180 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, 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 = 1,m =1. In every case where there is not inte- 
grate fecundity, that is, where m is not larger than M, the fraction 
(1 — 2c)m 
ay Ye Vg 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 = ae in which 
a is the first number of the progression, which in this case is 1, and ris 
: ; Gis ge : : —= 26m : 
the ratio of progression, which in this case is Coe the fraction 

oe ; I ; 4 
we are now considering. Supposing c = 5) the fraction will be 
9 
Be Soh 1S = =" Becomes! S = ee 258 
Sa gee 2 Le aa alee if 


=g. 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 nth 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 — Mc)"—!, A(M—Mc)*—! 
is a vanishing quantity, for MM — Mc is less than 1. Every form is, 
therefore, in time fused with other forms. But let us try higher 
I I 
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 
eae . That is, if m=M, and Mf < 
I 

I f Rete : 
Rea a fusion will in time 
become complete. 
