A Treatise on The Higher Plane Curves:Intended As A Sequel To A Treatise on Conic Sections Second EdPDF电子书下载

CHAPTER Ⅰ．COORDINATES 1

Descriptive and metrical theorems 1

General definition of trilinear coordinates 2

Relation between point(1,1,1)and line x+y+z=0 4

Particular cases of trilinear coordinates 5

circular coordinates 6

LINE COORDINATES 7

Their relation to trilinear coordinates 8

Particular cases of line-coordinates 9

Geometrical duality 9

CHAPTER Ⅱ．GENERAL PROPERTIES OF ALGEBRAIC CURVES 11

SECTION Ⅰ．NUMBER OF TERMS IN THE EQUATION 11

All forms which are general must have as many independent constants as the general equation 11

Number of terms in the general equation 13

Number of points which determine a n-ic 13

A single curve determined by these conditions 14

In what cases this number of points fails to determine a n-ic 15

If one less than this number of points be given,the curve passes through other fixed points 16

If of intersections of two n-ic-s,np lie on a p-ic,the remainder lie on a(n-p)-ic 16

Extension of Pascal＇s theorem 17

Steiner＇s and Kirkman＇s theorems on the hexagon 17

Theorems concerning the intersections of two curves 18

SECTION Ⅱ．MULTIPLE POINTS AND TANGENTS 20

Equation of tangent at origin 21

The origin,a double point 22

Three kinds of double points 22

Their relation illustrated 24

Triple points．their species 25

Number of nodes equivalent to a multiple point 26

Multiple point equivalent to how many conditions 27

Limit to number of nodes on a proper curve 27

Deficiency of a curve 28

Fundamental property of unicursal curves 29

Consideration of the case where the axis is a multiple tangent 30

Stationary tangents and inflexions 32

Two consecutive tangents coincide at a stationary tangent 32

Correspondence of reciprocal singularities 32

Curve crosses the tangent at an inflexion 33

Measure of inclination of curve to the axis 34

Points of undulation 35

Examples 36

Relation between points where tangents meet the curve again 37

Equation of asymptotes,how formed 38

SECTION Ⅲ．TRACING OF CURVES 40

Newton＇s process for determining form of curve at a singular point 44

Keratoid and ramphoid cusps 46

SECTION Ⅳ．POLES AND POLARS 47

Joachimsthal＇s method of determining points where a line meets a curve 47

Polar Curves 49

of origin 49

Every right line has(n-1)2 poles 49

Multiple points and cusps,how related to polar curves 50

SECTION Ⅴ．GENERAL THEORY OF MULTIPLE POINTS AND TANGENTS 50

All polars of point on curve touch at that point 51

Points of contact of tangents from a point how determined 51

Degree of reciprocal of a curve 52

Effect of singularities on degree of reciprocal 53

Discriminant of a curve 54

If the first polar of A has a double point B,polar conic of B has a double point A 54

Hessian and Steinerian defined 54

Conditions for a cusp 56

Number of points of inflexion 57

how affected by multiple points 58

Equation of system of tangents from a point 59

Application to the case of a cubic 60

Number of tangents from a multiple point 61

SECTION Ⅵ．RECIPROCAL CURVES 61

What singularities to be counted ordinary 62

Plücker＇s equations 63

Number of conditions,the same for curve and its reciprocal 64

Deficiency,the same for both 64

Cayley＇s modification of Plücker＇s equations 64

CHAPTER Ⅲ．ENVELOPES 65

Two forms in which problem of envelopes presents itself 65

Envelope of curve whose equation contains a single parameter 66

Envelope of a cos2θ+b sin2θ=c 67

Equation of parallel to a conic 68

Envelope of right line containing asingle parameter algebraically:its characteristics 68

Envelope of curve with related parameters 69

Method of indeterminate multipliers 70

Envelope of curve whose equation contains independent parameters 71

RECIPROCAL CURVES 72

Method of finding equation of reciprocal 73

Reciprocal of a cubic 74

Symbolical form of equation of reciprocal 74

Reciprocal of a quartic 75

Equation of system of tangents from a point 75

Equation of reciprocal in polar coordinates 76

CONDITION OE CONTACT[tact-invariant]of two curves 76

Its order in the coefficients of each curve 77

EVOLUTES 79

Defined as envelope of normals 79

Coordinates of centre of curvature 81

General expression for radius of curvature 83

Length of are of evolute 84

Radius of curvature in polar coordinates 84

Evolutes of curves given by tangential equation 86

Quasi-normals and quasi-evolutes 86

Quasi-evolute of conic 87

General form of equation of quasi-normal 88

Quasi-evolute when the absolute is a conic 89

Normal of a point at infinity 90

Characteristics of evolute 90

Deficiency of evolute 93

Condition that four consecutive points on a curve should be concircular 93

[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

CAUSTICS 95

Caustic by reflection of a circle 95

Quetelet＇s method 95

Pedal of a curve 96

Caustic by refraction of right line and circle 97

Evolute of Cartesian 98

PARALLEL OURVES AND NEGATIVE PEDALS 98

Cayley＇s formul？ for characteristics of parallel curves 99

Problem of negative pedals 102

Roberts＇s method 102

Method of inversion 103

Characteristics of inverse curve and of pedal 103

Caustic by reflexion of parabola 104

Negative Pedal of central and focal ellipse 104

CHAPTER Ⅳ．METRICAL PROPERTIES 105

Newton＇s theorem of constant ratio of rectangles 105

Carnot＇s theorem of transversals 106

Three inflexions of cubics lie on a right line 107

Two kinds of bitangents of quartic 108

DIAMETERS 109

Newton＇s generalization of notion of diameters 109

Theorem concerning intercept made between curves and asymptotes 110

Curvilinear diameters 110

Centres 112

POLES AND POLARS 112

Cotes＇s theorem of harmonic means of radii 112

Polar curves 113

Polars of points at infinity 113

Pole of line at infinity 114

Mac Laurin＇s extension of Newton＇s theorem 114

FOCI 115

Pole of line at infinity with respect to curve of nth class 116

Its metrical property:centre of mean distances of contacts of parallel tangents 116

General definition of foci 117

Number of foci possessed by a curve 118

Antipoints 119

Coordinates of foci how found 119

Locus of a point the tangents from which make with a fixed line angles whose sum is constant 120

Every focus of a curye is a focus of its evolute 121

Theorems concerning focal perpendiculars on tangents 121

concerning focal distances of point on curve 122

concerning angles between focal radii and tangent 123

Locus of double focus of circular curve determined by N-3 points 124

Locus of focus of curve of nth class determined by N-1 tangents 124

Miquel＇s theorem as to foci of parabolas touching 4 of 5 given lines 125

CHAPTER Ⅴ．CUBICS 126

General division of cubics 126

SECTION Ⅰ．INTERSECTION OF A CUBIC WITH OTHER OURVES 127

Tangential of a point and satellite of a line defined 127

Asymptotes meet curve in 3 collinear finite points 128

Three points of inflextion lie on a right line 128

Four points of contact of tangents from any point on curve,how related 129

Maclaurin＇s theory of correspondence of points on a cubic 130

Coresidual of four points on a cubic 131

To draw a conic having four-point contact and elsewhere touching a cubic 132

Conic of 5-point contact,how constructed 132

Sextactic points on cubic,how found 132

Sylvester＇s theory of residuation 133

Two coresidual points must coincide 134

Two systems coresidual to the same are coresidual to each other 135

Analogues in theory of cubics to anharmonic theorems of conics 137

Locus of common vertex of two triangles whose bases are given and vertical angles eqnal,or having a given difference 139

SECTION Ⅱ．POLES AND POLARS 139

Construction for polar of a point with respect to a triangle 140

Construction by the ruler for polar of a point with respect to a cubic 140

Anharmonic ratio constant of pencil of four tangents from any point on a cubic 141

Two classes of non-singular cubics 142

Sixteen foci of a circular cubic lie on four circles 142

Chords through a point on cubic cut harmonically by polar conic 143

Harmonic polar of point of inflexion 143

All cubics through nine points of inflexion have these for inflexions 145

Correspondence of two points on Hessian 146

Steinerian of a cubic identical with its Hessian 147

Cayleyan,different definitions of 148

Polar line with respect to cubic of point on Hessian touches Hessian 149

Common tangents of cubic and Hessian 149

Stationary tangents touch the Hessian 149

Tangents to Hessian at corresponding points meet on Hessian 150

Three cubics have common Hessian 150

Rule for finding point of contact of any tangent to Cayleyan 151

Points of contact of stationary tangents with Cayleyan 152

Coordinates of tangential of Point on cubic,how found 153

Polar conic of line with respect to cubic 153

How related to triangle formed by tangents where line meets cubic 154

Double points,how situated with regard to polar conics of lines 154

Polar conic of line infinity 155

Another method of obtaining tangential equation of cubic 155

Polar conic of a line when reduces to a point 155

Points,whose polar with respect to two cubics are the same 156

Critic centres of system of cubics 157

Locus of nodes of nodal cubics through seven points 157

Plücker＇s classification of cubics 158

SECTION Ⅲ．CLASSIFICATION OF CUBICS 159

Every cubic may be projected into one of five divergent parabolas and into one of five central cubics 161

Classification of cubic cones 162

No real tangents can be drawn from oval 164

Unipartite and bipartite cubics 165

Species of cubics 166

Newton＇s method of reducing the general equation 174

Plücker＇s groups 175

SECTION Ⅳ．UNICURSAL CUBICS 176

Inscription of polygons in unicursal cubics 178

Cissoid,its properties 179

Acnodal cubic has real inflexions,crunodal imaginaty 181

Construction for acnode given three inflexional tangents 181

SECTION Ⅴ．INVARIANTS AND COVARIANTS OF CUBICS 182

Canonical form of cubic 182

Notation for general equation 182

General equation of Hessian and of Cayleyan 183

Invariant S,and its symbolical form 184

Invariant T 185

General equation of reciprocal 186

Calculation of invariants by the differential equation 187

Discriminant expressed in terms of fundamental invariants 189

Hessian of λU+μH 189

Condition that general equation should represent three right lines 190

Reduction of general equation to canonical form 191

Expression of discriminant in terms of fundamental invariants 192

Of anharmonic ratio of four tangents from any point on curve 192

Covariant cubics expressed in form λU+μH 193

Sextic covariants 194

The skew covariant 195

Equation of nine inflexional tangents 196

Equation of Cayleyan in point coordinates 196

Identical equation in theory of cubics 198

Conic through five consecutive points on cubic 200

Equation expressed in four line coordinates 202

Condition that cubic should represent conic and line 202

Discriminant of cubic expressed as determinant 203

Hessian of PU and of UV 204

CHAPTER Ⅵ．QUARTICS 206

Genera of quartics 206

Special forms of quartics 207

Illustration of the different forms 208

Distinction of real and imaginary 210

Flecnodes and biflecnodes 210

Quartic may have four real points of undulation 211

Quartics may be quadripartite 212

Inflexions of quartics,how many real 213

Classification of quartics in respect of their infinite branches 213

THE BITANGENTS 214

Discussion of equation UW=V2 215

There are 315 conics passing through eight contacts of bitangents 218

Scheme of these conics 221

Hesse＇s algorithm for the bitangents 221

Geiser＇s method of connecting bitangents with solid geometry 222

Cayley＇s rule of bifid substitution 223

Bitangents whose contacts lie on a cubic 225

Aronhold＇s discussion of the bitangents 225

From 7 bitangents the rest can be found by linear constructions 227

Aronhold＇s algebraic investigation 229

BINODAL AND BICIRCULAR QUARTICS 231

Tangents from nodes of a binodal are homographic 232

Foci of bicircular quartic lie on four circles 233

Casey＇s generation of bicircular quartics 234

Two classes of bicircular quartics 236

Relations connecting focal distances of point on bicircular 237

Confocal bicirculars cut at right angles 288

Hart＇s investigation 240

Cartesians 241

The limacon and the cardioide 242

Focal properties obtained by inversion 242

Inscription of polygons in binodal quartics 243

UNICURSAL QUARTICS 244

Correspondence between conics and trinodal quartics 245

Tangents at or from nodes touch the same conic 247

Tacnodal and oscnodal quartics 249

Triple points 250

INVARIANTS AND COVARIANTS OF QUARTICS 251

General quartic cannot be reduced to sum of five fourth powers 252

Covariant quartics 256

Examination of special case 257

Covariant conics 261

CHAPTER Ⅶ．TRANSCENDENTAL CURVES 263

The cycloid 263

Geometric investigation of its properties 264

Epi-cycloids and epi-trochoids 266

Their evolutes are similar curves 269

Examples of special cases 270

Limacon generated as epi-cycloid 270

Steiner＇s envelope 271

Reciprocal of epicycloid 271

Radius of curvature of roulettes 272

Trigonometric curves 273

Logarithmic curves 274

Catenary 275

Tractrix and syntractrix 277

Gurves of pursuit 278

Involute of circle 278

Spirals 279

CHAPTER Ⅷ．TRANSFORMATION OF CURVES 282

LINEAR TRANSFORMATION 283

Anharmonic ratio unaltered by linear transformation 284

Three points unaltered by linear transformation 285

Projective transformation 286

Homographic transformation may be reduced to projection 287

INTERCHANGE OF LINE AND POINT COORDINATES 289

Metbod of skew reciprocals 291

Skew reciprocals reduceable to ordinary reciprocals 294

QUADRIC TRANSFORMATIONS 296

Inversion,a case of quadric transformation 298

Applications of method of inversion 300

RATIONAL TRANSFORMATION 301

Roberts＇s transformation 301

Cremona＇s rational transformation 304

If three curves have common point their Jacobian passes through it 307

Deficiency unaltered by Cremona transformation 309

Every Cremona transformation may be reduced to a succession of quadric transformations 310

TRANSFORMATION OF A GIVEN CURVE 312

Rational transformation between two curves 312

Deficiency unaltered by rational transformation 314

Transformation,so that the order of the transformed curve may be as low as possible 316

Expression of coordinates by means of elliptic functions when D=1 317

and by means of hyper-elliptic functions when D=2 318

Theorem of constant deficiency derived from theory of elimination 319

CORRESPONDENCE OF POINTS ON A CURVE 319

Collinear correspondence of points 319

Correspondence on a unicursal curve 320

Number of united points 321

Correspondence on curves in general 322

Inscription of polygons in conics 325

in cubits 326

CHAPTER Ⅸ．GENERAL THEORY OF CURVES 328

Cayley＇s method of solving the general problem of bitangents 329

Order of bitangential curve 330

Hesse＇s reduction of bitangential equation 332

Bitangential of a quartic 337

Formation of equation of tangential curve 340

Application to quartic 344

POLES AND POLARS 345

Jacobian properties of 345

Steiner＇s theorems on systems of curves 348

Tact-invariants 349

Discriminant of discriminant of λu+μv,and of λu+μv+νw 349

Condition for point of undulation 350

For coincidence of double and stationary tangent 350

Steinerian of a curve 351

Its characteristics 352

The Cayleyan or Steiner-Hessian 352

Its characteristics 353

Generalization of the theory 353

OSCULATING CONICS 356

Aberrancy of curvature 356

Investigation of conic of 5-point contact 358

Determination of number of sextactic points 360

SYSTEMS OF CURVES 360

Chasles＇method 361

Characteristics of systems of conics 362

Number of conics which touch five given curves 363

Zeutheu＇s method 365

Degenerate curves 365

Cayley＇s table of results 368

Number of conics satisfying five conditions of contact with other curves 370

Professor Cayley＇s note on degenerate forms of curves 371