Mathematical Contributions to the Theory of Evolution. 149 



Table of Heredity for divers Values of y. 



Value of y. 



7. 



0-9. 



1. 



1-2. 



2-35. 



« . 



Parental corre- 



-2485 

 '1243 



-0621 



0-2851 

 -1425 



0-0713 



-3000 

 '1500 



0-0750 



-3248 

 -1624 



-0812 



0-4000 

 -2000 



0-1000 



0-5C00 

 -2500 



0-1750 



Grandparental 



Great grand- 

 parental cor- 

 relation 





Fraternal corre- 



-2899 



-3065 



-4000 



0-4586 



-6596 



1 -0000 





Regression on 

 nth. midparent 



Correlation with 

 nth midparent 



-4970 

 0-4970(0 -7071;" 



0-5701 

 0-5701(0 •7071)" 



-6000 

 -6000(0 •7071)» 



0-6497 

 -6497(0 •7071)» 



0-7999 

 0-7999(0-7071)'' 



1 -0000 

 (0-7071)" 



(6) Difficulties arising when we apply these Results for Blended Inheritance. 

 — Now the above table shows us that by varying y sufficiently we can 

 obtain a considerable range of values for the correlation of characters 

 in kindred. But these values are limited by two serious considerations, 

 namely : — 



(i) The ancestral correlation is halved at each stage. 



(ii) The fraternal correlation appears to become perfect as we 



approach the upper limit of parental correlation, i.e.^ 0-5. 



Now actual determinations of grandparental correlation in the cases 

 of eye-colour in man, of coat-colour in horses, and of coat-colour in 

 hounds, which I have recently made, do not as a rule seem to 

 justify the statement that the grandparental is half the parental cor- 

 relation. Further, in two of these cases, the average parental 

 correlation is quite 0*5, but the fraternal correlation is, while larger 

 than 0*4, still a good deal short of perfect. Hence I am bound to 

 conclude that : — 



(i) These characters do not obey the laws of blended inheritance as 



deduced from the law of ancestral heredity ; or, 



(ii) The laws of blended inheritance, as deduced from the law of 



ancestral heredity, would be largely modified if we considered 

 the influence of assortative mating, or 



(iii) The fundamental assumption that if all the midparents right 



away back had the same amount of the character, the average 

 offspring would have also the same amount, is not justified. 

 Thus the result /5 = l/( ^2(1-1-7) ) Equation (xxiii), per- 

 haps, is unnecessary, or there may be two independent con- 

 stants of inheritance. 



