452 
PROF. KARL PEARSON ON SKEW VARIATION. 
Since E is the junction of several types, we turn to consider Type III. which is the 
curve found along the critical line 
2/3 2 —3/3 x —6 = 0. 
It passes throrfgh the Gaussian point G, and its equation is 
y = y 0 (l +x/a) p e~ px,a . 
It is fully discussed in my first memoir; see ‘Phil. Trans.,’ A, vol. 186, p. 373, 
et seq. 
From G to the exponential point E, p ranges from co to zero, which latter value 
provides the exponential curve. After the exponential point p becomes negative and 
we reach Type IIIj, a J-curve with range limited in one direction only. This curve 
separates the doubly limited curves of Type Ij from curves of Type VIj, which lie 
below the line 2/3 2 —3/3i —6 = 0, and above the lower branch of the biquadratic loop. 
On this lower branch of the loop we have Type XI., or the form 
y = y,x~ m 
the range being from an arbitrary value b to ay and m ranging from co to 5. This 
type is fully discussed in the present memoir; see p. 444. It continues right awajr 
along this branch of the biquadratic, but at (3 2 — 22725 and /3 X = 972, the eighth 
moment of the theoretical curve would become infinite, and accordingly the probable 
error of the fourth moment coefficient would become theoretically infinite. Thus since 
the fitting of the curve depends on the fourth moment its constants would cease to 
be reliable measures of the distribution. We enter at this point the “heterotypic 
area,” for this type of curve.* We have now two further areas to clear off, namely 
those between the Type III. line and the lower branch of the biquadratic loop. 
Above the former and below the latter we have the range of double limited frequency 
curves, i.e., Type I L , or 
y = 7/o (1 + x/a 1 ) m (l-x/a 2 ) m \ 
This curve was fully discussed in my first memoir (‘ Phil. Trans.,’ A, vol. 186, 
p. 376, et seq.) m 1 and m 2 are both positive, and experience has shown that probably 
the bulk of all frequency distributions cluster into this area. 
Above the biquadratic loop and below the line 2/3 2 — 3/3 x — 6 = 0, we have curves of 
Type VIj, or 
v = _ V £_ 
x q ' (x—a) q - 
witli range from x = a to x = oo. 
* Of course, by using the actual eighth moment of the data, instead of the eighth moment of the 
theoretical curve, the standard deviation of the fourth moment would be finite, but this procedure 
would really indicate that, as far as the high moments are concerned, curve and data were discordant, and 
that we should not really be finding the probable error of a constant of our theoretical frequency curve. 
