180 APPENDIX I — DIVERGENT EVOLUTION. 



Method of using Table III (set- p. 179). 



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

 different values to M, in, 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 .1/ = in, and for convenience in computation let 

 us make M = 1, m == 1. In every case where there is not inte- 

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



(1 — 2c)m . , , . , , _ , . , 



, , , , 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 ease is 1 , and r is 



( 1 — ic) in 

 the ratio of progression, which in this case is \*_ [ i r the fraction 



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



f 2 "1 



I 1 — — I 1 



IO J 8 ,; . c I 2_ 



1 — — 1 



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 »th generation when n is a high number. 

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

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



pure forms, each being equal to — of A (M -Mc) n — X . A (M Mc) n — 1 



is a vanishing quantity, for M - Mc is less than 1. Every form is, 

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



1 - - c ^ 1 — c 



become complete. 



