[Figure 36]

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 _{~GREEK SMALL LETTER KAPPA~}_ through S.  We thus obtain two projective pencils, a, b, c, d, ... and a’, b’, c’, d’, ... at S, which meet the conic in the points _{~GREEK SMALL LETTER ALPHA~}_, _{~GREEK SMALL LETTER BETA~}_, _{~GREEK SMALL LETTER GAMMA~}_, _{~GREEK SMALL LETTER DELTA~}_, ... and _{~GREEK SMALL LETTER ALPHA~}’_, _{~GREEK SMALL LETTER BETA~}’_, _{~GREEK SMALL LETTER GAMMA~}’_, _{~GREEK SMALL LETTER DELTA~}’_, ... (Fig. 36).  Take now _{~GREEK SMALL LETTER GAMMA~}_ as the center of a pencil projecting the points _{~GREEK SMALL LETTER ALPHA~}’_, _{~GREEK SMALL LETTER BETA~}’_, _{~GREEK SMALL LETTER DELTA~}’_, ..., and take _{~GREEK SMALL LETTER GAMMA~}’_ as the center of a pencil projecting the points _{~GREEK SMALL LETTER ALPHA~}_, _{~GREEK SMALL LETTER BETA~}_, _{~GREEK SMALL LETTER DELTA~}_, ....  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 ({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER ALPHA~}’, {~GREEK SMALL LETTER GAMMA~}’{~GREEK SMALL LETTER ALPHA~}) to ({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER BETA~}’, {~GREEK SMALL LETTER GAMMA~}’{~GREEK SMALL LETTER BETA~}).  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