322 DIVERGENT EVOLUTION THROUGH SEGREGATION. 
Second rule.—The nth generation of pure-breeds =A(M—Mc)"=A(M—Mc)"™1x 
(M—Mc); and the nth generation of half-breeds— Ame(M—Me)"! multiplied by the 
(1—2e)m 
M—Me ae 
by the number of the generation, 7. ¢., containing » terms, of which the first is 1; 
sum = of the series 1+~, ., containing as many terms as that expressed 
“p= le a ae Se ): H being the number of half-breeds, and 
P being the number of pure-breeds. 
Third rule.—To correct this formula, so that it shall indicate the proportions that 
will result when the relative vigor of pure and cross breeds is considered, we must 
substitute MV for M, and mv for m; V being the proportion of each generation of 
pure breeds that grow to maturity and propagate, and v being the proportion of 
half-breeds that do the same. 
METHOD OF USING TABLE III. 
By supposing » to be an indefinitely high number, and by giving 
different values to M, m, and c, we shall have the means of contrasting 
the number of the pure-breeds with that of the half-breeds, when the 
process has been long continued under different degrees of positive 
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 m is not larger than M the 
.. (L—2e)m _ ; b R 
fraction M= veo less than unity, and the sum of the geometrical 
progression of our formula will fall within the limits of a number that 
can be easily computed by the well-known formula S= in which 
al 
tq 
a is the first number of the progression, which in this case is 1, and ¢ 
is the fraction we are now considering. Supposing c= ‘5, the freer 
will be Soe =, becomes S=;—3=9—3=?. This 
number 9 is therefore equal to the sum of this progression and can, 
therefore, be used as the value of the infinite progression in the formula 
for the nth generation when » is a very high number. Substituting 
these values 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)r-—. A (M—Me)"—1 is a vanishing quantity, for M—Mc is less 
than 1. Every form is therefore in time fused with othec forms. But 
let us try higher degrees of segregation. If we make c= 79 OF zo 00) 
we still find that half-breeds=pure-breeds, while the latter are con- 
stantly decreasing, which shows that imperfect positive segregation, 
without the aid of some quality like segregate fecundity, can not pre- 
vent a species being finally fused with other species as long as the 
whole number of each successive generation does not increase. 
Let us now consider cases in which the Segregation is incomplete, but 
Segregate Fecundity comes in to modify the result. Let M=2, m=1, 
¢= >. Substituting these values in our formula, we shall find that 
