Mathematical Contributions to the Theory of Evolution. 403 



It seems to me that they are equally well fulfilled by the series 



provided the sum of this series is equal to unity, 

 or 70'/ (1—7/3') = 1, that is 7 0' = J. 



But 7/3'* = -2» |- = 7 /» X ( vS) 1 by (x), 



or 0' = \/ 20 of our previous notation. 



Hence the conditions laid down are fulfilled by our general solu- 

 tion (x) provided 



^ = 57i (xYii) * 



I do not assert that such a law is more probable than Mr. Galton's, 

 or indeed as simple. But it throws back the theory of inheritance 

 on at least one arbitrary constant 7, and therefore while covering 

 Mr. Galton's law of ancestral heredity (7 = l), allows a greater 

 scope for variety of inheritance in different species. 



It seems worth while to notice the changes that result in ancestral 

 correlation when we put on a total " tax " 7. As a numerical illus- 

 tration, take this tax at 10 per cent., then 7 = t 9 q. We find 



,8 = 0*39284, and by (xii) and (xiii) : 

 <z = 074639. c = 0*58953. 



From these values Ave can form a table exactly like that on p. 397. 

 On examination of it, we see that the effect of a " general tax " is to 

 increase sensibly all the correlations. In particular the more distant 

 ancestry play a relatively greater part than they would do under Mr. 



Table of Heredity. Tax 10 per cent. 





Individual parent. 



Mid-parent. 



Order. 

















Correlation and regression. 



Correlation. 



Regression. 



1 



2 

 3 

 4 

 5 

 6 



-3111 

 -1642 

 0-0867 

 -0457 

 -0241 

 -0127 



-4400 

 0-3284 

 -2451 

 -1830 

 0-1366 

 0-1019 



0-6223 

 0-6569 

 -6933 

 -7319 

 0-7725 

 0-8154 



qt\\ 



0-5895(0- 5278)2 



0-5895(0-7464)2 



0-5895(1-0556)2 



2 g 2 



