PROF. KARL PEARSON ON SKEW VARIATION. 
437 
Beyond the exponential point, our biquadratic branch lias entered the area of 
Type VI. curves,* and m, will now again be negative. 
Now the equation to 'Type VI. is 
y 
(x—a) g2 
Vo : -- 
5 
and the range from x— a to oo. The special case of this along the branch of the 
biquadratic occurs when q 2 — 0, leading tof 
l - (h = ej = /’ 1, 
or 
y = yjx' h 
where 
= 2(5ft-6/9,- 9) 
which is positive, since q x is now beneath both the lines 
5/30—6/3, —9 = 0 and 2/3 2 —3/3] —6 = 0. |. 
This curve, which will be more fully considered below, has a range from a certain 
value a to o°. [t thus starts with a finite ordinate and asymptotes to zero. It is 
a transition curve extending from the exponential point along the lower limb of the 
biquadratic loop. I call this curve Type NI. The biquadratic never cuts the cubic 
along which Type V. lies and no further change occurs in Type XI. 
I now pass to the consideration of the B-line or 5/3,— 6/3] — 9 = 0. 
The general differential equation § to the type of frequency curve under con¬ 
sideration is 
Idy = - {\ /3i (/3, + 3) + (10/3, -12(3,-18) x/a) 
V dx rr { 4/3, - 3/3]) + Vfa (& -+ 3) xfa + (2/3, - 3 fa - 6) x 2 l<r 2 } ’ 
the origin being at the mean. 
Hence if 5/3, —6/3] —9 = 0, the term in x/a disappears from the numerator, and we 
can further get rid of fa by substituting 1 (6/3]+ 9) for it. Making this substitution, 
we reach 
1 c ly = _ -2y / A_ 
y d x or (3 +fa) — (T {Vft 1 —x/rr) 2 
* ‘ Phil. Trans.,’ A, vol. 197, p. 449. ■ 
t Loc. cit., Equations, bottom of p. 449. 
» f As we pass outwards from the exponential point along the biquadratic qi ranges from oo to 5, which 
it reaches at the asymptote to the biquadratic fa = 50, or when fa — oo, fa = 50. 
§ “ Mathematical Contributions to the Theory of Evolution, XIV. On the General Theory of Skew 
Correlation and Non-Linear Regression,” p. 6, ‘ Drapers’ Company Research Memoirs,’ Cambridge 
University Press. 
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