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