326 Proceedings of tlie Royal Society of Edinburgh. [Sess. 
In the more general case in which the number or the species is not 
stationary but increases in each generation in the ratio m :1, m being near 
to unity, the series will be 
VT 
and /(cc) = e™(^~^^. The chance of extinction of a single gene in one genera- 
tion is e~'^, where m is near to unity. In other species in which an 
individual may survive for many breeding seasons, or in which the 
generation is of indeterminate length, the form of the function f{x) will 
be modified : it is sufficiently clear, however, that if we consider that stage 
in an animal’s or plant’s life - history at which reproduction is about to 
commence, the form of the function will not be very different, and the 
chance of extinction of a particular gene, thus far established in the species, 
will be 
e-\ 
where ^ is a small number not greatly different from unity.* The 
arbitrary element thus introduced into the question of the survival of a 
mutant gene is due to the fact that in the first place its survival depends 
on that of the individual in which it occurs, and this chance is variable 
from species to species ; once, however, it has reached the point of existing 
in an adult individual capable of leaving many offspring, the conditions of 
its survival are closely similar in all cases. While it is rare, its survival 
will be at the mercy of chance, even if it is well fitted to survive. Using 
the above expression, 
f{x) = e^-\ 
it may be seen that only about 2 per cent, will survive 100 generations, 
while those that do will on the average be represented in some 50 indivi- 
duals. Only when the number of individuals affected becomes large will 
the effect of selection predominate over that of random survival, though 
even then only a very small minority of the population may be affected. 
3. Factobs not acted on by Selection. 
If p be the proportion of any gene, and q of its allelomorph in a 
dimorphic factor, then in n individuals of any generation we have 2np) 
genes scattered at random. Let 
cos 0=\ - 
where 0 lies between 0 and tt. 
* An upper limit can be set to I by the mere fact of segregation, for in the case of the 
most uniform possible reproduction, when each individual bears 2 offspring the chance of 
extinction of any gene is so that I cannot exceed D4. 
