Locus of Points whence Tangents to two Circles have a given Ratio

Radical Centre of three Circles

99

99

Properties of system of Circles having common radical Axis

100

The limiting Points of the system

101

Properties of Circles cutting two Circles at right Angles, or at constant Angles

102

Locus of centre of Circle cutting three given Circles at equal Angles

To describe a Circle touching three given Circles (see also pp. 115, 130, 279)

Mr. Casey's Solution of this Problem.

110

113

Relation connecting common Tangents of four Circles touched by same fifth

Method of Inversion of Curves

Quantities unchanged by Inversion

Equation of Circle cutting three at right Angles (see pp. 128, 347)

Locus of Point such that Area of Triangle formed by feet of Perpendiculars

from it on sides of Triangle may be given

Equation of Tangent to circumscribing Circle at any vertex

Tangential Equation of circumscribing Circle

Conditions that general Equation should represent a Circle

Radical Axis of two Circles in Trilinear Co-ordinates

Equation of Circle inscribed in a Triangle

Its Tangential Equation

Equation of inscribed Circle derived from that of circumscribing

Length of Tangent to a Circle in Trilinear Co-ordinates

Feuerbach's theorem, that the four Circles which touch the sides of a Triangle

Tangential equation to Circles whose Centre and Radius are given

Distance between two Points expressed in Trilinear Co-ordinates

Determinant Notation

Determinant expressions for Equations of Circles through three Points, or

cutting three at right Angles

Relation connecting mutual distances of four points in a plane

Proof of Mr. Casey's theorems

CHAPTER X.

GENERAL EQUATION OF SECOND DEGREE.

Number of conditions which determine a Conic

Transformation to Parallel Axes of Equation of second Degree

131

132

Discussion of Quadratic which determines Points where Line meets a Conic

Harmonic Property of Polars (see also p. 284)

Polar properties of inscribed Quadrilateral (see also p. 307)

Equation of pair of Tangents from given Point to a Conic (see also p. 257)
Rectangles under segments of parallel chords in constant ratio to each other
Case where one of the lines meets the Curve at infinity

Condition that a given Line should touch a Conic (see also pp. 255, 328)
Locus of centre of Conic through four Points (see also pp. 243, 260, 290, 308)

CHAPTER XI.

CENTRAL EQUATIONS.

Sum of Squares of Reciprocals of Semi-diameters at right Angles is constant

Sum of Squares of conjugate Semi-diameters is constant

154

154

Ordinates of Ellipse in given ratio to those of concentric Circle

157

Equilateral Hyperbola: its Properties

161

CONJUGATE DIAMETERS: their Properties (see also p. 154)

Length of central Perpendicular on Tangent

164

Angle between Conjugate Diameters

164

Locus of intersection of Tangents which cut at right Angles (see also pp. 161,

258, 339).

166

Supplemental Chords

166

To construct a pair of Conjugate Diameters inclined at a given Angle

166

Relation between intercepts made by variable tangent on two parallel tangents

PAGE

NORMAL: its Properties

To draw a Normal through a given Point (see also p. 323)

Chord subtending a right Angle at any Point on Conic passes through a fixed

Point on Normal (see also pp. 259, 273)

168

169

170

Tangents at any Point equally inclined to Tangent to Confocal conic through the

Point

176

Angle subtended at the Focus by a Chord, bisected by Line joining Focus to its

Pole (see also pp. 244, 272)

177

Line joining Focus to Pole of a Focal Chord is perpendicular to that Chord

Polar Equation, Focus being Pole (see p. 309).

Origin of names Parabola, Hyperbola, and Ellipse (see also p. 316)

180

ASYMPTOTES: how found

180

Intercepts on Chord between Curve and Asymptotes are equal

181

Lines joining two fixed to variable Point make constant Intercept on Asymptote

Constant area cut off by tangent

Expression for Parameter of Parabola given by general Equation

ditto, given lengths of two Tangents and contained Angle (see also

p. 203)

Parabola the limit of the Ellipse when one Focus passes to infinity

Intercept on Axis by two Lines, equal to projection of distance between their

Poles

Locus of intersection of Tangents which cut at right Angles (see also 273, 339)

Angle between two Tangents half that between corresponding Focal Radii

Circle circumscribing Triangle formed by three Tangents passes through Focus

(see also pp. 203, 263, 273, 308)

If a Chord of a Conic pass through a fixed Point O, then tan PFO.tan P'FO is

constant (see also p. 319)

199

PAGE

Locus of intersection of Normals at extremities of a Focal Chord (see also p. 323) 200

Expression for angle between tangents to ellipse from any point

Radii Vectores through Foci have equal difference of Reciprocals

Examples on Parabola

Three Perpendiculars of Triangle formed by three Tangents intersect on

201

201

201

Directrix (see also pp. 236, 263, 278, 329)

201

Area of Triangle formed by three Tangents

201

Radius of Circle circumscribing an inscribed or circumscribing Triangle

Locus of intersection of Tangents which cut at a given Angle (see also pp.

202

Locus of intersection of Normals at the extremities of Chords passing through
a given Point (see also p. 326)
Given three Points on Equilateral Hyperbola, a fourth is given (see also pp. 278,
329)

Circle circumscribing any self-conjugate Triangle with respect to an Equilateral
Hyperbola passes through centre (see also p. 310)

Locus of intersection of Tangents which make a given Intercept on a given

Locus of centre, given four Tangents (see also pp. 243, 256, 327)

ditto, given three tangents and sum of squares of axes

Locus of Foci given four Points

ECCENTRIC ANGLE

Construction for Conjugate Diameters

Radius of Circle circumscribing an inscribed Triangle (see also p. 321)
Area of Triangle formed by three Tangents, or three Normals

Condition that Conics should be similar, and similarly placed
Properties of similar Conics

Condition that Conics should be similar, but not similarly placed

CONTACT OF CONICS

214

Conics having double contact

215

Osculating Circle defined

216

Expressions and construction for Radius of Curvature (see also pp. 223, 231,

356)

If a Circle intersect a Conic, chords of intersection are equally inclined to

the Axes (see also p. 223)

Condition that four Points of a Conic should lie on a Circle

Relation between three Points whose osculating Circles meet Conic again

in the same Point

217

Three values of k for which it represents right lines

Equation of Conic passing through five given Points

Equation of osculating Circle

Equations of Conics having double contact with each other

Every Parabola has a Tangent at infinity (see also p. 317)

221

PAGR

if concentric, touch at infinity

Similar and similarly placed Conics have common Points at infinity

All Circles have imaginary common Points at infinity (see also p. 313)
Form of Equation referred to a self-conjugate Triangle (see also p. 242)
Conics having same Focus have two imaginary common tangents (see also pp.
308, 340)

Method of finding Co-ordinates of Foci of given Conie (see also p. 340)

Relation between Perpendiculars from any point of Conie on sides of inscribed

quadrilateral

Anharmonic Property of Conics proved (see also pp. 240, 276, 306)

Extension of Property of Focus and Directrix

229

229

Result of substituting the Co-ordinates of any Point in the Equation of a Conic

Diameter of Circle circumscribing Triangle formed by two Tangents and their

Chord.

Property of Chords of Intersection of two Conics, each having double contact

with a third

Diagonals of inscribed and circumscribed Quadrilateral pass through the same

Point

231

If three Conics have each double contact with a fourth, their Chords of
Intersection intersect in threes

Brianchon's Theorem (see also pp. 268, 304)

If three Conics have a common Chord, their other Chords intersect in a Point

Pascal's Theorem (see also pp. 268, 289, 304, 307)

233

234

Steiner's Supplement to Pascal's Theorem (see also p. 360)

235

Circles circumscribing the Triangles formed by four lines meet in a Point

When five Lines are given, the five Points so found lie on a Circle

Given five Tangents to find their Points of Contact

235

235

236

Mac Laurin's Method of generating Conics (see also p. 287)

Given five Points on a Conic to construct it, find its centre, and draw Tangent at

any of the points

236

Equation referred to two Tangents and their Chord

237

Corresponding Chords of two Conics intersect on one of their Chords of Intersec-

tion (see also pp. 232, 234)

238

Locus of Vertex of Triangle whose sides touch a given Conic, and base Angles

move on fixed Lines (see also pp. 307, 336)

To inscribe in a Conic a Triangle whose sides pass through fixed Points (see

also pp. 261, 269, 289).

Generalizations of MacLaurin's Method of generating Conics (see also p. 288)

Anharmonic properties of Points and Tangents of a Conic (see pp. 229, 276, 306) 240

Anharmonic ratio of four Points on a Conic {abcd} = {a'b'c'd'}, if the Lines aa',

&c. meet in a Point; or if they touch a Conic having double contact

with the given one

239

Envelope of Chord joining corresponding Points of two homographic systems on

a Conic (see also p. 291)

Locus of Pole of a given Line with regard to a Conic passing through four fixed

points (see also pp. 148, 256, 260, 290)

or touching four right Lines (see also pp. 256, 265, 269, 309, 327)

243

Focal properties of Conics

244

Locus of Intersections of Tangents to a Parabola which cut at a given Angle (see

also pp. 207, 273)

245

Self-conjugate Triangle common to two Conics (see also pp. 335, 347)

245

when real, when imaginary

245