ancestry beyond the parents in no way alters our judgment as 

 to the size of organ or degree of characteristic probable in the offspring nor 

 its variability."* This remark and the proof apply equally of course to 

 gametic and to somatic characters if the correlation be of the above form. 



(5) Accordingly there remains not the least antinomy between the 

 Mendelian theory and the Law of Ancestral Heredity, if we confine our 

 attention to gametic constitution. The Mendelian ancestry is correlated 

 with the offspring in a series descending in a geometrical progression, and 

 the regression is linear. The values of the correlation coefficients are 



* " Regression, Heredity, and Panmixia," 'Phil. Trans.,' A, vol. 187, 1896, p. 306. 



