Projective geometry, by Oswald Veblen and John Wesley Young.

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PROJECTIVE CONICS

307

two conics as generated by the flat pencils at A and B and at A"
and i. The correspondence established between the two flat pencils
at B by letting correspond lines joining B to homologous points of
the two conics is perspective because the line I corresponds to itself.
Hence there is a pencil of
planes whose axis, b, passes
through B and whose planes
contain homologous pairs
of lines of the flat pencils
at B. The correspondence \ A
established in like manner
between the flat pencil at A
and the flat pencil at A" may
be regarded as the product
of the projectivity between
the two planes, which carries the pencil at A to the
pencil at A', followed by
FIG. 110
the projectivity between the
pencils at A' and A" generated by the second conic. Both of these
projectivities determine parabolic projectivities on I with B as invariant point. Hence their product determines on 1 either a parabolic
projectivity with B as invariant point or the identity. This product
transforms the tangent at A into the line A"A'. As these lines meet
I in the same point, the projectivity determined on I is the identity.
Hence corresponding lines of the projective pencils at A and A" meet
on 1, and hence they determine a pencil of planes whose axis is a = AA".
The axial pencils on a and b are projective and hence generate a
regulus the lines of which, by construction, pass through homologous
points of the two conics. We are therefore able to supplement
Theorem 11 by the following
COROLLARY 1. The lines joining corresponding points of two projective conics in different planes form a regulus, if the two conics
have a common tangent and point of contact and the projectivity
determined between the two planes by the projectivity of the conies
transforms their common tangent into itself and has the common
point of the two conies as its only fixed point.