132.  Involution of rays. The whole theory thus far developed may be dualized, and a theory of lines in involution may be built up, starting with the complete quadrilateral.  Thus,

The three pairs of rays which may be drawn from a point through the three pairs of opposite vertices of a complete quadrilateral are said to be in involution.  If the pairs _aa’__ and __bb’__ are fixed, and the line __c__ describes a pencil, the corresponding line __c’__ also describes a pencil, and the rays of the pencil are said to be paired in the involution determined by __aa’__ and __bb’__._

133.  Double rays. The self-corresponding rays, of which there are two or none, are called double rays of the involution.  Corresponding rays of the involution are harmonic conjugates with respect to the double rays.  To the theorem of Desargues (§ 125) which has to do with the system of conics through four points we have the dual: 

The tangents from a fixed point to a system of conics tangent to four fixed lines form a pencil of rays in involution.

134. If a conic of the system should go through the fixed point, it is clear that the two tangents would coincide and indicate a double ray of the involution.  The theorem, therefore, follows: 

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._