PROF. KARL PEARSON ON SKEW VARIATION. 
441 
In concluding our discussion of this curve we may note that, perhaps, the easiest 
way of tracing the biquadratic is to calculate ft and ft from 
o _ 4 (2 y— l)- (y + l) f, _ 3 (y + 1 + ft)3 (y + 1) (lGy~ — 13y 4 3 ) 
Pl 3y — 1 ’ /3 3 — y (3y— 1) (3 — y) 
by giving a succession of values to y. 
For y = 0'5 to 3 we get the points on the lower branch of the loop ; for y = 0'5 to 
0'3 we obtain the points on the upper branch of the loop. It will be seen that this 
amounts to taking the origin at ft= —3, ft = —4, and rotating a line through this 
point round it to intersect the curve. The slope of this line to the ft axis is 3/(3 —y). 
The cubic, it may be here noted, which gives the Type V. curve may be traced from 
ft = 4 (y 2 —1), ft = 3 (y +1 +A) = 3 (y+ _ l) , (4y-3) 
3 y 3—y 
Here y must be given values from 1 to 3. 
The Type III. line, which passes through the Gaussian point, also passes through 
ft = —3 and ft=—4, and the above means of getting at the points on the cubic 
corresponds to finding the points in which a straight line passing through (— 4, — 3) 
and rotating from the position of the Type III. lhie cuts the cubic—-its slope in any 
position being as before 3/(3 —y). 
Actually if B be the angle between the above line from (-4,-3) to the cubic, i.e., 
tan 6 = 3/(3 — y), 
r — 12 (sec B — cosec 0), 
but to use this polar equation has not been found a very ready manner of plotting the 
cubic. # 
(5) Frequency Curve. Tripe IX .— 
Range from x — —a to x = 0 ; y is zero at one end of the range and equal to y 0 at 
the other. 
The analysis proceeds precisely as in the case of the curve of Type VIII., except 
that m is now opposite in sign. We have 
y 0 = N (1 +rn)/a, 
x (= distance of mean from point x = —a) = a (rn+ l)/(m + 2), 
rr 2 — f u — a 2 (m+l)/{(m + 3)(m + 2) 2 }, 
// : > = — 2enn (rn + 1 )/! (ni + 4) (m v 3) (w. + 2 )” ]-, 
/u i = 3ft 1 {w, +1) (3wi 2 + 5m + 4)/{(wr + 5) (vn + 4) (ni + 3) {rn + 2) 4 i , 
* The parts of the cubic and the quartic lying in the other three quadrants have been plotted by 
Miss B. C. B. Cave. Geometrically the interrelations of the two curves, their asymptotic and other 
critical lines are of much interest, but until some interpretation can be put on imaginary values of the 
moment coefficients, these interrelations have no statistical bearing. 
