148. Axes of conics. At the close of the last chapter (§ 140) we gave the theorem: A conic determines at every point in its plane an involution of rays, corresponding rays [pg 89] being conjugate with respect to the conic. The double rays, if any exist, are the tangents from the point to the conic. In particular, taking the point as the center of the conic, we find that conjugate diameters form a system of rays in involution, of which the asymptotes, if there are any, are the double rays. Also, conjugate diameters are harmonic conjugates with respect to the asymptotes. By the theorem of the last paragraph, there are two conjugate diameters which are at right angles to each other. These are called axes. In the case of the parabola, where the center is at infinity, and on the curve, there are, properly speaking, no conjugate diameters. While the line at infinity might be considered as conjugate to all the other diameters, it is not possible to assign to it any particular direction, and so it cannot be used for the purpose of defining an axis of a parabola. There is one diameter, however, which is at right angles to its conjugate system of chords, and this one is called the axis of the parabola. The circle also furnishes an exception in that every diameter is an axis. The involution in this case is circular, every ray being at right angles to its conjugate ray at the center.
149. Points at which the involution determined by a conic is circular. It is an important problem to discover whether for any conic other than the circle it is possible to find any point in the plane where the involution determined as above by the conic is circular. We shall proceed to the curious problem of proving the existence of such points and of determining their number and situation. We shall then develop the important properties of such points.
150. It is clear, in the first place, that such a point cannot be on the outside of the conic, else the involution would have double rays and such rays would have to be at right angles to themselves. In the second place, if two such points exist, the line joining them must be a diameter and, indeed, an axis. For if F and F' were two such points, then, since the conjugate ray at F to the line FF' must be at right angles to it, and also since the conjugate ray at F' to the line FF' must be at right angles to it, the pole of FF' 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.
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 [pg 91] 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.
