K. Pearson 
203 
q=ac, "The representation, though sensibly less satisfactory than that by the 
[previous] solutions, is still pretty close." 
A method by which a fundamental constant of the distribution may take any 
value between 0 and oo and still give a "pretty close" representation, must I think 
condemn itself Such a statement demonstrates effectively that the author has 
not yet determined numerically or even approximately in his own mind the 
probable errors of the constants he uses. 
It will be seen that as far as the three illustrations Kapteyu himself gives go 
he has not advanced the matter beyond the Galton-McAlister curve. That curve 
fits reasonably (according to Kapteyn) all his three series. But the skewnesses of 
the three series are respectively "72, 1'8 and "32. I have calculated these roughly, 
but I think there can be no doubt of their approximate correctness. In every one 
of these cases the skewness sensibly exceeds the maximum limit of skewness, 
i.e. •21, possible for the Galton-McAlister curve which Kapteyn applies to 
them*. 
C. (v) The General Results loliich floio without the Third Gaussian Axiom. 
It seems to me accordingly that very grave objections can be raised not only 
from the theoretical but from the practical standpoint to the methods I have 
discussed which attempt to allow for asymmetry, i.e. 
(i) The Galton-McAlister Geometrical Mean Law 
(ii) The Galton-Fechner use of Half Gaussian Curves, 
(iii) The Edgeworth-Kapteyn use of transformed Gaussian Curves. 
All these experienced statisticians differ in toto from the opinion of Ranke and 
Greiner — that we need not trouble about descriptive curves for asymmetrical 
distributions — but their methods seem to me unsatisfactory theoretically and 
insufficient practically, because they still make a fetish of the Gaussian axioms. 
They do not return to the Laplace-Poisson method of replacing those fundamental 
axioms by more general conceptions. If a Gaussian curve does not tit, they will 
consent to deduce their own curves from a truncated Gaussian curve, which some 
shadow variable of the mathematician is supposed to follow, and of which we have 
no experience in any organic characters hitherto measured. Indeed if we had 
such experience, it would at once negative the very axioms on which the Gaussian 
curve is based. 
Now it seems to me that all these attempts, whether embodied in the general 
method of Edgeworth or in the special hypotheses of Galton-McAlister or 
Kapteyn, amount to abolishing the third of the Gaussian assumptions, namely that 
small increments of the variable or the character are independent of the total 
already reached. That is to say that they amount to saying that increments of the 
* I am unable to say how far the general form of Kapteyn allows for the requisite range of skewness 
and kurtosis, because neither the modal difference, nor the standard deviation, to say nothing of the 
higher moments, can in general be evaluated. 
