330 Proceedings of the Koyal Society of Edinburgh. [Sess. 
the third term being evidently negligible compared to the first. For 
equilibrium, therefore, 
Jc=:p/^ i 1 cot lp 7 r+ i . 
^ y In 27 rln } 
0)2 
Remembering that k~j—, we have 
2tT p 
Hence cot |-^7r is of the order— and is very small, so that p is near to 1. 
Jn 
Then 
4:n 
This is a very slow rate of diminution, a population of 7i individuals 
breeding at random would require generations to reduce its variance in 
the ratio 1 to e, or 2*8 generations to halve it. As few specific groups 
contain less than 10,000 individuals between whom interbreeding takes 
place, the period required for the action of the Hagedoorn effect, in the 
entire absence of mutation, is immense. It will be noticed that since I is 
always less than 1‘4 in species stationary in number, the solution above 
makes jp slightly greater than I, which strictly would indicate negative 
frequencies at the extremes : the value of h is, however, connected with 
the curvature in the central portion of the curve, and the small distortion 
at the extremes, where the assumptions, upon which our differential 
equation is based, break down, will not affect its value. (Fig. 2 shows the 
distribution of z = \o^—.) 
^ q ^ 
The number by which the number of factors current is reduced in each 
A 
generation is and since this number depends on the general form of the 
distribution curve, it will not be diminished by a number of mutations of 
the same order. The effect of such very rare mutations would merely be 
to adjust the terminal of the curve until the rate of extinction is increased 
sufficiently to counterbalance the additional mutations. It is probable, 
however, that /x is always far greater than is necessary to make this state 
of affairs impossible, save in the case of a small colony recently isolated 
from a very variable species. In this case, with 7i small and A large, p, 
might for a time be of the order An~^, rather than of the order An~^\ 
or An~'^. 
In the case of a population with A factors, with a supply of fresh 
