_Two conics or none may be drawn through a fixed point to be tangent to four fixed lines._ *135. Double correspondence.* It further appears that two projective pencils of rays which have the same center are in involution if two pairs of rays correspond to each other doubly. From this it is clear that we might have deemed six rays in involution as six rays which pass through a point and also through six points in involution. While this would have been entirely in accord with the treatment which was given the corresponding problem in the theory of harmonic points and lines, it is more satisfactory, from an aesthetic point of view, to build 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 points in the involution __ being conjugate with respect to the conic. The double points, if any exist, are the points where the line meets the conic._ *140.* The dual theorem reads: _A conic determines at every point in the plane an involution of rays, corresponding rays being conjugate with respect to the conic. The double rays, if any exist, are the tangents from the point to the conic._