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d the theory of lines in involution on its own base. The student can show, by methods entirely analogous to those used in the second chapter, that involution is a projective property; that is, six rays in involution are cut by any transversal in six points in involution.
*136. Pencils of rays of the second order in involution.* We may also extend the notion of involution to pencils of rays of the second order. Thus, _the tangents to a conic are in involution when they are corresponding rays of two protective pencils of the second order superposed upon the same conic, and when they correspond to each other doubly._ We have then the theorem:
*137.* _The intersections of corresponding rays of a pencil of the second order in involution are all on a straight line __u__, and the intersection of any two tangents __ab__, when joined to the intersection of the corresponding tangents __a'b'__, gives a line which passes through a fixed point __U__, the pole of the line __u__ with respect to the conic._
*138. Involution of rays determined by a conic.* We have seen in the theory of poles and polars (§ 103) that if a point P moves along a line m, then the polar of P revolves about a point. This pencil cuts out on m another point-row _P'_, projective also to P. Since the polar of P passes through _P'_, the polar of _P'_ also passes through P, so that the correspondence between P and _P'_ is double. The two point-rows are therefore in involution, and the double points, if any exist, are the points where the line m meets the conic. A similar involution of rays may be found at any point in the plane, corresponding rays passing each through the pole of the other. We have called such points and rays conjugate with respect to the conic (§ 100). We may then state the following important theorem:
*139.* _A conic determines on every line in its plane an involution of points, corresponding po