212 
Skew Variation, a Rejoinder 
If we assume it to be F{x) = 2 (a,-*'') as I have done, we fall back on the normal 
curve when a,, for r > 0 is zero within the limits due to its probable error. Ranke, 
if he wishes to demonstrate the Gaussian law as general, must show this to be the 
case. It has been over and over again demonstrated that a^, a^, etc. differ 
significantly from zero for a great variety of series. Another advantage of the form 
F {x) = 'S^{a,.x''') is that it covers as I have shown discrete as well as continuous 
varia,tion. Considering a = a^^J F {x) as the standard deviation of the "instantaneous 
Gaussian curve," we see that the "instantaneous Gaussian curve" varies from one 
position to a second, like the " instantaneous ellipse " of the astronomers. A 
reasonable first hypothesis to make is that the local mean square deviation cr- is 
independent of x, we obtain the Gaussian curve. A next assumption is that it is 
a linear function of — perhaps it would be better to say that its mean local value 
is a linear function of x, i.e. the mean square of the local variability a- is correlated 
lineally with x. This gives my curve of Type III. The next easiest assumption 
is to suppose the regression line of a" on x to be parabolic. In this case we obtain 
the remainder of the curves treated in my II. and XI. memoirs. If we stop at aq 
we have what I have termed the skew frequency curves of the gth order*, and we 
see that this involves a regression curve between the square of the mean local 
variability and the character of the (j'th order*. I see, however, at present no 
practical necessity for proceediiig beyond skew curves of the 2nd order, although 
I propose shortly to publish a discussion of skew curves of the 3rd order illustrating 
some theoretical points which arise in their discussion. 
To sum up I think Ranke's criticism fails (a) because he has disregarded the 
universally recognised need of modern statistical science for asymmetrical frequency 
curves, (/3) because he has not appreciated the mathematical transformation by 
which a number of finite terms are replaced by an integral expression, (7) because 
he has not realised that modern theories of heredity lead directly to discontinuous 
skew distributions, (S) because continuity does not depend upon infinity of 
fundamental cause-groups, and lastly (e) because, and this may be due to my fault 
in the first deduction of my curves, he has quite failed to see either their scope or 
their real generality. 
* "Mathematical Contributions to the Theory of Evolution, XIV. On the Theory of Skew 
Regression." Dulau and Co. 
