STUDIES IN STATISTICAL REPRESENTATION. 49} 
; ‘a 4 (us — Ys) = 
Yi 
This condition will be fulfilled if y; < y,y;, for then the 
left hand side will be essentially positive. | 
or | yah a =e YY, — 2 yi) 
1 
If, however, ¥; > "y; then we must have 
4 (ys — ry)” 
2 
Yi 
ae BI Peg in2 ‘ 
YY = YY; — 242) > 
1 
3 
‘ Biel 2 
that is, yw — a WwYs — 2y2) > 2 (ya = Ws)” numerically, 
: | 
Y 
or in other words y; cannot be between the values 
ya(3 yy — 298) + 2(v8 — Ys)? wereeeeeeees (45) 
Yi 
The condition for real roots shews, therefore, that if 
yz > yyy; there isa certain portion of the straight line whose 
distance from the axis of y (to which it is parallel) is 4° 
which cannot be cut by a curve of the form y = ba + ca! 
when p and q are real. In other words, there is what may 
be termed an “‘impossible region’’ about any point in which 
no curve of this form cutting three other points can lie. 
Reverting to the equation (47a) it has been seen that the 
roots must be not only real but positive. Consequently 
YY; — yz must have the same sign as yy, — y; and the 
opposite sign from yy; — yy. This still further limits the 
possible values of 4. . 
It is of interest to note what happens when y;=y,y;- In 
this case (47a) degenerates from a quadratic to a linear 
equation | 
E(YxYs — Yiys) + (YoY, — Y2) = O 
Ys — YYs _ Ys — VIYs Ys 
Yos — Yrs Y3V YiYs — Yrs 
* iu YXY3V Yo TF OVA) ies V Ys 
We YW(YsV/ Ys a ay Ys) Vn 
aie, is Ys 
Nn n ¥ 
Consequently 4” = 
