any point C, which we will also call R’.  We must show that the corresponding point C’ must also coincide with the point B.  Join all the points to S, as before, and it appears that the points {~GREEK SMALL LETTER ALPHA~} and _{~GREEK SMALL LETTER PI~}’_ coincide, as also do the points _{~GREEK SMALL LETTER ALPHA~}’{~GREEK SMALL LETTER PI~}_ and _{~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER RHO~}’_.  By the above construction the line _{~GREEK SMALL LETTER GAMMA~}’{~GREEK SMALL LETTER RHO~}_ must meet _{~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER RHO~}’_ on the line joining ({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER ALPHA~}’, {~GREEK SMALL LETTER GAMMA~}’{~GREEK SMALL LETTER ALPHA~}) with ({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER PI~}’, {~GREEK SMALL LETTER GAMMA~}’{~GREEK SMALL LETTER PI~}).  But these four points form a quadrangle inscribed in the conic, and we know by § 95 that the tangents at the opposite vertices _{~GREEK SMALL LETTER GAMMA~}_ and _{~GREEK SMALL LETTER GAMMA~}’_ meet on the line v.  The line _{~GREEK SMALL LETTER GAMMA~}’{~GREEK SMALL LETTER RHO~}_ is thus a tangent to the conic, and C’ and R are the same point.  That two projective point-rows superposed on the same line are also in involution when one pair, and therefore all pairs, correspond doubly may be shown by taking S at one vertex of a complete quadrangle which has two pairs of opposite sides going through two pairs of points.  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.