324 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
maintain equilibrium. That such is the actual state of the case in 
mankind may be inferred from the fact that the frequency distribution 
of the numerical proportion of the allelomorphs, calculated on the 
assumption of selection maintained in equilibrium by occasional mutation, 
leads to the value of the Dominance Ratio which is actually observed. 
In all cases it is worth noting that the rate of mutation required varies 
as the variance of the species, but diminishes as the number of individuals 
is increased. Thus a numerous species, with the same frequency of 
mutation, will maintain a higher variability than will a less numerous 
species : in connection with this fact we cannot fail to remember the 
dictum of Charles Darwin, that “ wide ranging, much diffused and common 
species vary most” (1, chap. ii). 
1. Equilibrium under Selection. 
Let the three phases of a dimorphic factor be born in any generation 
in the proportion 
P : 2Q : R, 
then the proportion of the two allelomorphic genes will be 
P + Q : Q + R, or 'p \ q\ 
if by selection those that become parents are in the proportion 
aP : 26Q : rR, where aP + 26Q + cR= 1, 
then the proportion born in the next generation will be 
(aP + Z>Q)2 : 2(aP + 6Q)(6Q + cR) : (6Q + cUf ; 
equilibrium is thus only possible if Q^ = PR, i.e. P=p^ Q==pq^ R = 9'^ and 
if aP + 6Q=p, bQ + cU = q. 
Hence it follows that, if 
w = 1 + Cl, 6=1+^, 6' = 1 + y, 
_ ^ = 1 - 
pq (f 
specifies the condition of equilibrium. 
If selection favours the homozygotes, no stable equilibrium will be 
possible, and selection will then tend to eliminate whichever gene is below 
its equilibrium proportion ; such factors will therefore not commonly be 
found in nature : if, on the other hand, the selection favours the heterozy- 
gote, there is a condition of stable equilibrium, and the factor will continue 
in the stock. Such factors should therefore be commonly found, and may 
explain instances of heterozygote vigour, and to some extent the deleterious 
effects sometimes brought about by inbreeding. 
