PROF. KARL PEARSON ON SKEW VARIATION. 
445 
Thus for m — 9 we find /3 X = 972, /3 2 = 22725, which satisfy the equation 
8/3 2 —l5/3 1 — 36 = 0 of the heterotypic line. 
m may he found from 
= 2 (5j9,—6j3, —9) 
2&-3A-6 ’ 
or from /3 1 alone by the cubic 
m 3 (4- i 8 1 )+m 2 (9/3 1 -12)-24/3 1 m+16/3 1 = 0, 
then 
b= ±.( m _2),\/|5|, 
and 
Vo = N6” l_1 (ra—l), 
while the mean x = b (m— l)/( m— 2) enables us to place the curve on the observations. 
There is no discontinuity in the form of the curve down to m = 5, but only 
discontinuity after m = 9 in the probable errors of its moment-coefficients. 
The curve starts with a finite ordinate and meets that ordinate at a finite angle; it 
asymptotes to the cc-axis at x — o° , and has no point of inflexion except at infinity. 
(8) Frequency Curve, Type XII .— 
_ / V {V 3 +& + \//b) -\- x \\/ 
y ~ v C(v^ +a--/a)-J 
This J-curve arises along the R-line, or 5/8 2 — 6/b~ 9 = 0. Its range is from 
x — cr(.\/s + /3 l — \//3 1 ) to x = — a (\/3 + /3] + \//3j), and then its mean is the origin. 
When & is zero it degenerates into a rectangle (be., at the rectangle point). 
In order to illustrate the nature of the curve more fully let us start from the 
general equation which arises when the denominator of the differential equation has 
real roots, # be., 
y = ;7 l (l+bAh)'" 1 (1 -x/a 2 ) m , 
where 
_ N F (m] + m 2 + 2) 
(mj + m 2 ) nil+nH T (m x + 1) T (rn 2 + 1) 
and 
m \ _ m 2 _ wii + m 2 
the origin being the mode and b the range. 
Transferring to the mean as origin! this becomes 
= V* f b(nh + l) [ V n 7 b (m 3 +1) _ V** 
aC'aX 1 \m l + m 2 + 2 ) \m x + m 2 + 2 / 
* ‘ Phil. Trans.,’ A, vol. 186, p. 369. 
f Loc. cif., p. 370. 
3 P 
VOL. CCXVI.-A. 
