*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. [Figure 42] FIG. 42