*152. Discovery of the foci of the conic.* We know that on one axis no such points as we are seeking can lie (§ 150). The involution of points _PP’_ on this axis can therefore have no double points. Nevertheless, let _PP’_ and _RR’_ be two pairs of corresponding points on this axis (Fig. 43). Then we know that _P_ and _P’_ are separated from each other by _R_ and _R’_ (§ 143). Draw a circle on _PP’_ as a diameter, and one on _RR’_ as a diameter. These must intersect in two points, _F_ and _F’_, and since the center of the conic is the center of the involution _PP’_, _RR’_, as is easily seen, it follows that _F_ and _F’_ are on the other axis of the conic. Moreover, _FR_ and _FR’_ are conjugate normal rays, since _RFR’_ is inscribed in a semicircle, and the two rays go one through _R_ and the other through _R’_. The involution of points _PP’_, _RR’_ therefore projects to the two points _F_ and _F’_ in two pencils of rays in involution which have for corresponding rays conjugate normals to the conic. We may, then, say: _There are two and only two points of the plane where the involution determined by the conic is circular. These two points lie on one of the axes, at equal distances from the center, on the inside of the conic. These points are called the foci of the conic._ *153. The circle and the parabola.* The above discussion applies only to the central conics, apart from the circle. In the circle the two foci fall together at the center. In the case of the parabola, that part of the investigation which proves the existence of two foci on one of the axes will not hold, as we have but one axis. It is seen, however, that as _P_ moves to infinity, carrying the line _q_ with it, _q_ becomes the line at infinity, which for the parabola is a tangent line. Its pole _Q_ is thus at infinity and also the point _P’_, so that _P_ and _P’_ fall together at infinity, and therefore one focus of the parabola is at infinity. There must therefore be another, so that _A parabola has one and only one focus in the finite part of the plane._ [Figure 44] FIG. 44 *154. Focal properties of conics.* We proceed to develop some theorems which will exhibit the importance of these points in the theory of the conic section. Draw a tangent to the conic, and also the normal at the point of contact _P_. These two lines are clearly conjugate normals. The two points _T_ and _N_, therefore, where they meet the axis which contains the foci, are corresponding points in the involution considered above, and are therefore harmonic conjugates with respect to the foci (Fig. 44); and if we join them to the point _P_, we shall obtain four harmonic lines. But two of them are at right angles to each other, and so the others make equal angles with them (Problem 4, Chapter II). Therefore _The lines joining a point on the conic to the foci make equal angles with the tangent._