The square of any semi-ordinate to the axis is equal to the rectangle under the latus rectum and the abscissa. Hence, Ququ; and since the same may be proved for any ordinate, it follows, that A diameter bisects all its own ordinates. Cor. 2. Let Rr be the parameter to the diameter PW. Then, by Prop. III. Cor. 3. X Hence the Proposition may be thus enunciated: The square of the semi-ordinate to any diameter is equal to the rectangle under the parameter and abscissa. It will be seen, that Prop. vII. is a particular case of the present proposition. ELLIPSE. DEFINITIONS. 1. AN ELLIPSE is a plane curve, such that, if from any point in the curve two straight lines be drawn to two given fixed points, the sum of these straight lines will always be the same. 2. The two given fixed points are called the foci. Thus, let ABa be an ellipse, S and H B Pi 4. The distance from the centre to either focus is called the eccentricity. 5. Any straight line drawn through the centre, and terminated both ways by the curve, is called a diameter. 6. The points in which any diameter meets the curve are called the vertices of that diameter. 7. The diameter which passes through the foci is called the axis major, and the points in which it meets the curve are called the principal vertices. 8. The diameter at right angles to the axis major is called the axis minor. Thus, let AB be an ellipse, S and H B p Then, C is the centre, CS or CH the eccentricity, Pp is a diameter, P and its vertices, Aa is the major axis, Bb is the minor axis. 9. A straight line which meets the curve in any point, but which, being produced both ways, does not cut it, is called a tangent to the curve at that point. 10. A diameter drawn parallel to the tangent at the vertex of any other diameter, is called the conjugate diameter to the latter, and the two diameters are called a pair of conjugate diameters. 11. Any straight line drawn parallel to the tangent at the vertex of any diameter and terminated both ways by the curve, is called an ordinate to that diameter. 12. The segments into which any diameter is divided by one of its own ordinates are called the abscisse of the diameter. 13. The ordinate to any diameter, which passes through the focus, is called the parameter of that diameter. Thus, let Pp be any diameter, and Tt a tangent at P. Draw the diameter Dd parallel to Tt. Take any point Q in the curve, draw Qq parallel to Tt, cutting Pp in v. Through S draw Rr parallel to Tt. Then, Dd is the conjugate diameter to Pp. Qg is the ordinate to the diameter Pp, corresponding to the point Q. Pv, vp are the abscissæ of the diameter Pp, corresponding to the point Q. Rr is the parameter of the diameter Pp. T 14. Any straight line drawn at right angles to the major axis, and terminated both ways by the curve, is called an ordinate to the axis. 15. The segments into which the major axis is divided by one of its own ordinates are called the abscissæ of the axis. 16. The ordinate to the axis which passes through either focus is called the latus rectum. (It will be proved in Prop. Iv., that the tangents at the principal vertices are perpendicular to the major axis; hence, definitions 14, 15, 16, are in reality included in the three which immediately precede them.) 17. If a tangent be drawn at the extremity of the latus rectum and produced to meet the major axis, and if a straight line be drawn through the point of intersection at right angles to the major axis, the tangent is called the focal tangent, and the straight line the directrix. Thus, from P any point in the curve, draw PMp perpendicular to Aa, cutting B P HD CMS Then, Pp is the ordinate to the axis, corresponding to the point P LT is the focal tangent. Nn is the directrix. 18. A straight line drawn at right angles to a tangent from the point of contact, and terminated by the major axis, is called a normal. The part of the major axis intercepted between the intersections of the normal and the ordinate, is called the subnormal. The sum of two straight lines drawn from the foci to any point in the curve is equal to the major axis. That is, if P be any point in the curve, |