Fig. 37
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 [pg 79] 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'.