§ 82. The polars, with regard to a conic, of points in a row p form a pencil P projective to the row (§ 66). This pencil cuts the base of the row p in a projective row.
If A is a point in the given row, A′ the point where the polar of A cuts p, then A and A′ will be corresponding points. If we take A′ a point in the first row, then the polar of A′ will pass through A, so that A corresponds to A′—in other words, the rows are in involution. The conjugate points in this involution are conjugate points with regard to the conic. Conjugate points coincide only if the polar of a point A passes through A—that is, if A lies on the conic. Hence—
A conic determines on every line in its plane an involution, in which those points are conjugate which are also conjugate with regard to the conic.
If the line cuts the conic the involution is hyperbolic, the points of intersection being the foci.
If the line touches the conic the involution is parabolic, the two foci coinciding at the point of contact.
If the line does not cut the conic the involution is elliptic, having no foci.
If, on the other hand, we take a point P in the plane of a conic, we get to each line a through P one conjugate line which joins P to the pole of a. These pairs of conjugate lines through P form an involution in the pencil at P. The focal rays of this involution are the tangents drawn from P to the conic. This gives the theorem reciprocal to the last, viz:—
A conic determines in every pencil in its plane an involution, corresponding lines being conjugate lines with regard to the conic.
If the point is without the conic the involution is hyperbolic, the tangents from the points being the focal rays.