Theory of HomograpJiic Division. 381
EXEBCISES.
1. Given two homologous point pairs of an involution, show how to
find the central point and the double points.
2. A system of conies passing through four fixed points cuts any trans-
versal in involution.
For, let 1S, S' be two fixed conies passing through the points, then 8 •+ /£<S"
will denote a variable conic through them.; and if 8, S' be given by their
general equations, then, if the transversal be the axis of x, the point pairs
in which they are intersected by the transversal are given by the equations
ax~ + 2gx + c, ax1 + 2/# + c', and as? + 2gv 4- c + k (a'x? + 2g'x-f c').
Hence (§ 300, Cor. 3), they are in involution.
3. The three pairs of opposite sides of a quadrangle are cut in involution
by any transversal.
4. Coaxal circles are cut in involution by a transversal: the points of
contact of the circles of the system which touch the transversal being the
double points, and the central point that in which the radical axis meets it.
5. A system of conies having a common self-conjugate triangle cut in
involution any line passing through a summit of the triangle.
6. For every two projective rows on different lines there exist two points,
for each of which the rows are isogonal, that is, the angles subtended by
one row are respectively equal to those subtended by the other.
(TOWNSEND.)
7. If aa'j bb'j cd be three point pairs in involution—
ab'. be', ca! + a'b. b'e.ca = 0. (874)
ab'. lc. c'a'+ a'b . b'c'. ca - 0. (875)
ab . b'e'. ea'+ a'b'. be . c'a - 0. (876)
ab. b'c. c'a'+ a'b'. be', ca - 0. (877)
8. A common tangent to any two of three circumconics of a quadrilateral
is cut harmonically by the third.
9. Show that the folloAving are special cases of Ex. 8:—
1°. If through the intersection of common chords of two conies a tangent
be drawn to one of them, it is cut harmonically by the other.
2°. If through any point on the chord of contact of two tangents to a
conic a third tangent be drawn intersecting both, it is divided harmonically
by the tangents and the point and chord of contact.