《A Treatise on The Higher Plane Curves:Intended As A Sequel To A Treatise on Conic Sections Second Ed》求取 ⇩

CHAPTER Ⅰ．COORDINATES1

Descriptive and metrical theorems1

General definition of trilinear coordinates2

Relation between point(1,1,1)and line x+y+z=04

Particular cases of trilinear coordinates5

circular coordinates6

LINE COORDINATES7

Their relation to trilinear coordinates8

Particular cases of line-coordinates9

Geometrical duality9

CHAPTER Ⅱ．GENERAL PROPERTIES OF ALGEBRAIC CURVES11

SECTION Ⅰ．NUMBER OF TERMS IN THE EQUATION11

All forms which are general must have as many independent constants as the general equation11

Number of terms in the general equation13

Number of points which determine a n-ic13

A single curve determined by these conditions14

In what cases this number of points fails to determine a n-ic15

If one less than this number of points be given,the curve passes through other fixed points16

If of intersections of two n-ic-s,np lie on a p-ic,the remainder lie on a(n-p)-ic16

Extension of Pascal＇s theorem17

Steiner＇s and Kirkman＇s theorems on the hexagon17

Theorems concerning the intersections of two curves18

SECTION Ⅱ．MULTIPLE POINTS AND TANGENTS20

Equation of tangent at origin21

The origin,a double point22

Three kinds of double points22

Their relation illustrated24

Triple points．their species25

Number of nodes equivalent to a multiple point26

Multiple point equivalent to how many conditions27

Limit to number of nodes on a proper curve27

Deficiency of a curve28

Fundamental property of unicursal curves29

Consideration of the case where the axis is a multiple tangent30

Stationary tangents and inflexions32

Two consecutive tangents coincide at a stationary tangent32

Correspondence of reciprocal singularities32

Curve crosses the tangent at an inflexion33

Measure of inclination of curve to the axis34

Points of undulation35

Examples36

Relation between points where tangents meet the curve again37

Equation of asymptotes,how formed38

SECTION Ⅲ．TRACING OF CURVES40

Newton＇s process for determining form of curve at a singular point44

Keratoid and ramphoid cusps46

SECTION Ⅳ．POLES AND POLARS47

Joachimsthal＇s method of determining points where a line meets a curve47

Polar Curves49

of origin49

Every right line has(n-1)2 poles49

Multiple points and cusps,how related to polar curves50

SECTION Ⅴ．GENERAL THEORY OF MULTIPLE POINTS AND TANGENTS50

All polars of point on curve touch at that point51

Points of contact of tangents from a point how determined51

Degree of reciprocal of a curve52

Effect of singularities on degree of reciprocal53

Discriminant of a curve54

If the first polar of A has a double point B,polar conic of B has a double point A54

Hessian and Steinerian defined54

Conditions for a cusp56

Number of points of inflexion57

how affected by multiple points58

Equation of system of tangents from a point59

Application to the case of a cubic60

Number of tangents from a multiple point61

SECTION Ⅵ．RECIPROCAL CURVES61

What singularities to be counted ordinary62

Plücker＇s equations63

Number of conditions,the same for curve and its reciprocal64

Deficiency,the same for both64

Cayley＇s modification of Plücker＇s equations64

CHAPTER Ⅲ．ENVELOPES65

Two forms in which problem of envelopes presents itself65

Envelope of curve whose equation contains a single parameter66

Envelope of a cos2θ+b sin2θ=c67

Equation of parallel to a conic68

Envelope of right line containing asingle parameter algebraically:its characteristics68

Envelope of curve with related parameters69

Method of indeterminate multipliers70

Envelope of curve whose equation contains independent parameters71

RECIPROCAL CURVES72

Method of finding equation of reciprocal73

Reciprocal of a cubic74

Symbolical form of equation of reciprocal74

Reciprocal of a quartic75

Equation of system of tangents from a point75

Equation of reciprocal in polar coordinates76

CONDITION OE CONTACT[tact-invariant]of two curves76

Its order in the coefficients of each curve77

EVOLUTES79

Defined as envelope of normals79

Coordinates of centre of curvature81

General expression for radius of curvature83

Length of are of evolute84

Radius of curvature in polar coordinates84

Evolutes of curves given by tangential equation86

Quasi-normals and quasi-evolutes86

Quasi-evolute of conic87

General form of equation of quasi-normal88

Quasi-evolute when the absolute is a conic89

Normal of a point at infinity90

Characteristics of evolute90

Deficiency of evolute93

Condition that four consecutive points on a curve should be concircular93

[The condition of Art．114 is immediately obtained by equating to nothing the differential of the value of the radius of curvature(Art．102)subject to the condition Ldx+Mdy=0．]93

CAUSTICS95

Caustic by reflection of a circle95

Quetelet＇s method95

Pedal of a curve96

Caustic by refraction of right line and circle97

Evolute of Cartesian98

PARALLEL OURVES AND NEGATIVE PEDALS98

Cayley＇s formul？ for characteristics of parallel curves99

Problem of negative pedals102

Roberts＇s method102

Method of inversion103

Characteristics of inverse curve and of pedal103

Caustic by reflexion of parabola104

Negative Pedal of central and focal ellipse104

CHAPTER Ⅳ．METRICAL PROPERTIES105

Newton＇s theorem of constant ratio of rectangles105

Carnot＇s theorem of transversals106

Three inflexions of cubics lie on a right line107

Two kinds of bitangents of quartic108

DIAMETERS109

Newton＇s generalization of notion of diameters109

Theorem concerning intercept made between curves and asymptotes110

Curvilinear diameters110

Centres112

POLES AND POLARS112

Cotes＇s theorem of harmonic means of radii112

Polar curves113

Polars of points at infinity113

Pole of line at infinity114

Mac Laurin＇s extension of Newton＇s theorem114

FOCI115

Pole of line at infinity with respect to curve of nth class116

Its metrical property:centre of mean distances of contacts of parallel tangents116

General definition of foci117

Number of foci possessed by a curve118

Antipoints119

Coordinates of foci how found119

Locus of a point the tangents from which make with a fixed line angles whose sum is constant120

Every focus of a curye is a focus of its evolute121

Theorems concerning focal perpendiculars on tangents121

concerning focal distances of point on curve122

concerning angles between focal radii and tangent123

Locus of double focus of circular curve determined by N-3 points124

Locus of focus of curve of nth class determined by N-1 tangents124

Miquel＇s theorem as to foci of parabolas touching 4 of 5 given lines125

CHAPTER Ⅴ．CUBICS126

General division of cubics126

SECTION Ⅰ．INTERSECTION OF A CUBIC WITH OTHER OURVES127

Tangential of a point and satellite of a line defined127

Asymptotes meet curve in 3 collinear finite points128

Three points of inflextion lie on a right line128

Four points of contact of tangents from any point on curve,how related129

Maclaurin＇s theory of correspondence of points on a cubic130

Coresidual of four points on a cubic131

To draw a conic having four-point contact and elsewhere touching a cubic132

Conic of 5-point contact,how constructed132

Sextactic points on cubic,how found132

Sylvester＇s theory of residuation133

Two coresidual points must coincide134

Two systems coresidual to the same are coresidual to each other135

Analogues in theory of cubics to anharmonic theorems of conics137

Locus of common vertex of two triangles whose bases are given and vertical angles eqnal,or having a given difference139

SECTION Ⅱ．POLES AND POLARS139

Construction for polar of a point with respect to a triangle140

Construction by the ruler for polar of a point with respect to a cubic140

Anharmonic ratio constant of pencil of four tangents from any point on a cubic141

Two classes of non-singular cubics142

Sixteen foci of a circular cubic lie on four circles142

Chords through a point on cubic cut harmonically by polar conic143

Harmonic polar of point of inflexion143

All cubics through nine points of inflexion have these for inflexions145

Correspondence of two points on Hessian146

Steinerian of a cubic identical with its Hessian147

Cayleyan,different definitions of148

Polar line with respect to cubic of point on Hessian touches Hessian149

Common tangents of cubic and Hessian149

Stationary tangents touch the Hessian149

Tangents to Hessian at corresponding points meet on Hessian150

Three cubics have common Hessian150

Rule for finding point of contact of any tangent to Cayleyan151

Points of contact of stationary tangents with Cayleyan152

Coordinates of tangential of Point on cubic,how found153

Polar conic of line with respect to cubic153

How related to triangle formed by tangents where line meets cubic154

Double points,how situated with regard to polar conics of lines154

Polar conic of line infinity155

Another method of obtaining tangential equation of cubic155

Polar conic of a line when reduces to a point155

Points,whose polar with respect to two cubics are the same156

Critic centres of system of cubics157

Locus of nodes of nodal cubics through seven points157

Plücker＇s classification of cubics158

SECTION Ⅲ．CLASSIFICATION OF CUBICS159

Every cubic may be projected into one of five divergent parabolas and into one of five central cubics161

Classification of cubic cones162

No real tangents can be drawn from oval164

Unipartite and bipartite cubics165

Species of cubics166

Newton＇s method of reducing the general equation174

Plücker＇s groups175

SECTION Ⅳ．UNICURSAL CUBICS176

Inscription of polygons in unicursal cubics178

Cissoid,its properties179

Acnodal cubic has real inflexions,crunodal imaginaty181

Construction for acnode given three inflexional tangents181

SECTION Ⅴ．INVARIANTS AND COVARIANTS OF CUBICS182

Canonical form of cubic182

Notation for general equation182

General equation of Hessian and of Cayleyan183

Invariant S,and its symbolical form184

Invariant T185

General equation of reciprocal186

Calculation of invariants by the differential equation187

Discriminant expressed in terms of fundamental invariants189

Hessian of λU+μH189

Condition that general equation should represent three right lines190

Reduction of general equation to canonical form191

Expression of discriminant in terms of fundamental invariants192

Of anharmonic ratio of four tangents from any point on curve192

Covariant cubics expressed in form λU+μH193

Sextic covariants194

The skew covariant195

Equation of nine inflexional tangents196

Equation of Cayleyan in point coordinates196

Identical equation in theory of cubics198

Conic through five consecutive points on cubic200

Equation expressed in four line coordinates202

Condition that cubic should represent conic and line202

Discriminant of cubic expressed as determinant203

Hessian of PU and of UV204

CHAPTER Ⅵ．QUARTICS206

Genera of quartics206

Special forms of quartics207

Illustration of the different forms208

Distinction of real and imaginary210

Flecnodes and biflecnodes210

Quartic may have four real points of undulation211

Quartics may be quadripartite212

Inflexions of quartics,how many real213

Classification of quartics in respect of their infinite branches213

THE BITANGENTS214

Discussion of equation UW=V2215

There are 315 conics passing through eight contacts of bitangents218

Scheme of these conics221

Hesse＇s algorithm for the bitangents221

Geiser＇s method of connecting bitangents with solid geometry222

Cayley＇s rule of bifid substitution223

Bitangents whose contacts lie on a cubic225

Aronhold＇s discussion of the bitangents225

From 7 bitangents the rest can be found by linear constructions227

Aronhold＇s algebraic investigation229

BINODAL AND BICIRCULAR QUARTICS231

Tangents from nodes of a binodal are homographic232

Foci of bicircular quartic lie on four circles233

Casey＇s generation of bicircular quartics234

Two classes of bicircular quartics236

Relations connecting focal distances of point on bicircular237

Confocal bicirculars cut at right angles288

Hart＇s investigation240

Cartesians241

The limacon and the cardioide242

Focal properties obtained by inversion242

Inscription of polygons in binodal quartics243

UNICURSAL QUARTICS244

Correspondence between conics and trinodal quartics245

Tangents at or from nodes touch the same conic247

Tacnodal and oscnodal quartics249

Triple points250

INVARIANTS AND COVARIANTS OF QUARTICS251

General quartic cannot be reduced to sum of five fourth powers252

Covariant quartics256

Examination of special case257

Covariant conics261

CHAPTER Ⅶ．TRANSCENDENTAL CURVES263

The cycloid263

Geometric investigation of its properties264

Epi-cycloids and epi-trochoids266

Their evolutes are similar curves269

Examples of special cases270

Limacon generated as epi-cycloid270

Steiner＇s envelope271

Reciprocal of epicycloid271

Radius of curvature of roulettes272

Trigonometric curves273

Logarithmic curves274

Catenary275

Tractrix and syntractrix277

Gurves of pursuit278

Involute of circle278

Spirals279

CHAPTER Ⅷ．TRANSFORMATION OF CURVES282

LINEAR TRANSFORMATION283

Anharmonic ratio unaltered by linear transformation284

Three points unaltered by linear transformation285

Projective transformation286

Homographic transformation may be reduced to projection287

INTERCHANGE OF LINE AND POINT COORDINATES289

Metbod of skew reciprocals291

Skew reciprocals reduceable to ordinary reciprocals294

QUADRIC TRANSFORMATIONS296

Inversion,a case of quadric transformation298

Applications of method of inversion300

RATIONAL TRANSFORMATION301

Roberts＇s transformation301

Cremona＇s rational transformation304

If three curves have common point their Jacobian passes through it307

Deficiency unaltered by Cremona transformation309

Every Cremona transformation may be reduced to a succession of quadric transformations310

TRANSFORMATION OF A GIVEN CURVE312

Rational transformation between two curves312

Deficiency unaltered by rational transformation314

Transformation,so that the order of the transformed curve may be as low as possible316

Expression of coordinates by means of elliptic functions when D=1317

and by means of hyper-elliptic functions when D=2318

Theorem of constant deficiency derived from theory of elimination319

CORRESPONDENCE OF POINTS ON A CURVE319

Collinear correspondence of points319

Correspondence on a unicursal curve320

Number of united points321

Correspondence on curves in general322

Inscription of polygons in conics325

in cubits326

CHAPTER Ⅸ．GENERAL THEORY OF CURVES328

Cayley＇s method of solving the general problem of bitangents329

Order of bitangential curve330

Hesse＇s reduction of bitangential equation332

Bitangential of a quartic337

Formation of equation of tangential curve340

Application to quartic344

POLES AND POLARS345

Jacobian properties of345

Steiner＇s theorems on systems of curves348

Tact-invariants349

Discriminant of discriminant of λu+μv,and of λu+μv+νw349

Condition for point of undulation350

For coincidence of double and stationary tangent350

Steinerian of a curve351

Its characteristics352

The Cayleyan or Steiner-Hessian352

Its characteristics353

Generalization of the theory353

OSCULATING CONICS356

Aberrancy of curvature356

Investigation of conic of 5-point contact358

Determination of number of sextactic points360

SYSTEMS OF CURVES360

Chasles＇method361

Characteristics of systems of conics362

Number of conics which touch five given curves363

Zeutheu＇s method365

Degenerate curves365

Cayley＇s table of results368

Number of conics satisfying five conditions of contact with other curves370

Professor Cayley＇s note on degenerate forms of curves371