We mention, but without proving it, that any two projective rows may be placed so as to form an involution.
An involution may be said to consist of a row of pairs of points, to every point A corresponding a point A′, and to A′ again the point A. These points are said to be conjugate, or, better, one point is termed the “mate” of the other.
From the definition, according to which an involution may be considered as made up of two projective rows, follow at once the following important properties:
1. The cross-ratio of four points equals that of the four conjugate points.
2. If we call a point which coincides with its mate a “focus” or “double point” of the involution, we may say: An involution has either two foci, or one, or none, and is called respectively a hyperbolic, parabolic or elliptic involution (§ 34).
3. In an hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci.
For if A, A′ be two conjugate points, F1, F2 the two foci, then to the points F1, F2, A, A′ in the one row correspond the points F1, F2, A′, A in the other, each focus corresponding to itself. Hence (F1F2, AA′) = (F1F2, A′A)—that is, we may interchange the two points AA′ without altering the value of the cross-ratio, which is the characteristic property of harmonic conjugates (§ 18).
4. The point conjugate to the point at infinity is called the “centre” of the involution. Every involution has a centre, unless the point at infinity be a focus, in which case we may say that the centre is at infinity.
In an hyperbolic involution the centre is the middle point between the foci.
5. The product of the distances of two conjugate points A, A′ from the centre O is constant: OA · OA′ = c.