HEREDITY 



541 



SECTION XVII BREEDING TRUE, OR STABILITY OF A 

 CHARACTER ESTABLISHED BY SELECTION 



It is the object of this section to show that if an improvement 

 has been made in a population, or if a breed has been developed 

 by selection, the offspring will not degenerate if allowed to 

 breed among themselves without selection ; that is to say, if by 

 selection a certain per cent of a character has been established 

 on the average, the offspring will breed true to that amount of 

 the character which has been established. 



For instance, if selection has been made for six generations 

 of a character H^ the amount of this character appearing in the 

 offspring, after this selection, is given by || H v if we assume 

 mediocrity back of the six generations of selected ancestry. 

 Now if these offspring with || of the desired character be 

 allowed to breed together without further selection, their off- 

 spring will exhibit the character H^ in the amount given by 



/,^,*/, IT r/ 



i + ~ H\ + Q -H i H ----- \- H^ = H l \ 

 2 \04/ 48 2' 64 



so that the first generation of offspring after selection has 

 ceased will exhibit the character to exactly the extent that their 

 parents exhibited it. 



Let us carry this forward another generation. Then the 

 character will be exhibited in the amount given by 



4_ 77 , 77 . J7 , // 77 



+ 4^] 1 + ^ 1 + ^~ 4 1 + ' + 2^ 1 = 6^ 1 ' 



which again shows the offspring unchanged so far as the amount 

 of the character is concerned ; and it is easily seen that this 

 would be true if we should allow breeding to go on for any 

 number of generations without further selection as to the char- 

 acter in question. 1 



For the sake of completeness and generality let us consider 

 the case where selection for a deviation H^ has been made for 

 n generations, and where the offspring so produced are allowed 



1 It is of course assumed that all forms of natural selection are also excluded. 



