and at each step we shall have P and P representing volumes which are diminishing and which are approaching fixed volumes as their limits. By § 96 we arrive at the conclusion that NOTE. -The principle just deduced is the third of the three great principles of elementary geometry heretofore alluded to. Exercise. Show how a numerical representative of a rectangular parallelopiped may be determined. Suggestion.-Pursue the method of § 57. CHAPTER XIV. AN INTRODUCTION TO THE STUDY OF THE PLANE SECTIONS OF A RIGHT CIRCULAR CONE. 156. If we conceive of a plane tangent to a right circular cone (§ 136), and the plane through the element of tangency and the axis be represented by the page on which we have the figure, these planes will be perpendicular to each other (§§ 107, 112). If a plane be passed parallel to the tangent plane, it will intersect one nappe of the cone and not the other. This line of intersection will be a plane curve which is called a parabola. The curve will change its form depending upon the T M P FIG. 346. distance of the secant plane from the tangent plane. There are many properties peculiar to the parabola, a few of which will be deduced in this chapter. THEOREM. The ratio of the distances of every point of a parabola from a certain point in its plane and from a certain straight line, also in the plane, equals 1. Within the cone, tangent to the cone and also tangent to the secant plane, conceive a sphere to be constructed. It will be tangent to the cone on one of its small circles, the plane of which will be perpendicular to the axis of the cone; and for that reason, perpendicular to the plane on which the figure is represented. The plane of the small circle, and the plane which intersects the surface of the cone in a parabola, are both perpendicular to the plane of the picture; and so (§ 111, Ex. 4) their line of intersection (DM) is perpendicular to the plane of the picture. Let P be any point on the parabola. Join it to F (the point of tangency of the sphere and the secant plane), to V (the vertex of the cone), and draw PM ¦ to DM. Through P pass a plane perpendicular to the axis of the cone; it will intersect the surface of the cone in the OTPT, and the plane of the picture in TT. The plane of the parabola intersects the plane of the picture in QD. or PF PR (tangents to a sphere from P); (portions of elements between parallel planes); (AAQT and AND are isosceles); (opposite sides of a rectangle are equal). NOTE. Sections of a cone made by planes not parallel to a tangent plane will be considered after a few of the simpler properties of the parabola have been deduced. Most of the properties of the Conic Sections are best developed by the methods of Analytic Geometry and Calculus; but because of their importance in Nature and in Art, few of the simpler properties are here deduced. 157. Definitions. By reason of the property deduced in the preceding article, the parabola is often described as: The locus of a point moving in a plane so that the ratio of its distances from a fixed point and a fixed straight line equals 1. The fixed point is called the focus. The fixed straight line is called the directrix. The FD is called the axis, because the curve is symmetrical with respect to this line: PQ PQ, as may be = show from the figure in § 156. A is called the vertex. QP is called an ordinate. The double ordinate through Fis called the parameter or latus rectum. M Exercise. - Show that the ordinate at the focus equals FD. 158. PROBLEM. Having given the fixed point and the fixed line in a plane, to construct the parabola. distance of the first line from AB as a radius, construct an arc, intersecting the first line. These two points (C and D) of intersection are two points on the parabola. Repeat the process, using the second line in the same way as the first was used, thus determining the points E and G. In the same way H and K and any number of points may be determined. Through these points draw a smooth curve; it will approximate closely to the required parabola. M T P A mechanical construction may be affected by using the edge of a ruler as the directrix, against which one perpendicular side of a right triangle is caused to slide, while against the other perpendicular side the marking point P presses a string, one end of which is secured at F and the other so fastened that KP+PF = KM. FIG. 349. P will be a point on a parabola, because in any proper position PF-PM. Exercise. Construct parabolas, having given the distance of F from the directrix, 1, 3, 10, and 20 centimetres, S |