*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 _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 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.