PAEN

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

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

Criterion whether a Point be within or without a Conic

249

250

Discriminant of Tangential Equation

250

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 252

Malfatti's problem

Tangent and Polar of a Point with regard to a Conic given by the general Equation 254

DISCRIMINANTS, defined; discriminant of a Conic found (see also pp. 72, 144,

Condition that a Line should tonch a Conic (see also pp. 147, 328)

255

Hearn's Method of finding Locus of Centre of a Conic, four conditions being given 256

Equation of pair of Tangents through a given Point (see also p. 144)

257

Property of Angles of a circumscribing Hexagon (see also p. 277)

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

259

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

fixed Point

Locus of the latter Point when Point on Curve varies

259

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

Point not on Curve

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

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

259

260

260

260

Equation of Asymptotes of a Conic given by general Equation (see also p. 328)

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

260

261

261

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

pp. 239, 269, 289)

Co-ordinates of Focus of a Conic given three tangents (see also pp. 228, 340)

Directrix of Parabola passes through intersection of Perpendiculars of circum-

PAGE

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

Degree of Polar Reciprocal in general

Pascal's Theorem and Brianchon's mutually reciprocal

Polar of one Circle with regard to another

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

given one

270

Reciprocation of Theorems concerning Angles at Focus

272

Envelope of Asymptotes of Hyperbolas having same Focus and Directrix

Reciprocals of equal Circles have same Parameter

273

274

Relation between Perpendiculars on Tangent from Vertices of circumscribing

Quadrilateral

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. 307)

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

Asymptote

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

Anharmonic property of Tangents to Parabola

286

287

How any Tangent cuts two Parallel Tangents

287

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

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

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

Conic (see also p. 345)

Envelope of Line joining corresponding Points of two Homographic Systems

Criterion whether two Systems of Points be Homographic

Foci, how found when two Pairs of corresponding Points are given

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

293

294

294

295

296

297

298

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.

PAGE

298

299

301

301

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

Projective Properties of a Quadrilateral

Any two Conics may be projected into Circles

Projective proof of Carnot's Theorem (see also p. 277)

Projections of Properties concerning Foci

The six Vertices of two Triangles circumscribing a Conic, lie on the same Conic,
(see also p. 331)

Projections of Properties concerning Right Angles

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

The six Vertices of two self-conjugate Triangles lie on same Conic (see also p. 329)
Chord of a Conic passes through a fixed. Point, if the Angle it subtends at a
fixed Point in Curve, has fixed Bisector

Projections of Theorems concerning Angles in general

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

fixed Tangents

308

308

309

310

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

Line passes to Infinity

318

Orthogonal Projection

Determination of Focus of Section of a right Cone

Locus of Vertices of right Cones from which a given Conie may be cut
Method of deducing properties of Plane Curves from Spherical

Radius of Circle circumscribing inscribed Triangle

321

CHAPTER XVIII.

INVARIANTS AND COVARIANTS.

Equation of Chords of Intersection of two Conics

322

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

through a given Point

323

Condition that two Conics should touch

324

Criterion whether Conies intersect in two real and two imaginary Points or not 325

Equation of Curve parallel to a Conic

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

Criterion whether six lines touch the same conic

PAGE

Equation of pair of Tangents whose Chord is a given Line

327

Equation of Asymptotes of Conic given by Trilinear Equation

328

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

inscribed or circumscribed about another

328

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

329

Circle circumscribing self-conjugate Triangle cuts the director circle orthogonally 329

Centre of Circle inscribed in self-conjugate Triangle of equilateral Hyperbola

lies on Curve

329

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

circumscribed about another

How to find equations of Sides of self-conjugate Triangle common to two Conics
(see also p. 347)

335

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

337

General Tangential Equation of two Circular Points at infinity

Condition for Equilateral Hyperbola and for Parabola in Trilinear Co-ordinates

Every line through an imaginary Circular Point, perpendicular to itself

General Equation of Director Circle

Equation of reciprocal of two Conics having double contact

Condition that they should touch each other

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

other such Conics

343

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

touching three Lines, are touched by the same Conics

345

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

General equation of Jacobian

346

346

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

346

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

Conics

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

to two coincident Lines

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

347

348

348

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

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

CHAPTER XIX.

THE METHOD OF INFINITESIMALS.

Direction of Tangents of Conics

Determination of Areas of Conics

PAGE

352

353

Tangent to any Conic cuts off constant Area from similar and concentric Conic. 354

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

where met by its Envelope

Determination of Radii of Curvature

355

355

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

on Confocal Ellipse

357

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

Fagnani's Theorem

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

on Confocal Conics

359

NOTES.

Theorems on complete Figure formed by six Points on a Conic

On Mr. Casey's form of the Equation of a Conic having double contact with a

given one and touching three others

On the Problem to describe a Conic under five conditions

On systems of Conics satisfying four Conditions

366

367

368

ERRATA.

p. 129, last line, The determinants should be connected by the sign of multiplica-
tion, not addition.

p. 130, lines 1, 23, the words "rows" and "columns" should be interchanged.
34

p. Dor, at end of Art. 389, add "Mr. Burnside has expressed this invariant T in
terms of the elementary invariants :

T= 02123 −4 (1220133 + 211233 +03110322) + 12§ ;

where is the invariant, corresponding to 123, formed with the coefficients of the
reciprocal systems."