- DIVERGENT EVOLUTION THROUGH SEGREGATION. 319 
segregation, must lead to extinction. We have already seen that par- 
tial segregation can not by itself prevent the fusion of species. It 
therefore follows that in order to account for the continuance of diverg- 
ent races we must suppose either that the segregation is complete or 
that the divergent evolution is strong enough to more than counter- 
balance the influence of the occasional crossing, or that the partial seg- 
regation is aided by segregate fecundity or segregate vigor. 
Between the members of species belonging to different orders we find 
not only complete segregation, but complete sterility when attempts at 
crossing are made; but hope of gaining an explanation of how these 
characteristics have arisen is found, not in the study of those cases in 
which the process has been completed, but inthe study of the relations 
to each other of species and varieties that are characterized by partial 
segregation and mutual sterility, that is not complete. Here again 
mathematical analysis will help us in understanding the subject. 
Though I have not succeeded in constructing acomplete mathematical 
representation of all the grades of intermingling that will take place, I 
have found a general formula that gives a close approximation to the 
proportion in which two species will breed pure as contrasted with the 
proportion of first crosses and their descendants that will be produced, 
in any case in which the degree of segregation and the ratios of fertility 
for the pure and cross breeds are known. As my object is simply to 
show under what conditions the pure races will continue without being 
swamped by crossing, it is not necessary that I should follow the action 
and reaction between the three-quarter breeds. I wish however to 
call attention to the fact that when the number of the pure forms and 
of the half-breeds is constantly decreasing, without a general decrease 
in the sum of the descendants, it is evident that the three-quarter breeds 
and their descendants are increasing; and when a three-quarter breed 
on one side crosses with a three-quarter breed on the other side, the 
offspring will usually be about intermediate between the two species; 
therefore, where the two species are equally numerous, if we find that 
the pure forms will disappear through fusion, we may expect that the 
three-quarter breeds will also disappear through fusion. 
In constructing my formula it was found necessary to commence by 
placing in the Ist generation of the half-breeds a more or less arbi- 
trary symbol; for the true symbol in each case is the final one reached 
in the nth generation when » isa very high number. The chief inter- 
est therefore centers in what can be accomplished through the use of 
this formula for the nth generation. 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 fecundity, and to give results that would require thou- 
sands of years of continuous experimenting to reach in any other way. 
