must be at infinity in a direction at right angles to FF’.  The line FF’ is then a diameter, and since it is at right angles to its conjugate diameter, it must be an axis.  From this it follows also that the points we are seeking must all lie on one of the two axes, else we should have a diameter which does not go through the intersection of all axes—­the center of the conic.  At least one axis, therefore, must be free from any such points.

[Figure 42]

FIG. 42

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