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 = 1. In every case where there is not inte- 

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



^ /i- 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 S = , in which 



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



(I — '2C)in 

 the ratio of progression, which in this case is nf n /r~ the fraction 



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



r; .■. 5 = becomes 5 



I — — - I 



10 9 



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

 sion and can, therefore, be used as the value of the infinite progression 

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

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

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



pure forms, each being equal to — oiA{M — Mc) »— 1 . A (M — Mc) ^~ " 

 is a vanishing quantity, for M — Mc 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 -^—, vsTe 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 71f<-^ — fusion will in time 



At' I C 



become complete. 



