l8o APPENDIX I DIVERGENT EVOLUTION. 



METHOD OF USING TABLE III (seep. 179). 



By supposing n to be an indefinitely high number, and by giving 

 different values to M, m, and c, we shall have the means of contrast- 

 ing the number of the pure-breeds with that of the half-breeds, when 

 the process has been long continued under different degrees of posi- 

 tive segregation and segregate fecundity. 



In the first place, let us take a case in which there is no segregate 

 fecundity, that is M = m, and for convenience in computation let 

 us make M = i, m = i. In every case where there is not inte- 

 grate fecundity, that is, where m is not larger than M, the fraction 



M M~~ * s ^ ess than unity, and the sum of the geometrical pro- 

 gression of our formula will fall within the limits of a number that 

 can be easily computed by the well-known formula 5 = , in which 

 a is the first number of the progression, which in this case is i , and r is 

 the ratio of progression, which in this case is ,, A the fraction 



we are now considering. Supposing c = , the fraction will be 



f i ^ i 



I 1 IO J 8 = a 



! _ _L = 9 ~ ir 



10 9 



= 9. This number 9 is, therefore, equal to the sum of this progres- 

 sion andean, therefore, be used as the value of the infinite progression 

 in the formula for the wth generation when n is a high number. 

 Substituting these values in the last formula of the table, we find that 

 the nth generation of the half-breeds equals the wth generation of the 



pure forms, each being equal to of A (M Mc) n ~ l . A(M Mc) n ~ l 



is a vanishing quantity, for M Me is less than i . Every form is, 

 therefore, in time fused with other forms. But let us try higher 



degrees of segregation. If we make c = or - , we still find 



100 1000 



that half-breeds = pure-breeds, while the latter are constantly de- 

 creasing, which shows that imperfect positive segregation, without 

 the aid of some degree of segregate survival, can not prevent a species 

 being finally fused with other species. The pure-breeds must de- 

 crease as long as the whole number of each successive generation of 

 pure-breeds does not increase by a multiple equal to or larger than 



. That is, if m = M, and M < - fusion will in time 



i c i c 



become complete. 



= r; .'. 5 = r^TZ becomes S = - g - 9 _ 8 



