a 93 93 . 153, 154. The general equation to the Hyperbola is, Py? - Qx'= -1; or a® yo — 68 m2 = - a2 72 155_7. Discussion of the equation: The sq. on MP : rectangle AM, MA' :: sq. on BC : sq. on AC. 91 158. The equation to the equilateral hyperbola is y® – x? = -2 99 159. The results obtained for the ellipse are applicable to the hyperbola, by changing 62 into 62 160. The Latus Rectum is defined to be a third proportional to the transverse and conjugate axes 161-3. The focus; the eccentricity: The rectangle AS, SA' = the square on BC 93 164, SP= ex-a, HP = ex + a, HP - SP= A A' 165. To find the locus of a point the difference of whose distances from two fixed points is constant 166. The equation to the tangent is a’ y yi - bo x x = - - a2 62 93 167. The rectangle CT, CM = sq. on CA 94 168. The equation to the tangent, at the extremity of the Latus Rectum, is y = exThe 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, p2 = 2 a to 95 170. The locus of y is the circle on the transverse axis 171. The tangent makes equal angles with the focal distances, 62 tan. SPT= cy' 172. The perpendicular from the centre on the tangent, a 6 p= 97 drri . a. . 94 62r 96 Art. Page 174. If CE be drawn parallel to the tangent and meeting H P in E, then P E=AC 98 175–7. The equation to the normal is a2y 62 x 62x CG = ex”; CGʻ = •y ; M = 72 ♡ The rectangle PG, PG' = the rectangle SP, HP 98 178,9. The diameters of the hyperbola pass through the centre, but do not all meet 6 98 ۳۲ = rr, PG =وP; PG Q2 a the curve ; a line, whose tangent is being the limit 180, 1. There is an infinite number of pairs of conjugate diameters, 62 tan. & tan, = 99 · 102 182. The equation to the curve referred to conjugate axes is aje ye – 6,2 222 - ao bo 100 183. a, b;' = a' - 5 100 184. a, b, sin. ( - b) = ab 101 185. The sq. on QV : the rectangle PV, VP' :: sq. on CD: sq. on CP 102 187. The conjugate diameter is parallel to the tangent. The equations are a' ye y 62 x a.' = al b2 the tangent. a2 yy - 62 x x =0 the conjugate 102 188. The sq. on CD= the rectangle SP, HP 189. If PF be drawn perpendicular on CD, then ab a2 62 PF= or p2 = 102 u? - a2 + 12 190—2. If a and á be the tangents of the angles which a pair of supplemental 72 a2 103 193. There are no equal conjugate diameters in general. In the equilateral hyperbola they are always equal to each other 103 194—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 104 197. The hyperbola is the only one of the lines of the second order that has a rectilineal asymptote 105 198. Method of reducing an equation into a series containing inverse powers of a variable. The asymptotes parallel to the axes 105 199. Discussion of the equation bxy +f=0 106 200. Referring the curve to its centre and axes, the equations are a’ ya — 62x2 = - a262, the curve, 107 201. In the equilateral or rectangular hyperbola (y2 – x2 = - a2) the angle between the asymptotes is 90° 107 202, 3, Asymptotes referred to the vertex of the curve ; a line parallel to the asymptote cuts the curve in one point only 107 . Art. • 112 Page 204. Examples of tracing hyperbolas, and drawing the asymptotes . 108 206. Reduction of the general equation of the second order to the form xy = k2 a (tan. 1)2 + 6 tan. 6+c=0 109 207. To find the value of U 110 209. Examples. If c= a, the curve is rectangular 110 211. Given the equation xy=k%, to find the equation referred to rectangular axes, and to obtain the lengths of the axes 212. From the equation u?y2 – 622? = - al 62 referred to the centre and axes to obtain the equation referred to the asymptotes, a? +62 d'y' = 113 4 213. The parallelogram on the co-ordinates is equal to half the rectangle on the semi-axes 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 114 215. Given the conjugate diameters to find the asymptotes. If the asymptotes are given, the conjugate to a diameter is given 114 216. The equation to the tangent referred to the asymptotes Ny + y'x = 2k2. 115 217, 8. The two parts of any secant comprised between the curve and asymptote are equal. The rectangle SQ, QS = sq. on CD 219. The general polar equation is u? (y' t u sin. 8)2 – 22 (a! t u cos. 6) = - a2 32 116 a? (e2 – 1) 220. The pole at the centre, u? = 116 (cos. 3)2 1 a (e2 - 1) 221. The focus, the pole, r = 116 1 2bi? 222. rv = (r +r); (r+x)= 17 (); ) 116 223, 4. The conjugate hyperbola. The locus of the extremity of the conjugate diameter is the conjugate hyperbola. The equation is al y2 - 62x2 = a? 62. 117 . 115 e2 . e cos. A 4 a CHAPTER X. THE PARABOLA, 225, 6. The equation to the parabola referred to its axis and vertex is ye=px 118 227. Difference between a parabolic and hyperbolic branch . 118 228. The equation to the parabola deduced from that to the ellipse referred to its vertex, by putting AS=m 118 229. The principal parameter, or Latus Rectum, is a third proportional to any abscissa and its ordinate. In the following articles 4 m is assumed to be 119 . Art. . Page 230. To find the position of the focus 119 231. The distance of any point on the curve from the focus, SP = x + m 120 232. The equation to the tangent is yy=2m (+3) 120 1 233, 4. The subtangent MT= 2 AM, A y = MP. The tangent at the vertex coincides with the axis of y 120 235. The equation to the tangent at the extremity of the Latus Rectum is y = x + m 121 236. 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 = ENT 121 238, 9. The locus of y is the axis AY. The perpendicular Sy cuts the directrix on the point where the perpendicular from P on the directrix meets that line 122 240. The tangent makes equal angles with the focal distance and with a parallel yl 122 2.x 241. The equation to the normal is y y'= - (x - 2) 123 . . 2 m • 123 . . . 242. The subnormal is equal to half the Latus Rectum : SG = SP, and PG : = N Amr 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 new origin and to new axes 124 246. The new equation is yo=p'x; the new parameter p = 4SP 125 247. Transformation of the equation when the position of the new origin and axes is given 125 248. The ordinate through the focus = 4SP= the parameter at the origin 126 249. The equation to the tangent 126 250. Tangents drawn from the extremities of a parameter meet at right angles in the directrix 126 251. The general polar equation is (y + u sin. 6)2 = p (x + u cos. 6) 127 252. The pole, any point on the curve, p cos. 6 — 2 y' sin. 6 at the vertex 127 (sin. 0)2 (sin, 0) 253. The focus, the pole, P P 127 2 1 CHAPTER XI. THE SECTIONS OF A CONE. Page 128 Art. {a sin. B x - sin. («+B) x*} sin. a y2 = 2 ) . • 130 . 132 132 133 134 134 134 135 136 . . 257—264. Discussion of the cases arising from various positions of the cutting plane 265. On mechanical description, and that “by points" 266. Tracing the Ellipse by means of a string. 267—8. The elliptic compasses. Another method 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 3y2 + (1 - e2) x2 – 2 ms (1 +e)=0 280. From the equations to the ellipse, to deduce those of the hyperbola and parabola 281. The general polar equation is р 1 21 te cos. 6 282–4. 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 making a given angle be drawn to meet the curve, the rectangle contained tained by the segments of the other segments are equal 295. A systematic examination of all curves is impossible point from the extremities of the given line are in a given ratio 143 |