Theory of .Homographic Division. 379

DEP. — Two superposed protective rows in which homologous
points are permutable are said to be in involution.

298. CENTRAL POINT OP INVOLUTION. — Supposing #><0,

the equation (870) may he written xx' - m(x + xf) + c' = 0, or I
(x - m}(x' - m} = n. (871)
where n = m2 - cf. Then, taking the point whose ahscissa is m
as origin, denoting it hy 0, 0 is called the central point of the
involution, and equation (871) gives
OX.OX' = n, (872)
n heing a constant. We see that the central point is that
which corresponds to infinity (Jor J\ in the general case.
299. DOUBLE POINTS OP INVOLUTION. — When two homologous
points coincide in one, such a point is called a double point.
Now, if X, X' coincide in (872), we have OX = ± */n ; if n > 0, -
there are two double points, which are symetriques with
respect to the central point. In this case homologous point pairs
are situated at the same side of the central point, and the involu-
tion is said to be hyperbolic. If n < 0 the double points are
imaginary, and the involution is called Elliptic.
300. In an hyperbolic involution, any tioo homologous points
divide harmonically the distance between the double points.
Dem. — Let F, F' be the double points, then we have (872)
OX. OX! = n and OF* = OF'* = n ; .-. OX. OX' = OF-, but
0 being the middle point of FF*, this equality indicates that
X, X' are harmonic conjugates to FF1. Eeciprocally, all the
point pairs which divide harmonically a given segment FF'
belong to an involution.
Cor. 1. — If three point pairs
ax~ + 2hx + & = 0, a'x~ + 2h'x + b' = 0, a"x* + 2h"x + 1" = 0
form an involution, they have a common pair of harmonic con-
jugates. Hence the condition of involution is the determinant