l8o APPENDIX I DIVERGENT EVOLUTION. 



METHOD OF USING TABLE III (see p. 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 



( T 2C^ "Wi 



-^r-j j/f- is less 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 



f j o r ) 117 



the ratio of progression, which in this case is ~~jnr^ rr> the fraction 

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



r " _?_] r 



10 j g a j 9 



-7- -r; .-. S ------- becomes S = -^ = : 9 _ _ g 



i i - 



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 nth generation of the 



pure forms, each being equal to of A (M Me) n ~ ' . A (M Me) 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. 



