Projective geometry, by Oswald Veblen and John Wesley Young.

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CORRELATIONS

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that F (abc)= ABC. The points [Q] of c are transformed into the
lines [1] on C, and these meet c in a pencil [Q'] 1)rojective with [Q]
(fig. 99). Since A corresponds to B and B to A in the projectivity
[Q] - [Q'], this projectivity is an involution I. Tlle point Q0 in which
P
a\
(C p) b^ (bp)
P
C=Cq]
FIG. 99
C'P meets c is transformed by r into a line on the point cp; and
since QO and cp are paired in I, it follows that cp is transformed
into the line CQ = CP. In like manner, bp is transformed into BP.
Hence p= (cp, bp) is transformed into P = (CP, BP).
THEOREMi 4. Any projective collineation, TI, in a plane, a, is the
product of two polarities.
Proof. Let Ac be a lineal element of a, and let
H (Aa) = A'a', H (A'a') = -A".
Unless 1 is perspective, Aa may be so chosen that A, A', A" are not
collinear, aa'a" are not concurrent, and no line of one of the three
lineal elements passes through the point of another. In this case there
exists a polarity P such that P(AA'A") - a na'a, namely the polarity
defined by the conic with regard to which AA" (aa") is a self-polar triangle and to which at is tangent at A'. If II is perspective, the existence
of P follows directly on choosing Aa, so that neither A nor a is fixed.
We then have
PI (AA'caa) = a'aAA'A,
and hence the triangle AA!(caa) is self-reciprocal. Hence (Theorem 3)
PII = P is a polarity, and therefore II = PP1.