Analytical Geometry

182. PROP. IV. To find the equation for a cylindrical surface, the

co-ordinates being rectangular

183. PROP. V. To find the intersection of a plane with an oblique

cylinder

149

184. PROP. VI. To find the equation for a cone having a cir-

cular base, the origin of the co-ordinates being at the centre

of the base

185. PROP. VII. To find the equation for the curve line formed

by the intersection of a plane with the convex surface of

an upright cone having a circular base

a. The section of an oblique cone having a circular base is

shown to be a circle when the cutting plane makes an angle

on one side of the cone equal to that which the base makes

with the opposite side

186. DEF. 21. A sub-contrary section of a cone

187. PROP. VIII. To find the equations for a spheroid, an

hyperboloid, and a paraboloid

CURVE SURFACES OF THE SECOND ORDER WHICH ARE NOT PRODUCED

BY REVOLUTION.

188. General equation for a surface of the second order

189. A straight line can cut a surface of the second order in no

more than two points

190. PROP. I. If a plane bisect any three parallel chords not in

one plane, it will bisect every chord which is parallel to

192. COR. Each of three diametral planes will bisect the chords

which are parallel to the intersection of the two others - 159

193. DEF. 23. Conjugal diametral planes

194. Form of the general equation when the co-ordinate axes are

coincident with any system of conjugate diameters. One

system of conjugate diameters may be rectangular

- 159

Art.

Page

195. DEF. 24. The principal axes of a curve surface

196. Expression of the transformed equation in terms of any

three conjugate semi-diameters

197. PROP. II. To find the nature of the curve resulting from the

intersection of the figure by a plane parallel to one of the

co-ordinate planes, all the terms in the equation for the

surface being positive

198. DEF. 25. An ellipsoid

199. PROP. III. To find the form of the curve produced when the

surface is intersected by a plane parallel to one of the co-

ordinate planes, the sign of 2 in the equation for the

surface being negative

200. Equations for an asymptote to the curve surface

201. DEF. 26. An hyperboloid of one sheet

162

- 163

202. PROP. IV. To find the nature of the curves resulting from

the intersection of the figure by a plane parallel to one of

the co-ordinate planes; when, in the equation for the sur-
face, the signs of y2 and 22 are negative

203. DEF. 27. An hyperboloid of two sheets

204. Transformation of the equation for a curve surface from that

in which the origin of the co-ordinates is at the centre to

one in which that origin is at the extremity of one of the

205. PROP. V. To find the nature of the curve surface when its

centre is infinitely remote

206. Def. 28. An elliptical paraboloid

- 166

207. DEF. 29. An hyperbolic paraboloid

208. PROP. VI. To find the nature of the curve obtained by the

intersection of a plane with an ellipsoid; the plane having

any position

a. Nature of the section of an hyperboloid and of a parabo-

loid

209. PROP. VII. To find the equation for a plane touching a

curve surface

210. COR. The equations for a normal to a curve surface

[merged small][merged small][merged small][merged small][ocr errors][merged small]

212. Plane of comparison, and nature of the scales for the pro-

jections

170

213. PROP. I. To determine the intersection of a line given in

space, with either of the co-ordinate planes, and the inclina-

tion of the line to that plane

171

Art.

a. COR. To determine, on two co-ordinate planes, the pro-

jections of a line passing through a given point and parallel

to a line which is given in space

214. PROP. II. A plane in space, being conceived to pass through

three given points; to determine its intersections with, and

its inclinations to two co-ordinate planes

a. Formation of a scale of heights for a plane in space

215. PROP. III. To determine the angle at which two given

planes in space are inclined to one another

a. Schol. Nature of the scales for planes which are equally
inclined to a co-ordinate plane -

216. PROP. IV. To find the point in which a given line in space

intersects a given plane

217. PROP. V. To find the point in which a line drawn through

a given point perpendicularly to a given plane intersects

that line

218. PROP. VI. To determine the position of a plane which shall

pass through a given point, and be perpendicular to a given

line

219. PROP. VII. To determine the position of a plane which

shall pass through a given line in space, and be inclined

to one of the co-ordinate planes in a given angle -

220. PROP. VIII. A plane being given, and a straight line situ-

ated in it; it is required to determine a plane which shall

pass through that line, and make with the other plane a

221. PROP. IX. To determine a plane which shall pass through a

given point, and be parallel to a given plane

222. PROP. X. To determine a plane which shall pass through a

given line, and be perpendicular to a given plane

223. PROP. XI. To determine the positions of two planes which

shall be parallel to one another, and each of which shall

contain a given line; the lines not being in one plane

224. PROP. XII. A plane in space being given, and from a

point in its common section with one of the co-ordinate

planes a line being drawn in the latter plane; it is required

to determine, in the given plane, a line proceeding from

the same point, and making with the former line a given

angle

COR. 1. To determine a line, proceeding from the given

point in the line of section, which shall make with that

COR. 2. To determine a line, proceeding as before, which
shall make with the co-ordinate plane a given angle

225. PROP. XIII. To determine a line which shall be perpen-

dicular to two lines given, in space, and not lying in one

plane

226. PROP. XIV. Three points being given, in space, with their

several angular distances, taken at a fourth point (not in

the same plane as the others) from a vertical line passing

through it; to determine the position of that fourth point 188

Art.

Page

227. PROP. XV. To determine the position of a plane which,

passing through a given point on the convex surface of a

cylinder, may be a tangent to that surface

228. PROP. XVI. To determine the position of a plane which,

passing through a given point on the surface of a cone,

may be a tangent to that surface

229. PROP. XVII. To determine the position of a plane which,

passing through a given point beyond a cylindrical or

conical surface, may be a tangent to that surface

190

191

- 191

230. PROP. XVIII. To determine the position of a plane which,

passing through a given point on the surface of a sphere,

may be a tangent to that surface

231. Schol. Manner in which the projections of a normal to a

curve may be determined

192

- 193

232. PROP. XIX. To determine the projection, on one of the co-

ordinate planes, of the lines in which two cylinders may

intersect one another

233. Orthogonal projection of a groined vault formed by the inter-

section of two hemicylinders having equal semicircular

bases

234. PROP. XX. To determine the projections, on one of the co-

ordinate planes, of the curve lines in which two cones may

intersect one another

235. PROP. XXI. To determine the projections, on one of the co-

ordinate planes, of the curve lines in which a cylinder may

intersect a sphere

236. Nature of the faces and joints of voussoirs for cylindrical and
conical vaults

237. PROP. XXII. To describe the projections, on a vertical and

a horizontal plane, of the key-voussoir in a hemicylin-

drical vault, the vertical plane of projection being perpen-

dicular to the axis of the cylinder which, for simplicity, is

supposed to be circular

238. PROP. XXIII. To describe the projections, on a vertical and

a horizontal plane, of a voussoir forming part of a hemi-

239. PROP. XXIV. To describe the projections, on a vertical and

a horizontal plane, of a voussoir at an angle of a groined

vault, or one formed by the intersection of two hemicylin-

drical vaults

200

ANALYTICAL GEOMETRY.

SECTION I.

ANALYTICAL GEOMETRY OF TWO DIMENSIONS.

CHAPTER I.

POSITION AND CO-ORDINATES OF A POINT IN A PLANE. DISTANCE BETWEEN TWO POINTS.

ARTICLE 1. Analytical Geometry is the name given to that branch of mathematics in which Algebra is made the instrument of geometrical investigation. It is divided into two parts Analytical Geometry of Two Dimensions, and Analytical Geometry of Three Dimensions, according as the objects of investigation are situated in the same plane, or in any manner in space.

A mathematical question which admits of a limited number of solutions only, is said to be determinate; and indeterminate, if the number of solutions is unlimited.

Determinate problems have relation to the magnitude and position of geometrical quantities; indeterminate, to their figure.

In analytical geometry of two dimensions, the points of a line or of the contour of a figure are usually referred to two given straight lines, as x'ox, Yor', which inter- X sect in a point o. xx, ry are termed

axes. According as XOY is a right angle, or not a right angle, the axes are termed rectangular, or oblique.

Sometimes also the points of a line or of the contour of a figure are referred to their distances from a given point, and the an

gles which the straight lines on which these distances are

« Anterior Continuar »

Libros

Analytical Geometry: With the Properties of Conic Sections, and an Appendix ...