462 Proceedings of the Royal Society of Edinburgh. [Sess. 
or 
D 
D 
d 
+ 
+ 
+ 15 
B 
b 
B 
which when expanded gives 
225 
?? i +30 
DD 
+ 1 
DD 
BB | 
B b 
bb 
30 
I )d 1 
D d 
m | 
__ [ + 452 
B b 
+ 30 
BB | 
bb 
dd 
dd 
dd 
1 
^ +30 
+ 225 
BB 
B b 
bb 
Total 256 512 256 
Total 
256 
512 
256 
Hence — = 225 in place of 7. 
n 
The subsequent matings can be easily calculated by the application of 
7 
the form in expression (2). The first term is (225 + 15 + 15 + — 452) 2 , and 
the rest are found likewise. 
Applying the process seriatim with suitable approximations we have 
ad 
the successive values of given in Table I. 
Table I. 
Tr , , ad m 
v alue of 7 — or — 
be n ’ 
After first generation 
225 
„ second „ ... 
56 
„ third „ ... 
27 
,, fourth ,, . . 
19 
„ fifth „ ... 
14 
„ sixth „ ... 
11 
„ seventh ,, ... 
9-6 
„ eighth „ ... 
8-6 
„ ninth ,, ... 
8-3 
It is thus seen that stability is attained only after a considerable number 
of generations in a free-mating population if coupling exists. 
It is possible to introduce a shortened notation. In all circumstances 
these populations after one generation consist of numbers which are those 
of a perfect square. If we write this in the following way we can at once 
proceed to the full expression with little trouble. 
