Projective geometry, by Oswald Veblen and John Wesley Young.

288

TWO-DIMENSIONAL PROJECTIVITIES

[CHAP. X

The coordinates of any point which satisfy (1) and (2) also satisfy (3).
Hence all conics (3) pass through the points common to A2 and B2.
For the value X = - a, (3) gives the pair of lines

(4)

(a1- a3) x1- (a- a) x2 = 0,
I -CL)1 2""

which intersect in (0, 0, 1). The points of intersection of these lines
with (1) are common to all the conics (3).
The lines (4) are distinct, unless al = a or a2= a. But if a1 = a3,
any point (xl/, 0, x/) on the line x2= 0 has the polar x'x1 + x3x3= 0
both with regard to (1) and with regard to (2). The self-polar figure
is therefore of Type IV. In order that this figure be of Type I, the
three numbers al, a2, a3 must all be distinct. If this condition is
satisfied, the lines (4) meet the conics (3) in four distinct points.
I o)
I )
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IG.-1 1 00
degree. We have thus 100)
no other common self-polar pair of point and line), they intersect in
four distinct points (proper or improper). Any two conies of the
pencil determined by these points have the same self-polar triangle.
Dually, two such conies have four common tangents, and any two
Th atalcosrutonofte ont i oma rb",,/th ecn.Dually, two succh conzics have foulr common tangents, anzd anzy two