332 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
We may suppose that usually r is near to unity, and logr, which may 
be positive or negative, may be considered to be of the order of 1 per 
cent. Let logr = ct, then for different factors a will have different values, 
indifferently positive and negative, since we have no reason to suppose 
that the selection favours either dominant or recessive characters. The 
mean square value of a for different factors we shall write 
For any factor 
d T p 
— low £ = a: 
dT ^ q 
therefore dp 
dO 
dT ~ • 
The factors which in one generation are at 0, will in the next be 
2n^ 
scattered owing to two causes : (1) random survival causing variance. 
(2) selection causing variance, pg o-/( = f sin^ d . c 7 a^). The total variance 
at any point will be 
+ icrJ sin2 e ; 
and so long as cTa is small as we have supposed, the equilibrium distribution 
will be 
1 
U ^ 
[2^ + 
or nearly 
2 log (o-^V 8?^) 
^si 
sin2^ + 
n being large compared with — x , the effects of selection are, for the more 
(Tct 
important factors, much more influential than those of random survival. 
At the extremes, however, for very unequally divided factors the latter 
is the more important cause of variation. (The distribution of ;^ = lo 2 : - is 
shown in fig. 3.) 
The amount of mutation needed to maintain the variability with this 
amount of selection may be calculated from the terminal ordinate 
whence 
2 log (o-« 
'•V? 
