442 
PROF. KARL PEARSON ON SKEW VARIATION. 
leading to 
Thus 
,, _ 4 m ( 777 + 0 ) p. _ 3 (m + S) (3m + 5777 4-4) 
(m + 1) (m + 4) 2 ’ 1 2 (m+ l) (w + 4) (m+ 5) 
w 3 (/3j —4) + 3m 2 (3/Sj —4)4-24771/3!+16/3, = 0 
would give m, and a would be found from 
«■ 
+ rr ( 777 
777 + 3 
777 + I 
the sign being found from the observed value of /x 3 . Lastly 
__ N 777+ 1 / 777 + .1 
° rr '777 + 2 V r/7 + 3 ' 
Practically it is better to determine m from 
2 (5/3 2 —6/3! — 9) 
~ 3/3i —2/3, + 6 
which value of 777 substituted in the expression for /3i gives the biquadratic. 
Clearly since the lower branch of the biquadratic lies below the line 5/3, —6/3 x —9 = 0, 
rn is positive until the line 2/3 2 —3/3i —6 = 0 is reached, and in this section of the 
branch, i.e., from m — 0 to m = 00 , or from /3, — 1’8, /3i = 0 up to ,8 2 = 9, ,/3 x = 4 (the 
exponential point, E) occurs an interesting isolated point—the line-point L. When 
/3 2 = 2‘4, /3j = 0’32, then m = 1, and Type IX. degenerates into a sloping straight 
line, y = y (l (l +x/a), or the frequency line is 
V = 
2 N 
OfT 
X 
Up to the line-point, Type IX. curve rises at x = —a perpendicular to the axis,- 
of x, at the line-point it makes a finite angle less than 90 degrees, and after the line- 
point we start with contact at x — —a. 
It is interesting to note the sloping line arising as a case of these generalised 
frequency curves, and we observe that its locus is separated from the rectangle locus 
by a considerable interval along the biquadratic in which the curve of Type IX. is 
very trapezoidal in form. 
(6) Frequency Curve of Type X. The Exponential Curve. —Beyond the line-point, 
L at /3, = 2‘4, /3i = 0’32, we reach as rn steadily mounts a series of frequency curves 
which culminate in the exponential curve at E or /3, = 9. /3, — 4. 
Clearly 
