534 TRANSMISSION 



J-, J, J, -^g- , by which he accounts for the total heritage. This 

 fractional influence of the different generations, therefore, may 

 be accepted as the best general statement possible of the law 

 of ancestral heredity. The influence of individual ancestors, 

 waiving all considerations of special prepotency, would be found 

 by dividing these fractions by the number of ancestors of that 

 generation (J by 8 = g 1 ^ for great-grandparents). (See table, 

 page 527, for an extended statement of the fractional influence 

 of generations and separate ancestors.) 



The variability of the offspring of an ancestry selected for an 

 indefinitely large number of generations back is given by a 

 formula which is merely an extension of the formula for the 

 variability of the offspring of two selected parents. If we 

 assume Galton's coefficients in the law of ancestral heredity, 

 the formula for the general case may be written as follows : 



2 V2 (2 V2) 2 (2 V2) 3 (2 



in which r lt r 2 , r 3 , - -, r nt are the coefficients of correlation 

 between offspring and the first, second, third, . . ., nth mid- 

 parents. Use will be made of this formula in treating of the 

 reduction of variability by selection. 



SECTION XV LIMIT TO THE REDUCTION OF 

 VARIABILITY 



We often speak of " fixing " the type by selection, meaning 

 by that the reduction of variability. All recent studies, however, 

 go to show that we do not greatly reduce variability by selection, 

 however much we alter the type. 



In the records of corn breeding it will be remembered that, 

 while the protein and oil contents rapidly responded to selection, 

 yet the coefficients of variability changed but little ; 2 indeed, it 

 is the experience everywhere that variability is but slightly 

 reduced by selection. 



This experience accords with mathematical theory. It will 

 be shown in the Appendix that, in general, the variability of an 

 array is obtained from the standard deviation of offspring in 



1 See Pearson, Grammar of Science, p. 482. 2 See table, p. ^6. 



