Conditions that general Equation may represent a Circle,

that two Circles may be concentric,

Co-ordinates of Points where a given Line meets a given Circle,

Imaginary Points, . . .

Condition that Circle should touch either Axis,

Equation of Tangent to a Circle at a given Point,

Condition that a Line should touch a Circle,

Equation of Polar of a Point with regard to a Circle,

Length of Tangent to a Circle,

Line cut harmonically by a Circle, Point, and its Polar,

Equation of pair of Tangents from a given Point to a Circle,

Circle through three Points,

71

72

73

74

75

76, 78

78

79

81

82

82

82

Condition that four Points should lie on a Circle, and its Geometrical meaning, 84

Polar Equation of a Circle,

Condition that intercept by Circle on a given Line may subtend a right Angle at a

given Point, .

88

If two Chords meet in a Point, Lines joining their extremities transversely meet on

its Polar,

Distances of two Points from the centre, proportional to the distance of each from

Polar of other,

Lines joining Vertices of Triangle to Vertices of Polar Triangle meet in a Point,

Expression of Co-ordinates of Point on Circle by auxiliary Angle,

Theorem of Hexagon circumscribing a Circle,

Problems where a variable Line always touches a Circle,

Examples on Circle solved by Polar Co-ordinates,

THE CIRCLE.-ABRIDGED NOTATION.

Equation of Circle circumscribing a Quadrilateral,

Equation of Circle circumscribing Triangle a, ẞ, y,

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

on sides of Triangle may be given,

Equation of Circle inscribed in a Triangle,

b

97

99

99

100

101

CHAPTER IX.

PROPERTIES OF TWO OR MORE CIRCLES.

PAGE.

Equation of radical Axis of two Circles,

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

Radical Centre of three Circles, .

Properties of system of Circles having common radical Axis,

The limit Points of the system,

Properties of Circles cutting two Circles at constant Angles,

Common Tangent to two Circles,

Centres of Similitude,

Axis of similitude,

112

To describe a Circle touching three given Circles (see also p. 266), .

114

118

To describe a Circle cutting three at given Angles,

Equation of Circle cutting three at right Angles (see note, p. 324).

CHAPTER X.

GENERAL EQUATION OF SECOND DEGREE.

Number of Conditions which determine a Conic,

Transformation to Parallel Axes of Equation of second Degree,

Discussion of Quadratic which determines Points where Line meets a Conic, .

119

120

120

122

123

Polar properties of inscribed Quadrilateral (see also p. 310),

Equation of pair of Tangents from given Point to a Conic (see also p. 269),

Rectangles under 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 p. 251),

Locus of centre of Conic through four Points (see also pp. 246, 285),

CENTRAL EQUATIONS.

Transformation of general Equation to the centre,

Functions of the Coefficients which are unaltered by transformation,

Sum of Squares of Reciprocals of Semidiameters at right Angles is constant,

Polar Equation of Ellipse, centre being Pole,

PAGE.

.

Ordinates of Ellipse in given ratio to those of concentric Circle,

Figure of Hyperbola,

146

147

Expression for Angle between two Tangents to a Conic,

CONJUGATE DIAMETERS: their Properties (see also p. 144),

Equilateral Hyperbola: its Properties, .

149

150

151

152

154

Length of central Perpendicular on Tangent,

154

Angle between Conjugate Diameters,

155

Locus of intersection of Tangents which cut at right Angles,

155

Supplemental Chords,

156

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

157

NORMAL: its Properties,

158

To draw a Normal through a given Point,

160

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

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

Angle subtended at the Focus by a Chord, bisected by Line joining Focus to its Pole, 168

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

168

Polar Equation: Focus being Pole,

Segments of Focal Chord have constant Harmonic Mean,

Origin of names Parabola, Hyperbola, and Ellipse (see also p. 305),

ASYMPTOTES: how found, .

Intercepts on Chord between Curve and Asymptotes are equal,

Lines joining two fixed to variable Point make constant Intercept on Asymptote,. 173

Transformation of the Equation to the form y2 = px,

176

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

178

Expression for Parameter of Parabola given by general Equation,

179

ditto, given lengths of two Tangents and contained Angle (see also p. 195), 180

Intercept on Axis by two Lines equal to projection of distance between their Poles, 181

Subnormal Constant,

183

Locus of foot of Perpendicular from Focus on Tangent,

185

PAGE.

185

186

Locus of intersection of Tangents which cut at right Angles (see also p. 261),
Angle between two Tangents half that between corresponding Focal Radii,
Circle circumscribing Triangle formed by three Tangents passes through Focus (see
also pp. 195, 249, 261),

Polar Equation of Parabola,

CHAPTER XIII.

EXAMPLES ON CONICS.

Loci,

Focal Properties,

Locus of Pole with respect to a series of Confocal Conics,

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

constant (see also p. 315),

189

190

191

191

Locus of intersection of Normals at extremities of a Focal Chord,

Three Perpendiculars of Triangle formed by three Tangents intersect on Di-

rectrix (see also p. 249),

193

Radius of Circle circumscribing an inscribed Triangle, .

193

Locus of intersection of Tangents which cut at a given Angle (see also p. 234), 194 -

Locus of foot of Perpendicular from Focus on Normal, .

Locus of intersection of Normals at the extremities of Chords passing through

a given Point,

194

Given three Points on Equilateral Hyperbola, a fourth is given,

195

Circle circumscribing any self-conjugate Triangle with respect to an Equilateral

Hyperbola passes through centre,

195

Locus of Intersection of Tangents which make a given Intercept on a given Tangent, 196

ECCENTRIC ANGLE,

Radius of Circle circumscribing an inscribed Triangle (see also p. 316), .

SIMILAR CONIC SECTIONS,

Condition that Conics should be similar, and similarly placed,

202

Properties of similar Conics,

203

Condition that Conics should be similar, but not similarly placed,

Expressions and construction for Radius of Curvature (for geometrical proofs

of these see p. 295),

207

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

208

Relation between three Points whose osculating Circles meet Conic again in

the same Point,

208

Co-ordinates of centre of Curvature,

209

Evolutes of Conics, .

210

Equation of Conic passing through five given Points,

Equations of Conics having double contact with each other,

Every Parabola has a Tangent at infinity (see also p. 306),

Similar and similarly placed Conics have common Points at infinity,

if concentric, touch at infinity,

PAGE.

211

212

213

213

214

all Circles have imaginary common Points at infinity, .

Anharmonic Property of Conics proved (see also p. 225),

Extension of Property of Focus and Directrix, ..

215

216

217

Result of substituting the Co-ordinates of any Point in the Equation of a Conic, . 217

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

a third, .

217

Diagonals of inscribed and circumscribed Quadrilateral pass through the same Point, 218

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

tact passes through a fixed Point,

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

intersect in threes,

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

Pascal's Theorem (see also p. 283, 309),

Steiner's Supplement to Pascal's Theorem (see also p. 317),.

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,
Locus of Vertex of Triangle whose sides touch a given Conic, and base Angles move
on fixed Lines (see also p. 309),

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

pp. 244, 284),

MacLaurin's Method of generating Conics (see also pp. 281, 283),

All Conics having same Foci have four imaginary common Tangents,

Locus of Intersection of Tangents to a Parabola which cut at a given Angle,

Solution of the Problem of finding Envelopes,

Equation of Conic having double contact with two given Conics,

Locus of a Point whence sum or difference of Tangents to two Circles is constant, 238
Equation of Conic touching four given Lines,

Locus of Centre in this case (see also p. 248),

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

238

239

240, 242