252 
KEY. J. T. GULTCK ON DIVERGENT EVOLUTION 
may expect that the three-quarter-breeds will also disappear 
through fusion. 
In constructing my formula, it was found necessary to com- 
mence by placing in the 1st generation of the half-breeds a more 
or less arbitrary symbol; for the true symbol in each case is the 
final one reached in the nth generation when n is a very high 
number. The chief interest therefore centres in what can be 
accomplished through the use of this formula for the ?ith gene- 
ration. It seems to me to furnish a method of reaching the 
final proportion of pure breeding that will be produced by any 
form of combination between Positive Segregation and Segregate 
Pecundity, and to give results that would require thousands of 
years of continuous experimenting to reach in any other way. 
Method of using Table III. (see p. 255). 
By supposing n to be an indefinitely high number, and by 
giving different values to M, m, and c, we shall have the 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 Pecundity. 
In the first place let us take a case in which there is no Segre- 
gate Pecundity, that is M =m ; and for convenience in computa- 
tion let us make M=l, m= 1. In every case where m is not 
larger than M the fraction 
(1— 2c) m 
M-M c 
is 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 a is the first number of the 
1-q 
progression, which in this case is 1, and q is the fraction we 
are now considering. Supposing c= t 1 q, the fraction will be 
1-^,- * S ’ l- 2 “comes iMs 
number 9 is therefore equal to the sum of this progression aud 
can therefore be used as the value of the infinite progression in 
the formula for the nth generation when n 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 — Mc) n_1 . A(M— Mc) n_1 is a 
vanishing quantity, for M — Me is less than 1. Every form is 
therefore iu time fused with other forms. But let us try higher 
