Miscellanea 
429 
It is very doubtful whether the series can be simplified any further and the question of con- 
vergency is a very difficult one, for no simple law can be given for any of the coefficients involved. 
The series has been tested with curves of Types I and III and the results are tabled below : 
Type 
{sk.f 
djD from 
formula 
djD 
(correct) 
Error 
I 
0-1 
0-1 
005 
0-05 
•1733333 
•4160000 
•2582456 
•4657500 
•715706 
•719087 
•696072 
•700078 
•715702 
■719048 
•696072 
•700181 
+ -000004 
+ ^000039 
■000000 
- -000103 
III 
0 
0 
0 
0 
0^1 
02 
0^5 
1^0 
•66871448 
•670909 
•678394 
•69402 
•66871460 
•670909 
■678347 
■69315 
- -00000012 
•000000 
+ ^000047 
+ -00087 
These show that the formula gives very good results even for high values of the skewness. 
It will be noted that in the ease of Type III curves we have quite good results even when the 
skewness is unity. If the equation of Type III be taken in the form 
?/ = 2/oe--(^l+^y 
then the square of the skewness is — ^ and when the skewness is unity -p is zero. But the valu? 
of d as given by (22) and (23) is 
d, dt dr. 
do - ' + - : - 1 + 
p p- p-' 
and this formula is of no great value if p Ije small. The pro?esses of § 5 give 
rf/ do' c/3' 
~ + + (p + (p + If + - 
by what is practically Montfort's Theorem, and the transformed series is simpler and of more 
value than the untransformed series. 
The formula given in § 3 for the area from the mode to the median has also been tested and 
gives results similar to those for the ratio of d to D. This may be useful if the area on either side 
of the mode be required. 
In conclusion I tender my thanks to Professor Pearson for his valuable criticism and help. 
IV. The Probable Error of a Mendelian Class Frequency. 
EDITORIAL. 
Dr Raymond Pearl in a recent paper entitled "The Probable Error of a Mendelian Class 
Frequency*," provides a striking illustration of how the capable biologist nee Is a long con- 
tinued training in the logic of mathematics before he ventures into the field of probability. 
Dr Pearl writes: "With the increased volume of Mendelian experimentation there is an ever- 
growing need for adequate and clearly understood tests for the statistical significance of diflferences 
between observed results and expectation." Now there is absolutely no obscurity about the test 
for a single class frequency in a Mendelian result. I cannot conceive that tlierc ever could be or 
ever has been such obscurity in the mind of a trained mathematician. If on the basis of Mendelian 
theory the relative frequency of that class be p and the relative frequency of all other classes 
he \ - p = q, then in observation samples of m, the frequencies of this particular class distribute 
themselves according to the binomial law N {j3 + qY" in N samples. To determine the impro- 
* American Naturalist, Vol. li. p. 144, March, 1917. 
