Fig. 35
128. Double correspondence. We have seen that corresponding points in an involution form two projective point-rows superposed on the same straight line. Two projective point-rows superposed on the same straight line are, however, not necessarily in involution, as a simple example will show. Take two lines, a and a', which both revolve about a fixed point S and which always make the same angle with each other (Fig. 35). These lines cut out on any line in the plane which does not pass through S two projective point-rows, which are not, however, in involution unless the angle between the lines is a right angles. For a point P may correspond to a point P', which in turn will correspond to some other point [pg 76] than P. The peculiarity of point-rows in involution is that any point will correspond to the same point, in whichever point-row it is considered as belonging. In this case, if a point P corresponds to a point P', then the point P' corresponds back again to the point P. The points P and P' are then said to correspond doubly. This notion is worthy of further study.
Fig. 36
129. Steiner's construction. It will be observed that the solution of the fundamental problem given in § 83, Given three pairs of points of two protective point-rows, to construct other pairs, cannot be carried out if the two point-rows lie on the same straight line. Of course the method may be easily altered to cover that case also, but it is worth while to give another solution of the problem, due to Steiner, which will also give further information regarding the theory of involution, and which may, indeed, be used as a foundation for that theory. Let the two point-rows A, B, C, D, ... and A', B', C', D', ... be superposed on the line u. Project them both to a point S and pass any conic κ through S. We thus obtain two projective pencils, a, b, c, d, ... and [pg 77] a', b', c', d', ... at S, which meet the conic in the points α, β, γ, δ, ... and α', β', γ', δ', ... (Fig. 36). Take now γ as the center of a pencil projecting the points α', β', δ', ..., and take γ' as the center of a pencil projecting the points α, β, δ, .... These two pencils are projective to each other, and since they have a self-correspondin ray in common, they are in perspective position and corresponding rays meet on the line joining (γα', γ'α) to (γβ', γ'β). The correspondence between points in the two point-rows on u is now easily traced.
130. Application of Steiner's construction to double correspondence. Steiner's construction throws into our hands an important theorem concerning double correspondence: If two projective point-rows, superposed on the same line, have one pair of points which correspond to each other doubly, then all pairs correspond to each other doubly, and the line is paired in involution. To make this appear, let us call the point A on u by two names, A and P', according as it is thought of as belonging to the one or to the other of the two point-rows. If this point is one of a pair which correspond to each other doubly, then the points A' and P must coincide (Fig. 37). Take now 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 α and π' coincide, as also do the points α'π and γρ'. By the above construction the line γ'ρ must meet γρ' on the line joining (γα', γ'α) with (γπ', γ'π). But these four points form a quadrangle inscribed in the conic, and we know by § 95 that the tangents at the opposite [pg 78] vertices γ and γ' meet on the line v. The line γ'ρ 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.