444 
PROF. KARL PEARSON ON SKEW VARIATION. 
and is therefore numerically the smaller root since r is negative ; it will he less than 
unity, and therefore m'—l = m will be negative if 
|(vV-4f+/’)<l, 
or 
e— r +1 >0, 
but this is the condition for the point /3,, /3 2 lying inside the loop of the quadratic. 
Thus in this case we reach the J-shaped curve of Type VII., or 
,r q ' (x—a) q - 
In order that the area of this curve and its moments should be finite, it is clearly 
needful that q 2 should be less than unity. 
(7) Frequency Curve. Type XI .—Beyond the exponential point the lower branch 
of the biquadratic is below the line 2/3.— 3/3] — G = 0, and consequently rn is again 
negative and the curve takes the form 
y = y«x~ m , 
where 
m = 2(5/3 2 -6A-9) 
2 / 3 . - 3 / 3 , -6 
The range is, however, only limited in one direction, it is from x = b to x = o°, say. 
This lower branch of the biquadratic loop tends to become vertical and asymptotic 
to the line /3i = 50. Hence m takes all values from co down to 5. 
Clearly, for moments about x = ?>, 
and these will be real and finite if p <m— 1, or only the fourth moment would fail 
at the limit /3 2 = o°, which indeed cannot in practice be reached. At the same time 
if we want, the probable error of the fourth moment to be finite, it is needful that // 8 
should be finite or we must have m > 9. Thus m — 9 must be where the curve passes 
into the heterotypic region and becomes of doubtful application. 
We easily find from the above result for y! 
x = 6(m—l)/(m —2), y 2 — F — b 2 (m— — 2) 2 (m — 3)}, 
y 3 — 2 b 3 m (m — 1)/{(m — 2) 3 (m — 3) (m — 4) }, 
U-i = 3& 4 (m— 1) (3m 2 — 5m + 4)/{(m— 2) 4 (m — 3) (m — 4) (m— 5)}, 
n _ 4 m 2 (m — 3 ) 
(m— 1) (m — 4)' 
3 (m— 3) (3nr — 5m + 4) 
(rn — 1) (m — 4) (m — 5) 
leading to 
