K. Pearson 
201 
perfectly correlated with each other, which is contrary to experience, it must follow 
that if Kapteyn's hypothesis were correct large quantities of characters distributed 
in truncated Gaussian curves ought to appear when we deal with variation. The 
total absence of such characters is evidence that the ^-characters are shadow 
variables and of no biological import. 
(iii) The previous statements reduce Kapteyn's special choice of 
F(X) = {X + k)i 
to a mere artifice adopted to get an empirical curve of variation by Edgeworth's 
hypothesis. Many other functions are cl priori equally valuable, and might be 
adopted to get curves of limited range, e.g. 
F{X) = {X + K)i{X-K'y. 
The hypothesis gains nothing in logical consistency by its appeal to the Gau.ssian 
curve ; that appeal is one adopted for convenience of fitting, and the sole test of 
Kapteyn's curve is empirical goodness of fit. 
Practical: 
(iv) Every frequency curve should he a gradtiation formida. Kapteyn's method 
of fitting is by equating certain total frequencies in order to determine his four 
constants. They thus fail to successfully smooth any special causes tending to 
exaggerate any particular frequency group. Such screening of special causes of 
frequency deviation is far less likely to occur when we use the method of moments, 
which is a true method of graduation*. 
(v) We ought in every law of frequency distribution to he able to judge of the 
effect of the unit of grouping on the values of the constants. This has been 
satisfactorily achieved for, perhaps, the bulk of cases, when the method of 
moments is used by Sheppard's correctionsf. In Kapteyn's process we have no 
means of ascertaining the extent to which the size of the unit of grouping 
influences the constants of his disti'ibution. 
In his Example II., for instance, he takes his curve to accurately reproduce the 
total area of the group of houses under £10 annual value. What difference would 
■■■ Thus Kapteyn deals with some statistics of the vahies of house property in England fitted by me 
{Phil. Trans. Vol. 186 A, p. 396). I specially state that £20 was the limit to taxable value, and that 
accordingly the frequency of houses immediately below this value will be exaggerated. Kapteyn's 
method fails to indicate such a source of a priori recognised irregularity. For example, one of his 
conditions is that the houses of value less than £10, i.e. more than half the total frequency, shall 
be identical in his result with the observed frequency. He thus cuts away at once any possibility 
of smoothing this group or allowing for the large probable error in it due to random sampling even. 
His method leads to a limiting house value of £2. 2s., while mine leads to £4. 4s. Mine corresponds to 
a weekly rent of about 2s. ; his to a weekly rent of Is. The latter rent hardly occurs in England 
unless the house is given in part payment of wages, or in charity. Kapteyn says that his distribution 
starts with a zero frequency, and mine with an infinite ordinate. " It seems hardly admissible that 
the latter solution can be in accordance with nature (s/c) in this particular." Why not? An infinite 
ordinate may and does in my case give a finite frequency. 
t See reference, p. 187. 
Biometrika iv 
26 
