PROF. KARL PEARSON ON SKEW VARIATION. 
435 
as block-frequency, and represent them by the letter B; they correspond to points on 
the B-line. (See Diagram, Plate 1.) 
The most remarkable limiting case of this kind has been already referred to. It 
will be shown in practical examples in a memoir on “ small samples,” now nearly ready 
for press, that the correlation between two variates may be determined by sampling 
these populations in pairs, and merely observing, which can be usually done without 
measurement, whether the pair is positively or negatively correlated. The ratio of 
the two frequency “lumps ” easily provides the correlation.* 
(3) Let us now consider the nature of the frequency on the loop of the biquadratic. 
Taking the form of the curve to be 
y = y 0 (l + x/a 1 ) m (l —cc/a,) m2 
we know that m x and m 2 are the roots of the quadratic 
m 2 —m (r—2) + e — r + 1 = 0, 
where 
r = 6 (/3 2 -/3 1 -l)/(38 I -2&+6), 
and 
o 
_ r 
4+iA (r + 2) 2 /(r+l) 
Now e—r+1 = 0 provides the biquadratic 
ft (8/3,-9/^-12) (/3,+ 3) 2 -(l0/3,-12/3!-18) 2 (4/3,- 3ft) = 0 ; 
actually 
+ 1 = (4/3 2 —3/3i),(lO/L—12,Si —18) 2 —& (/3,+ 3) 2 (8/3, —9/3 x —12) 
(3fr - 2/3, + 6) {A (/3,+ 3 ) 2 + 4/3 x (4/3, - 3A) (3ft •- 2/3, + 6)} 
Now /?i, 4/3,—3/3 x and /3,+ 3 are by their nature essentially positive. Hence, 
provided 3/3 x —2/3,+ 6 is positive, i.e., as long as we deal with points above the line 
2/3,— 3/3 x — 6 = 0, i.e., the Type III. curve line, e— r+1 will be positive, if (/3 X , / 3 2 ) lie 
outside the loop of the biquadratic. But within the loop it is negative, or one value 
of m must be negative, or we reach an infinite ordinate at x = — cq or a 2 , i.e., a 
J-shaped curve. The other ordinate at x = a 2 or —a x is zero, because the other m must 
be a finite positive quantity. 
If e—r+1 = 0, i.e., along the biquadratic loop, one value of m is zero, and the 
other is positive if r be greater than 2, and negative if it be less than 2. But 
r O _ 2(5/3,—6/3, —9) . 
3/3i — 2/3,+ 6 
Accordingly above the line 5/3 ,—Gfr — 9•= 0, and above the line 2/3, —3/3! —6 = 0, r—2 
will be negative, but these lines do not meet in the positive quadrant. Hence all 
* See “ Student,” ‘ Biometrika,’ vol. VI., p. 304, and Fisher, ‘ Biometrika,’ vol. X., p. 508. 
