FIG. 41 *147. Pairs in an involution of rays which are at right angles. Circular involution.* In an involution of rays there is no one ray which may be distinguished from all the others as the point at infinity is distinguished from all other points on a line. There is one pair of rays, however, which does differ from all the others in that for this particular pair the angle is a right angle. This is most easily shown by using the construction that employs circles, as indicated above. The centers of all the circles through _G_ and _G’_ lie on the perpendicular bisector of the line _GG’_. Let this line meet the line _AA’_ in the point _C_ (Fig. 41), and draw the circle with center _C_ which goes through _G_ and _G’_. This circle cuts out two points _M_ and _M’_ in the involution. The rays _GM_ and _GM’_ are clearly at right angles, being inscribed in a semicircle. If, therefore, the involution of points is projected to _G_, we have found two corresponding rays which are at right angles to each other. Given now any involution of rays with center _G_, we may cut across it by a straight line and proceed to find the two points _M_ and _M’_. Clearly there will be only one such pair unless the perpendicular bisector of _GG’_ coincides with the line _AA’_. In this case every ray is at right angles to its corresponding ray, and the involution is called _circular_. *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 __ 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.