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IX. Mathematical Contributions to the Theory of Evolution. — XIX. Second 
Supplement to a Memoir on Skew Variation. 
By Karl Pearson, F.R.S. 
Received February 2,—Read February 24, 1916. 
[Plate 1.] 
(1) In a memoir presented to the Poyal Society in 1894, I dealt with skew variation 
in homogeneous material. The object of that memoir was to obtain a series of curves 
such that one or other of them would agree with any observational or theoretical 
frequency curve of positive ordinates to the following extent:—(i) The areas should 
be equal; (ii) the mean abscissa or centroid vertical should be the same for the two 
curves ; (iii) the standard deviation (or, what amounts to the same thing, the second 
moment coefficient) about this centroid vertical should be the same, and (iv) to (v) 
the third and fourth moment coefficients should also be the same. If fx s be the s th 
moment coefficient about the mean vertical, N the area, x be the mean abscissa, 
a- — v iu 2 the standard deviation, /3 1 = M/wf At — c-JuT then the equality for the two 
curves of N, x, <r, f3 l and (3 2 leads almost invariably in the case of frequency to 
excellency of fit. Indeed, badness of fit generally arises from either heterogeniety, 
or the difficulty in certain cases of accurately determining from the data provided the 
true values of the moment coefficients, e.g., especially in J- and U-shaped frequency 
distributions, or distributions without high contact at the terminals ; here the usual 
method of correcting the raw moments for sub-ranges of record fails. 
Having found a curve which corresponded to the skew binomial in the same manner 
as the normal curve of errors to the symmetrical binomial with finite index, it occurred 
to me that a development of the process applied to the hypergeometrical series would 
achieve the result I was in search of, i.e., a curve whose constants would be determined 
by the observational values of N, x, a-, and /3 2 . 
The hypergeometrical series was one not only arising naturally in chance problems, 
but covering in itself a most extensive range of functions. The direct advantage of 
the hypergeometrical series is that it abrogates the fundamental axioms on which the 
Gaussian frequency is based. The equality in frequency of plus and minus errors of 
the same magnitude is replaced by an arbitrary ratio, the number of contributory 
VOL. COXVI.-A 546. 3 N [Published June 6, 1916. 
