328 
Proceedings of the Royal Society of Edinburgh. [Sess. 
0 
When 0 is near to 0, p which is always equal to sin^ — , will be very nearly 
equal to Let 
t — sin ^0, 
then the number of genes in existence is 2nt^, and the chance of their 
extinction in one generation is 
This chance is negligible save when t is very small, and may be 
equated to Jd; hence the number of genes exterminated in any one 
generation 
Jo 
= 4: f 
Jd 
A 
In the stationary case y = — , and the number of genes exterminated 
7T 
will be 
A 2 y/2V_ 
^ ' JUn ’ 
if new mutations occur at a rate njUL, then this equilibrium will be possible 
if _ 
A S 
For species in this stationary state the variance will vary (1) as the 
rate of mutation, (2) as the number of the population raised to the power 
of f, (3) as \/l, a quantity which will seldom differ much from unity. 
Using the variate 0 = loge^, the distribution for this case is shown in fig. 1. 
5. The Hagedoorn Effect. 
In the absence of mutation, extinction will still go on, and the number 
of factors must diminish, hence we may put for this case 
y = Aq^- 
-&T 
. COS p 
^ ' 2 siu ^pir 
If 0 is small, 
cosp^^ - = cos JpTT sin cos ^j97t . . . 
= cos JpTT + 2p sin JpTT . t - cos ^p-n- . . 
so that the rate of extinction is 
