252 Proceedings of the Royal Society of Edinburgh. [Sess. 
from each parent). Let a be dominant over b, and d over c, and the 
grouping becomes — 
a , a 
a , a 
a, b 
b, b 
b, b 
1 
2 
+ 4 
+ 2 
1 
c, c 
* 
c, d 
c, d 
c, d 
5 
a, d 
1 a, b 
a, a 
b , b 
b , b 
2 
+ 
+ 
2 
| c, c 
5 
d, d 
c, c 
c, d 
Totals, 3 10 3 
It can be. at once deduced from these results that when there are 2 p pairs 
of elements in each parent determining stature, the subsequent grouping 
of the population is given by 
(H-4 + 6 + 4+1)*, 
where there is no dominance, 
and 
(3 + 10 + 3) p 
where dominance applies to all the pairs and is equally divided between 
the pairs for each race. The mathematics of the first is easy and has been 
considered in my previous paper. It leads at once, when mating is random, 
to a correlation coefficient between parent and offspring of r — *5. 
With regard to the second, there are several things to note. For 
convenience we may write the point binomial as (l + ^ + l) p . 
If we expand by the multinomial theorem, arrange the terms, and then 
determine the moments, we have : — 
the second moment /x 2 = — — , 
n+ 2 
the third moment /x 3 = 0 as the multinomial is symmetrical, 
the fourth moment 2pw + 4jp(3j9 l ). 
^ (n+ 2) 2 
The relationship of these moments is such that for all values of 
n > 4, ^ is > 3, or the resulting curve partakes of the qualities of Type IV. 
yUo 
rather than of the normal curve. But n = 4< is not the dividing point. 
When n — 2 the point binomial obviously is the square of (1 + 1) and there- 
fore is of the type which gives rise to the normal curve. In this case 
im 2 = P and ^ We may therefore take it that this curve represents 
^ jb 
the dividing line, and that when n is greater than 2, a curve with moment 
relationships more nearly those of the symmetrical form of Type IV. 
