400 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The forms which the different point binomials tend to assume are 
interesting. As already shown by Professor Pearson, (g + l) n has the same 
slope as the curve given by his Type III., namely, by 
y = ^1 + - ^ e yX , where y == E m 
This curve has for the criterion 
2/? 2 - 3 y 8 j -6 = 0 , 
where 
/h = an( f P2 = b . 
When the criterion is greater than zero, Type IV. arises. As just shown, the 
second and fourth moments of (g+lF are the same as those of (g + l) TO 
(1 + q) n , when m + n=p ; so that if the criterion be zero for the curve which 
represents the limit of (g + l)y it will be greater than zero for the curve 
representing (g + l) m (1 + q) n , since the third moment of the latter is less 
than that of the former in the ratio m — n to or, in other words, a 
curve having the same moment relationships as Type IV. will represent 
the result of mixed dominance. 
I have not been able to prove that the limit of (q-\- l) m (1 + q) n is 
represented by 
y = 
Vo 
, - v tan - 1 - . 
1+ 5 
but I think that it is highly probable that it is so, and, in addition, even if 
the limit is different, the curve must be nearly the same as that given by 
the Type IV. equation. This at once renders futile the class of criticism 
which, ignorant of mathematical principles, condemns Type IV. as inappli- 
cable to practical problems on account of the imaginary roots in the 
denominator. 
The form (1 + l) n leads to the normal curve, as is well known. 
The form (l+'7i + l) p has the moment relationships of Type IV., as 
shown in my former paper, and finally (a + 1 + a) p those of Type II. 
The foregoing lemmas apply directly to the groupings which are possible 
on the Mendelian hypothesis. Four possibilities or combinations of possi- 
bilities have till now been experimentally ascertained or premised as the 
result of experiment : — 
1. Blending; 
2. Dominance ; 
3. Partial dominance ; 
4. Coupling or duplication of parts. 
and 
