K. Pearson 
211 
my generalised probability curves*. A very few terms of the expansion, however, 
suffice for describing practical frequency distributions. If we keep only three 
terms, we see that the same system of curves suffices to describe continuous and 
discrete variates — an important point. If I lay stress upon this method here, it 
is because Ranke insists on an infinity of cause-groups and supposes no continuity 
can arise without them— 
..."a truth 
Looks freshewt in tlic fashion of the day." 
(viii) The important physical constants of a frequency distribution are those 
which can be determined with the least probable errors. The probable errors of 
the moment coefficients increase rapidly with the moments. Hence the important 
physical constants are those which depend on the low moment coefficients, i.e. on 
the early terms of the expansion of f{xja). Now these physical constants are 
(a) the mean, (6) the modal difference or distance of mean from mode, (c) the 
skewness, and (d) the kurtosis. We may replace either (6) or (c) by the standard 
deviation. Experience shows that these four physical constants are certainly 
independent. The constants of my skew curves directly give them and we are 
able to determine by their probable errors whether they are significant or not. 
(ix) With regard to the other theories discussed I have shown : 
(a) That the Galton-McAlister curve, ascribed by Ranke to Fechner, is not 
applicable to a great number of cases, for its kurtosis is a function of its skewness 
and its skewness cannot exceed '21. 
(6) That the double Gaussian curves due to Galton and Fechner are illogical, 
because they reach a Gaussian result by rendeiing invalid every one of the Gaussian 
principles. Further, the skewness is always a function of the kurtosis and the 
kurtosis cannot exceed "87, a degree which is exceeded in a great variety of 
data. 
(c) That the Edgeworth curves as developed by Kapteyn fail from the logical 
standpoint, for they appeal to a truncated Gaussian distribution which has never 
been observed in experience. They are not true graduation formulae, and are 
obtained in such manner that it is not possible to determine any one of the chief 
physical constants or evaluate their probable errors. Further in the examples 
given by Kapteyn they all sensibly reduce to the Galton-McAlister curve. But 
this curve has in every one of the cases dealt with by Kapteyn a skewness 
significantly less at a maximum than is required by every one of the statistical 
series involved. 
Finally it seems to me that all discussion of asymmetrical frequency must turn 
in one form or another on the proper form to be given to F {x) in the equation 
1 dy _ — X 
y dx~ ao^F{x)' 
See " Mathematical Contributiuiis to the Theory of Evolution, XIV." Dulau and Co., London. 
27—2 
