It follows that rays from a source of light at one focus are reflected by an ellipse to the other. *155.* In the case of the parabola, where one of the foci must be considered to be at infinity in the direction of the diameter, we have [Figure 45] FIG. 45 _A diameter makes the same angle with the tangent at its extremity as that tangent does with the line from its point of contact to the focus (Fig. 45)._ *156.* This last theorem is the basis for the construction of the parabolic reflector. A ray of light from the focus is reflected from such a reflector in a direction parallel to the axis of the reflector. *157. Directrix. Principal axis. Vertex.* The polar of the focus with respect to the conic is called the _directrix_. The axis which contains the foci is called the _principal axis_, and the intersection of the axis with the curve is called the _vertex_ of the curve. The directrix is at right angles to the principal axis. In a parabola the vertex is equally distant from the focus and the directrix, these three points and the point at infinity on the axis being four harmonic points. In the ellipse the vertex is nearer to the focus than it is to the directrix, for the same reason, and in the hyperbola it is farther from the focus than it is from the directrix. [Figure 46] FIG. 46 *158. Another definition of a conic.* Let _P_ be any point on the directrix through which a line is drawn meeting the conic in the points _A_ and _B_ (Fig. 46). Let the tangents at _A_ and _B_ meet in _T_, and call the focus _F_. Then _TF_ and _PF_ are conjugate lines, and as they pass through a focus they must be at right angles to each other. Let _TF_ meet _AB_ in _C_. Then _P_, _A_, _C_, _B_ are four harmonic points. Project these four points parallel to _TF_ upon the directrix, and we then get the four harmonic points _P_, _M_, _Q_, _N_. Since, now, _TFP_ is a right angle, the angles _MFQ_ and _NFQ_ are equal, as well as the angles _AFC_ and _BFC_. Therefore the triangles _MAF_ and _NFB_ are similar, and _FA : FM = FB : BN_. Dropping perpendiculars _AA_ and _BB’_ upon the directrix, this becomes _FA : AA’ = FB : BB’_. We have thus the property often taken as the definition of a conic: _The ratio of the distances from a point on the conic to the focus and the directrix is constant._ [Figure 47] FIG. 47 *159. Eccentricity.* By taking the point at the vertex of the conic, we note that this ratio is less than unity for the ellipse, greater than unity for the hyperbola, and equal to unity for the parabola. This ratio is called the _eccentricity_. [Figure 48] FIG. 48