PROF. KARL rEARSON ON SKEW VARIATION. 
4 43 
Further 
y 
\j <m + ‘J) il 
= ~ I i + ^ 
CT 
cr (m -|- 2) 
N 
— e 
the range being from x = 0 to — co, if we take the positive sign—and From x 
to + co, if we take the negative. It is thus sufficient to consider 
= 0 
!J 
_ N ±r/<r 
1 
(T 
with range from x = 0 to x = + 00. The first two moments of the area about x = 0 
are v \ = <7 and v 2 = V. Thus .r = a- and /x 2 = cr 2 , as it should. Lastly, = 2 o- :! and 
/x 4 = 9 a- 1 . . 
The fitting of the exponential curve presents no difficulty. 
The exponential point E is a transition point of great interest as being even more 
than the Gaussian point G—the meeting point of many types. At E, Type IX_, 
changes to Type XL, but at E the familiar Type III. passes from a zero ordinate 
at the limited end of the range to a J-curve with infinite ordinate. Further, E is a 
point at which the areas of Type I. (Type I L ) as a limited range with zero ordinates at 
its terminals, and as a limited range with one infinite ordinate at a terminal (Type Ij) 
meet. Finally, Type VI. area, which lies between Type III. line and Type V. cubic, 
is divided into two sections by Type XI., which lies along the lower branch of the 
biquadratic loop below E. Below the biquadratic, Type VI. takes the form 
V = !/o (x — a) q2 /x?\ 
with a range from x = a to 00 , g, and q 2 being both positive. In the area, however, 
below Type IIIj and above Type XL, type VI. takes the form VIj, or the J-shaped 
curve 
11 = _ h. _ 
y x q '{x-a) g *’ 
with a range from x = a to x — 00. In this case r = 6 (/3 2 — /fi— 1)/{S(3 1 — 2/L+ 6) will 
be negative, since we are below the line 2/fi —3/^ — 6 = 0. Further, t is negative 
since we are above the cubic or Type V. branch 
Thus our quadratic 
4(4/L —3A)(2/L-3fr-6) - + 3) 2 . 
m!~ — rm! + e = 0, 
corresponds of necessity to real roots, of which one will be negative and the other 
positive. The positive root will be 
i(v r —4e + r), 
