to the lines AB, BC, CD, DA. Shew also that the line EG passes through the intersection of the tangents at A and B, and of those at C and D. 21. Find the condition that the line λu + μv + vw = 0 may touch the conic section √(lu) +√(mv) + √√ (nw) = 0. 22. Give a geometrical interpretation of equation (1) in Art. 304, and shew that it is a particular case of the theorem in Art. 317. 23. Interpret the last equation in Art. 313; deduce the following theorem; if from any point of the circle which circumscribes a triangle, perpendiculars are drawn on the sides of the triangle, the feet of the perpendiculars lie in one . straight line. 24. If ellipses be inscribed in a triangle each with one focus in a fixed straight line, the locus of the other focus is a conic section passing through the angular points of the triangle. SECTIONS OF A CONE. CHAPTER XVI. ANHARMONIC RATIO AND HARMONIC PENCIL. Sections of a cone. 324. WE shall now shew that the curves which are included under the name conic sections, can be obtained by the intersection of a right cone and a plane. DEF. A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which remains fixed. The fixed side is called the axis of the cone. made by a plane through OH and OC will meet the cone in a line OB, which is the position OC would occupy after revolving half way round. Let a section of the cone be made by a plane perpendicular to the plane BOC; let AP be the section, A being the point where the cutting plane meets OC; we have to find the nature of this curve AP. Let a plane pass through any point P of the curve, and be perpendicular to the axis OH; this plane will obviously meet the cone in a circle DPE, having its diameter DE in the plane BOC. Let MP be the line in which the plane of this circle meets the plane section we are considering, M being in the line DE. Since each of the planes which intersect in MP is perpendicular to the plane BOC, MP is perpendicular to that plane, and therefore to every line in that plane. Draw AF parallel to ED, and ML parallel to OB; join AM. Let AM=x, MP=y, OA = c, HOC=α, OAM=0; the angle AML will be equal to the inclination of AM to OB, that is to π 0-2α. But, from a property of the circle, MP2 = EM. MD; If we compare this equation with that in Art. 282, we see that the section is an ellipse, hyperbola, or parabola, according as sin 0 sin (0+ 2a) is negative, positive, or zero, that is, according as + 2a is less than π, greater than π, or equal to π. Hence if AM is parallel to OB the section is a parabola, if AM produced through M meets OB the section is an ellipse, if AM produced through A meets OB produced through Othe section is an hyperbola. If c=0 the section is a point if 0+2a is less than π, two straight lines if 0+2a is greater than π, and one straight line if 0 + 2α=π. Anharmonic Ratio and Harmonic Pencil. 325. We will now give a short account of anharmonic ratios and harmonic pencils, which are often used in investigating and enunciating properties of the conic sections. Let there be four straight lines meeting in a point; then if any straight line ADCB be drawn across the system, Now suppose any other straight line A'D'C'B' drawn across the system, then since AOB and A'OB' are the same angle, and so on for the other angles, we have AB DB A'B' D'B' = AC DC A'C' D'C'' which proves the proposition. Similarly we can prove that each of the following is a constant ratio 326. DEFS. Any four lines meeting in a point form a pencil. A straight line drawn across a pencil is called a transversal. AB DB AB CB Any one of the constant ratios ACDC' AD AC BC AD BD is called an anharmonic ratio of the pencil. The pencil is called harmonic if AB.DC=AD.BC, that is, if the rectangle formed by the whole line (AB) and the |