PROF. KARL PEARSON ON SKEW VARIATION. 
431 
We have 
Vx +\ Vx _ 2 {ctj3y — 9 e£+X ( a (3 + fty + ya — Be — + (a + /3 + y — 6 — e — (()} 
~k { i / x+1 + 2/j:) { a /3y + +x(af3 + (3y + ya + 6 e + e£ + £ 6 ) + X 2 ( a . + fi + y + 6 + e +£) + 2 X 2 ) ’ 
and accordingly we get the curve approximating to the hypergeometrical of the higher 
order by putting 
1 dy _ quadratic function of x 
y dx cubic function of x 
a 0 + a x x + a.,x 2 r -\ 
=---o - -.(n) 
c 0 + CiX + CoX + c 3 ar 
where the six independent constants can be expressed in terms of the original six, 
a, (3, y, 6, e, It will be seen that a hypergeometrical of the second order will, in 
general, have two modes, the exception being when 
a + /3 + y = 0 + e + £;.(iii) 
in which case (ii) coincides with (i) the general equation to the fourth approximation 
of curves when /3i and /3 2 fall into the heterotypic area. It will thus be noted that such 
curves approximate to hvpergeometric series of the second order when the special 
condition (iii) holds ; always assuming the unimodal character of homogeneous material. 
It seems probable that for the most part bimodal frequencies would be those that lead 
to values of /3i and (3 2 lying in the heterotypic region, and such are excluded from 
practical statistics. 
In the original paper* four types of curves were dealt with beside the Gaussian 
curve corresponding to an isolated point. A supplementary memoir issued in 190It 
dealt with two further types, which had been overlooked until actual experience 
demonstrated their existence. I have now to confess the omission of five further 
types, not to speak of a horizontal straight line, as sub-groups of the J-section of 
curves, which are themselves in practice so rare, that the region of the /3 lf f3 2 plane in 
which they occur had not been very fully investigated. My attention was drawn 
to these curves while considering the frequency curves for the correlation of small 
samples. If we take a sample of four from uncorrelated material, the sample is equally 
likely to have every correlation from —1 to +Lj In this case, (3 1 — 0, j3 2 — 1 '8, and 
the frequency curve is a horizontal straight line. What would my series of curves 
give in this case ? I discovered that they also gave a rectangle of frequency or a 
horizontal straight line, and this discovery led me to a closer investigation of the 
sub-groups of curves in the neighbourhood of the J-curve area. The point in the 
* ‘Phil. Trans.,’ A, vol. 186 (1895), pp. 343-114. 
f ‘Phil. Trans.,’ A, vol. 196 (1901), pp. 443-459. 
| ‘ Biometrika,’ vol. VI., p. 306, and vol. X., p. 312. 
3 N 2 
