PROF. KARL PEARSON ON SKEW VARIATION. 
451 
On the upper branch of the biquadratic loop we reach curves of Type VIII., i.e ., 
V = Vo ( l+x/a )~ m , 
discussed on p. 444 of the present memoir. Here m is less than unity. 
We now pass into the loop of the biquadratic between the upper branch and the 
14-line. Here we have J-curves, Type I T , of the form 
y = y 0 (l +x/a 1 )~ m ' (l + x/a 2 )~ m2 
where m 2 is less than unity, and m l is less than m 2 . 
Coming to the ft-line, m x becomes equal to m 2 and we have Type XII., or 
a (\/3 + /3; + ) + ;r \ 
discussed on p. 446 of the present memoir. Below the B-line, we return to Type Ij, 
but m 1 is now greater than m 2 * 
We now reach the lower branch of the biquadratic loop. This is divided into three 
portions by three critical points. The first portion is from the rectangle-point (14) to 
the line-point L. In this portion we start from 14 with the curve of Type IX. or, 
V = V o (1 + xfa) m 
for m — 0, or the rectangle, and proceed from that value to m — 1, which gives us the 
line (or triangle); the range is —a to 0. Since m is always < 1, the curve rises 
perpendicularly at x =— a, and approximates to a trapezoidal form. The method of 
fitting is discussed in this memoir, p. 441. The fitting of the line curve 
y=y 0 (l+x/ct) 
is dealt with on p. 442. 
Beyond the line-point L we have Type IX 2 which differs in no way from Type IX 1} 
except that m is now greater than unity, and there is contact of a rapidly increasing 
order at x — — a. 
When m = o° we find Type X. the exponential curve, at the exponential point E. 
The fitting of this curve 
' <r 
has been discussed on p. 443. 
For example, at the point /3 2 = 4, /fi = 2, between the R4ine and upper branch, 
y = y o (i + 
X Vasias 
Ct-\ 
/ 
1 + - 
ct 9 
0-7123 
but at (3. 2 = 8, [3 X = 4, between the R line and the lower branch, 
V = Vo ( 1 + 
x V 4011 
a x 
/ 
1 + 
a--, 
0-4011 
