170 
Miscellmiea 
In my opinion, however, there is absolutely no important discovery here, Kapteyn's or 
rather Edgeworth's (i) and my (iii) are in my o2Jinion only convenient analytical ways of 
expressing the general relation (ii). My sole object in referring to the matter is to meet 
Professor Kapteyn's charge, that I have largely profited by his paper and the suggestion that 
I had invented (iii) as a differential equation to frequency distributions after the appearance 
of that paper. 
We now, having cleared oft' Professor Kapteyn's first two statements, come I think to the 
kernel of the matter. Neither (i) nor (iii) is more general than (ii), the whole problem 
turns on the pro^Jer and suitable choice of F {x) in (i) or f{x) in (iii) just as it turns on a 
proper choice of {x) in (ii). Up to this point neither party has made any real progress. 
Kapteyn selects i^(.j') = (.r + K)9, and I selected /(.&) = (^(c,,,*"). 
The test of the merits of the two selections must depend upon certain points which I 
will shortly consider. But first I would meet another remark of Kapteyn's. He says I stop 
at ^2, but he does not note why, although the reasons have been stated, i.e. 
(i) I have given the cxiiressions to deduce any c,^ whatever, but the higher c's depend 
upon the high moments, which I have shown are subject to large percentage probable errors. 
(ii) The c series converges in practice rapidly, the reducing factor being of the order of 
the skewness and the kurtosis, both of which are usually much smaller than unity. This is 
indicated by tlie geneiul rough approach of most statistics as a first aijproximation to a 
Gaussian curve, and as a second approximation to a point binomial, and as a third approxi- 
mation to the hypergeometrical series. 
(iii) The sufficiency with which f{x) = c^^+ClX-\■c■,x''■ gives actual frequency distributions. 
These are the justifications for my own choice of f{x). 
To not one of my criticisms of Professor Kapteyn's choice of F {x) does he make any 
reply whatever. I pointed out : 
(i) That a good frequency-curve must be a graduation formula, and that Kapteyn by 
making his result depend on certain total areas had shown that he failed to realise this 
essential condition*. 
(ii) That we ought in every frequency distribution to be able to realise the effect of the 
unit of grouping, but that Kapteyn's method wholly ignores this important point. 
(iii) That the probable error of every constant involved ought to be ascertainable, and 
this is not the case with Kapteyn's constants; he finds for one case that his constant q = 0 
or y=oc give both a "pretty close" representation. As the whole range of q must lie between 
these arithmetical values, it is clear that it cannot be an important constant which will 
enable us to effectively discriminate between two allied distributions t. 
* Further : constants deduced from class frequencies are never as accurate as those deduced from 
moments. In fact they often are very bad indeed. Thus suppose it necessary to find the standard 
deviations (1) by moments, (2) by areas, say from the quartiles. Sheppard (Phil. Trans. Vol. 192a, p. 134) 
has shown that if the total frequencies are n and n', the probable ei'rors are ■74728(T'Jn and -gigOS-r/v/^ 
respectively. Or, if n were 1000, n' would have to be 1513 or 50 p.c. larger to obtain as good a result. 
The errors resulting from this source are as serious as the failure of ' class ' fitting (when only the 
same number of classes are taken as constants to be determined) to graduate the observations. 
t Professor Kapteyn's reply to this criticism is given above and it is, I venture to think, no 
reply at all. He says that it only shows "how widely different forms may be made to represent with 
tolerable precision the same frequency-curves." This gives the whole theory away. Any frequency 
distribution of ?i classes is absolutely determined by its moment-coeflieients fi.;,, /x.^, fx^... fi,^. The class 
frequencies can be expressed in terms of the /m's (Thiele) if enough are taken. Any constant there- 
fore of the frequency distribution ought to be uniquely expressible in terms of these constants. After 
