. The principles of projective geometry applied to the straight line and conic . njugate involution. rays of an involution pencil. Let 8, T, Q, R be the four given Let s, t, q, r be the four given points and AA,BB, CC the pairs lines and aa, bb, cc the pairs of of points in which their connectors lines joining their intersections to are met by any transversal s. Let any point 8. Let sq . tr be the 8Q . TR be the point P. line p. Since the ranges obtained by Since the pencils obtained by projecting RPTA from 8 and from joining 8 to the points of inter- Q on to s are projective, section of s and
. The principles of projective geometry applied to the straight line and conic . njugate involution. rays of an involution pencil. Let 8, T, Q, R be the four given Let s, t, q, r be the four given points and AA,BB, CC the pairs lines and aa, bb, cc the pairs of of points in which their connectors lines joining their intersections to are met by any transversal s. Let any point 8. Let sq . tr be the 8Q . TR be the point P. line p. Since the ranges obtained by Since the pencils obtained by projecting RPTA from 8 and from joining 8 to the points of inter- Q on to s are projective, section of s and q with rpta are projective,{CABA) = (BACA) = {CABA). (caba) = (baca) = (caba). Therefore AA, BB, GC are Therefore aa, bb, cc are pairs pairs of conjugate points of an of conjugate rays of an involution involution. pencil. If the transversal be drawn as If the point 8 be situated at *S, Involution 99 s to pass through TS. QR and the point of intersection of ts. qrTQ. MS as in the figure, and tq. rs, b coincides with 6 and c with c.(C\AAA,) = (C,A,B,A,). .•. (caba) = (caba).. Hence Ai and A., arc harmonicconjugates of 7^i and Ci. Cor.: If A A, BB and C are given,C is fixed, since the points A A,BB, CO form an invohition. Hence, if Jive of the sides of aquadrangle pass through five fixedcollinear points, the sixth side mustpass through a fixced point collinearwith the other five points. (SeeArt. 13(6).) Hence a and (t are harmonicconjugates of c and b. Cor.: If aa, bb and c are given, c isfixed, since the lines aa, bb, ccform an involution. Hence, if five of the vertices ofa quadrilateral lie on five fittedconcurrent straight lines, the sixthvertex must lie on a fixed line con-current with the other five lines. (See Art. 13(6).)Graphic construction of conjugate elements of an involution. If two pairs of conjugate pointsA and A and B and B are given,the theorem of the last articleenables us to construct C theconjugate of any element G. If two pairs of conjugate raysa and a and b
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