Focal properties of Conics (see also pp. 267, 277, 281, 321, 839)

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

Self-conjugate Triangle common to two Conics (see also pp. 348, 361).

when real, when imaginary

Locus of Vertex of a Triangle inscribed in one Conic, and whose sides touch one

another (see also p. 349)

Formation of Trilinear Equation of a Conic from Tangential, and vice versa

Criterion whether a Point be within or without a Conic

260

261

Discriminant of Tangential Equation

262

Given two points of a Conic having double contact with a third, its Chord of

contact passes through one or two fixed Points

Equation of a Conic having double contact with two given Conics

Locus of a Point whence sum or difference of Tangents to two Circles is constant 263

Malfatti's problem

262

262

263

Tangent and Polar of a Point with regard to a Conic given by the general

Equation

DISCRIMINANTS defined; discriminant of a Conic found (see also pp. 72, 149,

Condition that a Line should touch a Conic (see also pp. 152, 340)

266

.

Condition that two Lines should be conjugate

267

Hearn's method of finding Locus of Centre of a Conic, four conditions being

given

Equation of pair of Tangents through a given Point (see also p. 149)

269

Property of Angles of a circumscribing Hexagon (see also p. 289)

270

Test whether three pairs of Lines touch the same Conic .

270

Equations of Lines joining to a given Point intersections of two Curves

Chord which subtends a right Angle at a fixed Point on Conic passes through a

Envelope of Chord subtending constant Angle, or subtending right Angle at

Given four Points, Polar of a fixed Point passes through fixed Point

Locus of intersection of corresponding Lines of two homographic pencils

Envelope of Pole of a given Point with regard to a Conic having double contact

with two given ones

271

271

271

Anharmonic Ratio of four Points the same as that of their Polars

Equation of Asymptotes of a Conic given by general Equation (see also p. 340).

Given three Points on Conic, and Point on one Asymptote, Envelope of other

Locus of Vertex of a Triangle whose sides pass through fixed Points, and base

Angles move along Conics

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

also pp. 250, 281, 301)

Coordinates of Focus of a Conic given three Tangents (see also pp. 239, 353) 275

Directrix of Parabola passes through intersection of Perpendiculars of circum-

scribing Triangle (see also pp. 212, 247, 290, 342)

Locus of Focus given four Tangents (see also p. 277)

Director Circles of Conics touching four Lines have a common radical Axis

Circles having for Diameters Diagonals of complete Quadrilateral have common

radical Axis

277

277

Locus of Point where Tangent meeting two fixed Tangents is cut in a given ratio 277

Degree of Polar Reciprocal in general

279

Radical Axes and Centres of Similitude of Conics having double contact with a

given one

282

Polar of one Circle with regard to another

283

Reciprocation of Theorems concerning Angles at Focus

284

Envelope of Asymptotes of Hyperbolas having same Focus and Directrix

Reciprocals of equal Circles have same Parameter

Tangential Equation of Reciprocal Conic

Trilinear Equation given Focus and either three Points or three Tangents

Reciprocation of Anharmonic Properties

Carnot's Theorem respecting Triangle cut by Conic (see also p. 319)

Reciprocal, when Ellipse, Hyperbola, or Parabola; when Equilateral Hyperbola

Axes of Reciprocal, how found

Reciprocal of Properties of Confocal Conics

To describe a Circle touching three given Circles

How to form Equation of Reciprocal

Asymptotes together with two Conjugate Diameters form Harmonic Pencil

Lines from two fixed Points to a variable Point, how cut any Parallel to

Parallels to Asymptotes through any Point on Curve, how cut any Diameter

298

Anharmonic Property of Tangents to Parabola

299

How any Tangent cuts two Parallel Tangents .

299

Proof, by Anharmonic Properties, of Mac Laurin's Method of Generating Conics,

of Newton's mode of Generation

299

Chasles's extension of these Theorems

300

To inscribe in a Conic a Polygon whose sides pass through fixed Points

301

To describe a Conic touching three Lines and having double contact with a given

Conic (see also p. 359)

301

Anharmonic proof of Pascal's Theorem

301

of Locus of Centre, when four Points are given

Envelope of Line joining corresponding Points of two Homographic Systems

Criterion whether two Systems of Points be Homographic (see also p. 383)

302

302

304

Analytic condition that four Points should form a Harmonic System

Locus of Point whence Tangents to two Conics form a Harmonic Pencil (see

Condition that Line should be cut Harmonically by two Conics

INVOLUTION

Property of Centre

of Foci

PAGE

305

310

310

311

313

313

Foci, how found when two Pairs of corresponding Points are given
Condition that six Points or Lines should form a system in Involution
System of Conics through four Points cut any Transversal in Involution
System of Conics touching four Lines, when cut a Transversal in Involution
Proof by Involution of Feuerbach's Theorem concerning the Circle through
middle Points of Sides of Triangle

CHAPTER XVII.

THE METHOD OF PROJECTION.

All Points at Infinity may be regarded as lying in a right Line

The six Vertices of two Triangles circumscribing a Conic, lie on the same Conic,

(see also p. 343)

fixed Point in Curve, has fixed Bisector

Locus of Pole of a Line with regard to a system of Confocal Conics

322

The six Vertices of two self-conjugate Triangles lie on same Conic (see also p. 341) 322
Chord of a Conic passes through a fixed Point, if the Angle it subtends at a

Projections of Theorems concerning Angles in general

Locus of Point cutting in given ratio intercept of variable Tangent between two

fixed Tangents

Proof that any Conic may be projected so as to become a circle, while a given

Line passes to Infinity

Locus of Vertices of right Cones from which a given Conic may be cut

Method of deducing properties of Plane Curves from Spherical

Locus of Intersection of Normals to a Conic at the extremities of Chords passing

through a given Point

335

Condition that two Conics should touch

336

PAGE

Criterion whether Conics intersect in two real and two imaginary Points or not 337

Equation of Curve parallel to a Conic

Equation of Evolute of a Conic

Meaning of the Invariants when one Coric is a pair of Lines

Criterion whether six lines touch the same conic

Equation of pair of Tangents whose Chord is a given Line

337

340

340

341

Equation of Asymptotes of Conic given by Trilinear Equation

Condition that a Triangle self-conjugate with regard to one Conic should be

inscribed or circumscribed about another

Six vertices of two self-conjugate Triangles lie on a Conic

Circle circumscribing self-conjugate Triangle cuts the director circle orthogonally 341

Centre of Circle inscribed in self-conjugate Triangle of equilateral Hyperbola
lies on Curve

Locus of intersection of Perpendiculars of Triangle inscribed in one Conic and

circumscribed about another

Condition that such a Triangle should be possible

Tangential equation of four Points common to two Conics

Equation of four common Tangents

How to find equations of Sides of self-conjugate Triangle common to two Conics

(see also p. 347)

Envelope of Base of Triangle inscribed in one Conic, two of whose sides touch

another

Locus of free Vertex of a Polygon all whose sides touch one Conic, and all
whose Vertices but one move on another
Condition that Lines joining to opposite vertices, Points where Conic meets Triangle
of reference should form two sets of three meeting in a Point

Every line through an imaginary Circular Point, perpendicular to itself

Condition for Equilateral Hyperbola and for Parabola in Trilinear Coordinates

General Tangential Equation of two Circular Points at infinity

General Equation of Director Circle

Equation of Directrix of Parabola given by Trilinear Equation

352

To draw a Conic having double contact with a given one, and touching three

other such Conics

356

Four Conics having double contact with S, and passing through three Points, or

touching three Lines, are touched by the same Conics

359

Condition that three Conics should have double contact with the same conic

Jacobian of a system of three Conics

Lines joining corresponding Points cut in involution by the Conics

To draw a Conic through four Points to touch a given Conic

361

Jacobian of three Conics having two Points common, or one of which reduces

To form the equation of the sides of self-conjugate Triangle common to two

Conics

362

PAGE

Area of common conjugate Triangle of two Conics

Mixed Concomitants

Condition that a line should be cut in involution by three Conics

Invariants of a system of three Conics

Condition that they should have a common Point

Condition that XU + μV + vW can in any case be a perfect Square

Three Conics derived from a single Cubic, method of forming its Equation.

CHAPTER XIX.

Tangent to any Conic cuts off constant Area from similar and concentric Conic .

Line which cuts off from a Curve constant Arc, or which is of a constant length

where met by its Envelope

373

374

Excess of sum of two Tangents over included Arc, constant when Vertex moves

on Confocal Ellipse

877

Difference of Arc and Tangent, constant from any Point on Confocal Hyperbola 377

Fagnani's Theorem

378

Locus of Vertex of Polygon circumscribing a Conic, when other Vertices move

on Confocal Conics

378

NOTES.

Theorems on complete Figure formed by six Points on a Conic

On systems of Tangential Coordinates

Expression of the Coordinates of a Point on a Conic by a single Parameter.

On the Problem to describe a Conic under five conditions

On systems of Conics satisfying four Conditions

Miscellaneous Notes