202 
Sliew Variation, a Rejoinder 
be made if the first group had included only houses under £5 ? We are unable 
to answer this question. 
(vi) Every frequency curve should be determined by constants of which the 
probable errors are easily deducible. The method of moments admits of the 
probable errors of the moments being easily determined (see Biometrika, Vol. ll. 
pp. 273 et seq.). My system of skew curves gives all the constants in terms of 
moments whose probable errors are known. 
The moments in Kapteyn's theory depend on the integration of : 
I {x + Myie--^"^'!"'' dx, 
J -M 
and there is no means of readily evaluating this integral. In fact the arithmetical 
mean*, the standard deviation, the shewness and the Jmrtosis, and the modal 
divergence are vnobtainable from the constants of Kapteyn's theory. This seems 
to me sufficient to deprive the method of any practical significance even as an 
empirical representation. 
It has further been shown by Sheppard that the probable errors of constants 
determined by class frequencies (partial areas) are higher than when these 
constants are determined by the method of moments. We may give the above 
statement a separate paragraph as : 
(vii) The fundamental physical constants of the frequency distribution are 
not determinable from Kapteyn's empirical curve. 
To illustrate the results of this want of a knowledge of the probable errors, 
I turn to the three illustrations given by Kapteyn. 
Example (i). Observations on the Threshold of Sensation. Kapteyn himself 
shows that his solution is hardly less satisfactory if he uses the Galton-McAlister 
curve (our equation (xi) p. 194). He does not therefore know whether 5- = "00 and 
q = — '04 differ within the probable error of q. 
Example (ii). Valuation of House Property. Kapteyn fits this with a 
Galton-McAlister curve for his q comes out '00. Owing to the difficulty in 
calculating moments, we cannot do more than approximate to the value of the 
skewness in these data. I make it 1'8. It is certainly well over unity. We 
have already seen that it is impossible for a Galton-McAlister curve to give a 
skewness above "21. The apparent agreement Kapteyn finds for the frequencies 
is not therefore sufficient evidence that the fundamental constants of the 
distribution will be really given by reasonable values. 
Example (iii). Foreheads of Garcinus moenas. Kapteyn fits these first with 
(2 = 2'21. "The agreement seems satisfactory." Then with q = 0, or a Galton- 
McAlister curve, " The representation is hardly less satisfactory." Then with 
* " The arithmetic mean of all the A"s cannot be (lencralhj found in a simple and rigorous way," 
Kapteyn, p. 4-i. 
