Mathematical Contributions to the Theory of Evolution. 401 



61, to ... . of the table. For example, in the case of Mr. Gralton's 

 Basset hounds, O5015, 0"2553, and 0*1459 were the coefficients to be 

 used, rather than 0*5, 0*25, and 0*125, when he proceeded to apply 

 the law to three generations. These give the proper allowance for 

 the ancestry beyond the pedigree. Thus the great-grandparents 

 ought to have been given about a fifth more weight. If we proceed 

 to six generations in pedigree stock of the latter type then the 

 offspring will be within 1*2 per cent, of the selected ancestry, i.e., 

 their stability as given by the last column = 0*9879. 



(iii) Now let us apply these results to the all-important problem 

 of panmixia and degeneration. Suppose a selection made of a par- 

 ticular character for n generations, starting from the general popula- 

 tion. Then the offspring in the (n + l)th generation will have 1 — — of 

 the character on the average. Now, stopping selection, let us breed 

 with a first generation of mid-parents with 1— ^ of the character. 

 The offspring will have : 



2 ( 1_ i«) + 4 + 8 + 16 + ^"2^ 



= (l — -'W-Yl — —\ = 1 — — of the character. 

 2 V 2'7 2 \ 2 n j 2' 1 



The ?i + 2th generation will have : 



H 1 -2^) + i( 1 -Jr) + i + r6+ • • • • 



2 \ 2 n J 4 V 2*/ 4 V 2 7 2 " 



of the character, and so on. The law is obvious ; the offspring will 

 always have the same amount of the character as had the generation 

 after selection ceased. If we start with pedigree stock with unknown 

 ancestry beyond the nth generation, we reach the same conclusion. 

 Thus, after three generations the offspring will have 0'9027 of the 

 selected parents' character. Now stop selection and the fourth 

 generation will have : 



0-9027 e A + 6 2 +6 3 -f 64 



= 0*4515 + 0-2507 + 0*1276 + 0*0729 = 0*9027, 



the fifth generation will have 



0'9027 (e x + 6 3 ) + e 3 + e 4 + 6 5 = 0-9027, 



again, and so on. The general law is obvious. 



VOL. LX1T. 



