The sq. on MP rectangle A M, M A':: sq. on BC: sq. on AC. 158. The equation to the equilateral hyperbola is 159. The results obtained for the ellipse are applicable to the hyperbola, by changing 160. The Latus Rectum is defined to be a third proportional to the transverse and conjugate axes 164. SP ex-a, HP ex+a, HP - SP=AA' 91 92 92 99 165. To find the locus of a point the difference of whose distances from two fixed 168. The equation to the tangent, at the extremity of the Latus Rectum, is y = ex -a. The distances of any point from the focus and from the directrix are in a constant ratio 169. The length of the perpendicular from the focus on the tangent, 94 171. The tangent makes equal angles with the focal distances, 95 96 96 97 Art. Page 174. If CE be drawn parallel to the tangent and meeting HP in E, then PE=AC 98 175-7. The equation to the normal is The rectangle PG, PG the rectangle SP, HP 178, 9. The diameters of the hyperbola pass through the centre, but do not all meet the curve; a line, whose tangent is being the limit a 180, 1. There is an infinite number of pairs of conjugate diameters, 98 98 66 182. The equation to the curve referred to conjugate axes is 185. The sq. on QV: the rectangle PV, V P':: sq. on CD: sq. on CP 187. The conjugate diameter is parallel to the tangent. The equations are a2 b2 the tangent. 188. The sq. on C D a2 y2 y — b2 x2 x′ = — a2 y y b2 x x = 0 the rectangle SP, HP a2 b2 or p2 = u2 · a2 + b2 190-2. If a and a' be the tangents of the angles which a pair of supplemental chords makes with the transverse axis, « α = 102 plemental chords. Conjugate diameters are parallel to sup. chords 193. There are no equal conjugate diameters in general. In the equilateral hyperbola they are always equal to each other 194—6. The Asymptotes. The equation to the asymptote is the equation to the curve, with the exception of the terms involving inverse powers of x. Curvilinear asymptotes 197. The hyperbola is the only one of the lines of the second order that has a rectilineal asymptote 103 198. Method of reducing an equation into a series containing inverse powers of a 200. Referring the curve to its centre and axes, the equations are 201. In the equilateral or rectangular hyperbola (y2 — x2 —— a2) the angle between the asymptotes is 90° 202, 3. Asymptotes referred to the vertex of the curve; a line parallel to the asymptote cuts the curve in one point only Art. Page 108 109 110 110 204. Examples of tracing hyperbolas, and drawing the asymptotes 206. Reduction of the general equation of the second order to the form xy=k2 a (tan. 6)2 + b tan. 6+c=0. 207. To find the value of U 209. Examples. If ca, the curve is rectangular 211. Given the equation x y = k2, to find the equation referred to rectangular axes, and to obtain the lengths of the axes 212. From the equation a2 y2 — b2x2 — — a2 62 referred to the centre and axes to obtain the equation referred to the asymptotes, 112 213. The parallelogram on the co-ordinates is equal to half the rectangle on the 114 114 214. The parts of the tangent between the point of contact and the asymptotes are equal to each other and to the semi-conjugate diameter 215. Given the conjugate diameters to find the asymptotes. If the asymptotes are given, the conjugate to a diameter is given 216. The equation to the tangent referred to the asymptotes . 115 217, 8. The two parts of any secant comprised between the curve and asymptote are equal. The rectangle S Q, QS' sq. on C D 223, 4. The conjugate hyperbola. The locus of the extremity of the conjugate diameter is the conjugate hyperbola. The equation is 116 (a2 y2 — b2 x)2 = a*b*, or x2y2 = k1 117 225, 6. The equation to the parabola referred to its axis and vertex is y2= px 227. Difference between a parabolic and hyperbolic branch 228. The equation to the parabola deduced from that to the ellipse referred to its 118 vertex, by putting A Sm 229. The principal parameter, or Latus Rectum, is a third proportional to any abscissa and its ordinate. In the following articles 4m is assumed to be the value of the Latus Rectum Art. Page 230. To find the position of the focus 231. The distance of any point on the curve from the focus, S P = x + m 232. The equation to the tangent is yy=2m (x+x") 233, 4. The subtangent MT 2 AM, A y = MP. The tangent at the vertex coincides with the axis of y 235. The equation to the tangent at the extremity of the Latus Rectum is 236. The Directrix. The distances of any point from the focus and directrix are equal 121 237. The length (Sy) of the perpendicular from the focus on the tangent =√m r SP: Sy::Sy:SA 238, 9. The locus of y is the axis A Y. The perpendicular Sy cuts the directrix on the point where the perpendicular from P on the directrix meets that line 240. The tangent makes equal angles with the focal distance and with a parallel 121 122 242. The subnormal is equal to half the Latus Rectum: SG SP, and PG4mr 123 243. The parabola has an infinite number of diameters, all parallel to the axis 123 244, 5. Transformation of the equation to another of the same form referred to a 246. The new equation is y2=px; the new parameter p = 4 SP 247. Transformation of the equation when the position of the new origin and axes 125 the directrix 248. The ordinate through the focus 4SP the parameter at the origin 249. The equation to the tangent 250. Tangents drawn from the extremities of a parameter meet at right angles in 251. The general polar equation is 252. The pole, any point on the curve, (y+ u sin. 6)2=p (x' +u cos. 6) 127 254. rr2 = 2 (r+r) . 128 257-264. Discussion of the cases arising from various positions of the cutting 269. Tracing the hyperbola by means of a string 270. Tracing the parabola by means of a string 271-3. Description of the ellipse by points 274-6. Description of the hyperbola by points. The rectangular hyperbola 277-8. To describe the parabola by points 279. From the position of the directrix and focus, and focal ratio, to find the equation to the curves of the second order y2 + (1 − e2) x2 — 2 mx (1 + e) = 0 280. From the equations to the ellipse, to deduce those of the hyperbola and parabola 282-4. Practical method of drawing tangents to the ellipse 285-6. Tangents to the hyperbola 287-8. Tangents to the parabola 289-292. An arc of a conic section being traced on a plane, to determine the section and the axes 293. If through any point within or without a conic section two straight lines making a given angle be drawn to meet the curve, the rectangle contained by the segments of the one will be in a constant ratio to the rectangle contained by the segments of the other 294. If a polygon circumscribe an ellipse, the algebraic product of its alternate segments are equal CHAPTER XII. ON CURVES OF THE HIGHER ORDERS. 295. A systematic examination of all curves is impossible, 296. To find a point P without a given straight line, such that the distances of the point from the extremities of the given line are in a given ratio |