436 
PROF. KARL PEARSON ON SKEW VARIATION. 
along the upper boundary of the loop one m is zero and the other negative. 
Accordingly, from the R-point round the upper boundary of the loop, we have the 
curve 
y = y 0 {i+x/ai)~ m \ 
I call this curve Type VIII. 
Since —m\/ai = m 2 la 2 , and m 2 is zero while m i and cq are finite, it follows that 
a 2 = 0 , and accordingly the range of frequency is from x = 0 to x — — cq. The curve 
is therefore a J-shaped curve with infinite ordinate at one end of the range and a 
finite ordinate at the other. 
Now consider the lower side of the loop. Here 5/3 2 —6/3i — 9 will be positive, for 
this side is below the R-line and 3/3i —2/3 2 +6 will also be positive until the point 
in which the line 2/3 2 — 3/3i — 6 = 0 meets the lower side of the loop, i.e., the point 
(3 x =4, (3 2 — 9. Hence from the R-point up to A = 4 , j 3 2 = 9, a point practically 
outside the range of the customary statistical frequencies, r —2 will be positive, or 
nil will be positive. Further m x and cq being finite and m 2 zero, it follows that a 2 is 
zero, or the curve is 
y = y 0 {3+x/ai) m \ 
In this case the curve has a zero ordinate at one end and a finite ordinate at the 
other. I term this curve Type IX. 
At the point where the line 2(3 2 — 3(3i — 6 = 0 meets the biquadratic, Type IX. 
agrees with my earlier Type III. 
The equation to that type is # 
where 
y = y 0 (1 +x/a) ya e yx , 
4 2 
yd — —- 1 and y — — 7 = • 
A oVA 
Hence for A = 4, ya — 0 , and y = l/<r. Thus a is zero and the curve becomes 
y = y^~ x] \ 
the range being from 0 to co, 
But in Type IX., since r has become infinite, m : is infinite and the limit to 
y = 2 /o(i+*M)" h 
is accordingly the exponential curve 
y = y^~ Kx > 
as we shall see shortly A must equal l/<r, where a is the standard deviation. 
I propose to call this exponential curve Type X., and the point A = 4, A = 9, E or 
the exponential point. 
* ‘Phil. Trans.,’ A, vol. 186, p. 373. 
