432 
PROF. KARL PEARSON ON SKEW VARIATION. 
[3i (3 2 plane for which (3± = 0, /3 2 = 1'8, I term the rectangle-point and denote by R. 
(See folding diagram, Plate 1, at end of paper.) 
The rectangle-point is the point of contact with the axis of [3 2 of the biquadratic 
A(8/3 2 -9A-12) (/ 3 2 + 3 ) 2 = (4&-3A) (10/3 2 —12/3, — 18) 2 , 
which bounds the area of J-curves. The novel curves are in part limiting curves 
which occur when the point /3 1; (3 2 lies on this biquadratic, i.e., transition curves from 
J-curves to U-curves and from J-curves to limited range curves, and in part a 
limiting curve which exists along the line 5/3 2 —6/3 1 — 9 = 0 which passes through the 
rectangular point and never again meets the biquadratic in the loop in the positive 
quadrant. It would be convenient to speak of this line as the axis of the biquadratic 
loop, but unfortunately the loop is not symmetrical about it, and to avoid misunder¬ 
standing 1 term it the R-line. 
Up to the present the minimum limit to the area of U-curves had not been given. 
Since /3 2 is > /3 1? half the positive quadrant was impossible, but a recent observation 
shows that frequency curves above the line (3 2 -(3 l — 1 = 0 are impossible. This limit 
was suggested in the following manner. When samples of three are taken from an 
indefinite population, the frequency curves for the correlation of any two variates of 
the three individuals sampled are U-shaped frequency curves, but when samples of 
two are taken the correlation must be either positive or negative, and accordingly 
the frequency is collected into two lumps or blocks as a limiting case of a U-shaped 
distribution. But for two such lumps /3 2 —— 1 = 0. In other words, along the line 
(3 2 —(3 l — 1 = 0, the U-shaped frequency either brings all frequency to an end, or 
passes through a transitional case. The former is the true state of affairs, for j3 2 
cannot be less than /3 X H-1. To demonstrate this, # let s p — S (x n p ), and let there be 
n quantities x u . Clearly, s 0 = n, and s x = 0. Now by Burnside and Panton, 
‘Theory of Equations,’ vol. II., p. 35, 
7 t 
N (•£{ u) (u u) / — 
r> s>t 
r, s, t = 1 
= So 
$19 
s 2 , 
n , 
0 , 
o , 
s 2 , 
^3 
^ 2 ? 
^35 
/ sjn 
s 2 / n 2 
W / n 2 
s 2 / n 3 
«' (s 2 s 4 -Sj 2 ) - s.j 
-l) = % 3 (- 2 - a 5-l) 
^2 fJ- 2 / 
= s 2 3 (/8 2 -A-1), 
■ * I owe this neat proof to the kindness of Mr. G. N. Watson. 
