The details we leave to the student. [Figure 37] FIG. 37 [Figure 38] FIG. 38 *131. Involution of points on a point-row of the second order.* It is important to note also, in Steiner’s construction, that we have obtained two point-rows of the second order superposed on the same conic, and have paired the points of one with the points of the other in such a way that the correspondence is double. We may then extend the notion of involution to point-rows of the second order and say that _the points of a conic are paired in involution when they are corresponding __ points of two projective point-rows superposed on the conic, and when they correspond to each other doubly._ With this definition we may prove the theorem: _The lines joining corresponding points of a point-row of the second order in involution all pass through a fixed point __U__, and the line joining any two points __A__, __B__ meets the line joining the two corresponding points __A’__, __B’__ in the points of a line __u__, which is the polar of __U__ with respect to the conic._ For take _A_ and _A’_ as the centers of two pencils, the first perspective to the point-row _A’_, _B’_, _C’_ and the second perspective to the point-row _A_, _B_, _C_. Then, since the common ray of the two pencils corresponds to itself, they are in perspective position, and their axis of perspectivity _u_ (Fig. 38) is the line which joins the point _(AB’, A’B)_ to the point _(AC’, A’C)_. It is then immediately clear, from the theory of poles and polars, that _BB’_ and _CC’_ pass through the pole _U_ of the line _u_. *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: