( 239 ) 
(a,—A b,) (a,,—4 b,,)- 
These parabolae intersect between the points 13 and 23 and on 
the right side of the point of intersection the first mentioned parabola 
lies higher than the second mentioned; the expression under conside- 
ration is there therefore positive. 
The graphical representation of the expression 
(a,,—4 Da) ian) Ni b) (a,,;—A Oa) 
has the following shape: 
pO Sain 
from which we see in the same way that these parabolae intersect 
between the points 13 and 23 and that this expression is positive for 
higher values of A. 
The third expression : 
(a,,—A b,,) (a,;—A4 bs) — (a,—A bo) (a,, —2 0,,) 
: ong . asta as a; 
is positive for 2 equal to —,—,— and and, when equated to 
ae ee ae b, 
zero, it will in general not yield a real root; at least not between 
12 and 3. 
The graphical representation of this third expression has the 
following form: 
where the parabola passing through the points 13 and 23 lies every- 
where higher than the other one. The first mentioned parabola would, 
if there should exist roots, descend below the second one between the 
points 13 and 23, and so the two roots would lie between those points. 
But in this case also the third expression is positive above a certain 
5 U: . . 
value of 4 below ae Or both parabolae might also intersect on the 
) 
23 
En. 
