*151.* Let now _P_ be a point on one of the axes (Fig. 42), and draw any ray through it, such as _q_. As _q_ revolves about _P_, its pole _Q_ moves along a line at right angles to the axis on which _P_ lies, describing a point-row _p_ projective to the pencil of rays _q_. The point at infinity in a direction at right angles to _q_ also describes a point-row projective to _q_. The line joining corresponding points of these two point-rows is always a conjugate line to _q_ and at right angles to _q_, or, as we may call it, a _conjugate normal_ to _q_. These conjugate normals to _q_, joining as they do corresponding points in two projective point-rows, form a pencil of rays of the second order. But since the point at infinity on the point-row _Q_ corresponds to the point at infinity in a direction at right angles to _q_, these point-rows are in perspective position and the normal conjugates of all the lines through _P_ meet in a point. This point lies on the same axis with _P_, as is seen by taking _q_ at right angles to the axis on which _P_ lies. The center of this pencil may be called _P’_, and thus we have paired the point _P_ with the point _P’_. By moving the point _P_ along the axis, and by keeping the ray _q_ parallel to a fixed direction, we may see that the point-row _P_ and the point-row _P’_ are projective. Also the correspondence is double, and by starting from the point _P’_ we arrive at the point _P_. Therefore the point-rows _P_ and _P’_ are in involution, and if only the involution has double points, we shall have found in them the points we are seeking. For it is clear that the rays through _P_ and the corresponding rays through _P’_ are conjugate normals; and if _P_ and _P’_ coincide, we shall have a point where all rays are at right angles to their conjugates. We shall now show that the involution thus obtained on one of the two axes must have double points. [Figure 43] FIG. 43