Ganguli, Theory of Plane Curves, Index

8/14/2019 Ganguli, Theory of Plane Curves, Index

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INDEX OF AU'l'HORS CITED[The numbers refer to page . )

Alkwith ... 112 Descartes 21Bacharach 35 Edward 58, 148, 241Basset 347,351 Elliot 83, 85, l O BBertini 188, 321, 327, 349 Euler 29,139Blia. ... 327 Ferrer 7Bobeck 318 Frost 227, 229, 243Booth 4 - Gergonne 32, 37, 157Brianchon 6 Goursat 302,305Brill 43, 281, 287, 308, 311, 317, Gregory 156

321,340,351,374- Gnccia ... 367Brusotti 298 Haase 290, 295Castelnuovo 274, 283 Halphen 50, 321, 327, 331, 339, 374Cauchy 337 Hamburger 327Cayley 8, 10, 29,34,61, 107, 121, Hamnet Holditch 162

148,151,158, 162,167, 182, Harnack .. . 306187, 209, 274, 281, 3 O B , 311, Hart 193317, 321, 325,332,369,372. Heath 167Chaslel\ 6,31,90, 123, 189, 308, Henrichi 151

312,365,372 Hensel 32701ebsch 67,96, 108, 113, 119, 134, Heise 107,108

193,290, 295,302, 306, Hill 151318, 374 Hilton 304,347

Coble 272 Hil"St 257; ote 86 Hobson 222Cramer 29, 271, 321, 323, 328 Humbert 297Cremona 107,721,250,272,321, Hurwitz 319

327, 361, 367 Jacobi 29Dnrboux .. . 298 Juel 351De Beanne 6 Kantor 283De Jonquiares 3 O B , 365, 372 Klein 276,351Desargues ... 11,12 Kotter '" 369


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880 INDEX OF AUTHORS CITEDlrroneoker 327Lagrange 158Landesbnrg 327Leibnitz 6Loria 298, 308Lnroth 287Ka.elanrin 91)(agnns 251Karia Gaetana 248KlIbinB 6, 251, 253Montesano .. 273Moniard .. 268Kukhopadhyay... 356,359,360Newton 227, 234, 236, 271, 328, 333NOther 36, 43, 271, Z74 , 281, 321,327, 329. 349

32, 33, 248, 208... 287... 192

12, 29, 142, 181, 191, 209,321,350

10, 11, 135306

... 333

... 15763.277... 319

PaecalPaaohpeazoPltickerPQDoeletPortllrPu,ilellXQueteletB.iema.nn&oIati

Rowe ... 287Salmon 5, 8, 10, 58, 89, 103, 110,137, 151, 170, 185,209,225,241, 253, 2'15, 2'17, 279, 363,

371,376Sohwartz ... 2'16Scott 1,3, 17, 141,253,257,281,

3 ( 1 1 , 321, 351, 369Segre 286, 318, 321, 332Severi ... 3188iebeck 220Smith 158,321,349Steiner 107, us, 121, 192, 197St. Laurent .. 156Stolz 332, 383Study 45Sylvester 37, 859Taylor 55, 175, 196.202,208Townsend 3S'1Transon 3MTllchirnhllWleIl 156lVaker 327lVieleitner 298lVilliamlon 166Zeuthen 28, 188. 318, 321, 339,

845, 872, 373Zimmermann ... 202, 220


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GENERAL INDEX[The number8 refer to page.]

Aberrancyaxis of,Transou'. theory of,radius of,centre of,angle of,

Aberrancy curveAbsoluteAcuodeAdjoint-adjoiut cnnes,

Affinity, linear ...Algebraic curves, notion of

3553553563563 6 6

3 6 8 , 3 6 08

5 7 , 3 2 544, 2 8 2 ;2 8 3 , 3 8 0

25121

Analysis of higher singnlarities 326Analytical treatment

Quadrio InversionAnalytical Triangle

of

properties of, 229use of, in three variables 232

Anautotomic curves 49Angle made by tangents with

any line 222'between ourves unalteredby circular inversion. ... 18

Anticaustio ]57Antipoints 226

269227

Approximate forms of ourves 2:27Newton's method of, 2'3

Areal co-ordinates, relationwith Cartesian co-ordinates 1

Ail'gand's diagram 331Asymptotes

circular, 244determination of, 2 3 9special methods of finding 240"'

Asymptotic curves 242circle ~

Autotomio curves '9Auxiliary conic .. , 19Axis of projection" of perspective

12 :15

Basis curve of reaidua-tdon 37, 44Base conic of Quadric Inver-sion 2 5 8

2807I2 9 92 4 9

73, ll82 '96183

4 t

Bicircular QuarticBiflecnodeBipartite curves ...Birational TransformationBitangents

of unicnrsal curves:Bitaugential ourveBooth

Applioation of Quadric Inver- . Branches with higher singn-lion 2 6 7 larities 2 8 7 . -


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382 GENERAL INDEXBrill-Nother's theorem on Circular cubic 269residuation 43 lines ~Cartesian equation of pedal ... 168 inversion 268

Oval 166 asymptotes 243Cardioid 180 curves, foci of, 225Cauchy 337 Circular points at iufiuity 7co-ordinates of, 10Caustics 165 properties of, 10

classes of, 156 Cissoid 154,208by reflection of a circle ... 158 Class of R . curve 101,319

st. line 158 Collineation 251equation of, ... 159 treated !\,eometrically 252tangential equation of, ... 163bitangents of, 161 Common elements of twointersection with the re- correspondences 314fleeting circle 162 Complex singularities 70,321singulariteis on 163 Conchoid 208by refraction of a. line 163 Conjuga.te point... 57, 51, 143" circle ... 166 Confocal curves ... 21f:iCayley-Brill's correspondence Conics with four-pointicformnla 3\7

Cayley on intersections ofcnrves 34

Cayley n )(\7,108,of a net of ourvesclass of,order of,characteristics of,

Centre of a curveaberrancy

mean distances

121, 170369

122, 123121193

...8,9035687

Characteristics of curves 18\of a system ofcurves 372

Chasles on intersection ofcubics 31definition of centre 90

eorr.~pondence formula... 123189,312,318Circuit of unioursal curves... 298

contact 354Constituents of singularities 351Co-ordinatesCartesiansystem of

as a specialhomoge-

neODS,of circular pointsin terms of elliptic func-tions

Oo-reaidualOorrespondence, theory of, ...

of pts. on a curveanalytical discussion of ...

Correspondence IndexCorresponding PointsCote's Theorem on harmonic

meanCovariant curves

of ternary forms

310

30037

3073073ag31310886

107370


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GENERAL INDEX 38aCramer's paradoxanalysistransformation

29245,271

328Cremona Transformation 250,

274, 286, 327272,274

36275212272l

... 23,2526

8 4 . 2 9 82162098847

67 , 2872991 8 9352363264

Cremons conditionsCritic centres of cn bicsCrunodeCurves, algebraic, notiou of,approximate forms of,transcendental,proper and degenerateintersections of,triangular symmetric,confocalfoci of,centre of,singular points ou,with zero deficiencywith unit defioiencywith same deficiencyof closest contactwhich touch a given curveparticular cases of,

Curvilinear asymptotesCusp

species of,

244-57,58 1 4 3

322Cuspidailndex 348, 350, 3151Cycle 332order of, 332

Deficiency of curves 63. 186Deficiency nnaltered by

Cremona TransformationDeficiency of transformed

curveDeficiency, curves with thesame

Deficieucies, point and Iiue

Degenerate curves 25Derived curves ... 135, 207Diametral carves 87Discriminant 56, 141, 151Discriminantal Index 337, 338Double cusps 323, 324Double points 48, 53, 81, 93, 130couditions for, 64

species of, 56limit to the number of,... 62discrimination of inflex-ions from, ... 130

Double focus .211I Doubly periodio fnnction 302Double tangent ...of reciprocal ourves

Dualistic TransformationDuality, principle of,Effects of inversion

sinzu 1arities ...

73143, 186253, 254

142on

265Effects of inversion on acune 266

Elliptic Functions, co-ordi-nates in terms of, 300

Envelopes 145, 147, 149Equivalent singnlarities 325Evolute 153. 164, 166of a parabola 153ellipse Hi3ciasoid 164normal of, 1 5 4 -characteristics of, 197

Expansion of a function 336in line co-ordinates 343

Extension of Pllicker's For-mules 350

Extension of residual theorem '"First Polar 100, 103, 112

275

280 I188187


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384FlfonodeFoci of curTessingular,dekrmination of,triple.,oo.ordinates of," nlp,w~keo.y w . ,of inverse curves.of circular curvesreeiproeal, 'W .1 '.t.

Folium of Descartes

7Q206211,1112

21721311142202 2 322522469

Funotions, representation of, 2 2expansion of, .. 336

Gauss planeG~.t:io~neGell"l'1!.tI9nof a curveR~monic mean .

polar .

33132,157... 365

87128

H'8IIIiian 107, 109, 114, 123, 100cha.ra.o~ristica of 191of a net of curves ... 3 6 9

Higher singularities 3 2 1Hurwitz Correspondence 319Inflexion, points of 49, 53, 74; 98,

ISO, 181number of; 129, 133, 134, 320

Illfl!!.xionalta.ngent 4 9 , 1 4 3 , 2 6 4lqfuUte branches 2 S 6Invariants of ternary forms 370Inverse ourves

proper,foci of,

l~se of a right lineInYerse of ijl61ine a~5nflilityof l!piIIli.l:pointa

1702 6 2223

. .,

Inversion, theory of 1'1oiroul6r, 268quadrio, 257effects on singnla.rities of, 265

" a ourve of, ... 266an"lytioal treatment of, .. 259

In~rsection8 of ourves 26at singular points 60at higher sing. points 339

Intersections with adjoints 284with a pencil of adjoints 284of oanstics with reflectingcircle 162

Irreducible function 23curve 326Isoleted point ...Iacptic looioh~oteristics. of,

57173207

Isotropic linesJacobian of three ourves

of a net of curvesmnltiple points on,

KatacausticKeratoid ousp ...Line at infinity

properties of,inverse of,

9, 10, 113 6 6

... 368... 367157, 158

322211263250251

381, 34iI.

Linear transformationaffinityb~ch

Line co-ordinatesexpansion of a funotion

in,

3

261 LimaQonMaolaurin's theoremMixed polarsMongian

180, 248, 96991'88859.


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!IIllltiplepointstangents

511 liS, 1In, 145' 1 3

3eB369369

Net of curvesof first polarsas JacobiaD

Newton's approximation 234parallelcgram 227

Nodal indea 351Node 6'1, 128, 143NlIther's transformation 271Number of points

a ourveof inflexiousdetermining

24129Order, of a oycle 332

a super1inear branch 331regarded 8S an envelope, :143of a CUrve 24

Orthoptio looi 175equation of, 176derived from polarequation 179

characteristics of, 202O8culating cirole. invel'B6of 269

curves 352, 353O~I-inflexion 323O!d of Descartes 166P.bolio branohes ~Parallel CllrVes 111

cbrdeterhltici of, 20iParadoil:of Cramer 29"Pan.metric 01 1 i 372

order of. familyof ourves 379represeutation 287in nne oo-ordinates 200

hatial branoh 333

Pascal's theorem onhetaffoli ... 32, 33

Pedal curves 167equation o t, Illscharacteristics of, 195

Pencil of o n - i c s M looniea, appli08.ion ofthe principles ofcharacteristics to 375

Perspective, figures in, 1 4 ,centre of, Iiiaxis of, 15analytical t rea tment of, 15

Pllicker'iI equations 18iPoint and line deficiencies... 187Polar curves 19, 78

of origin 91conic ... 112o t point of inflexlon 82reciprocal curves 135 , 137of a Btiperlinear braBeh ~47

Poles and polus, theory ot 77w.r.t. a triangle 99

Pole aad polar conics _ 2 56Poles of a tight iilie 95, 98Polara, mixed 83li6int equation deri:Ved from

tangential 140Principle of duality 142Principal equivalence 351Projection, theory of, 11

axis of, 12vertex of, 12analytical aspect of, 18

Quaidrioinversion 281Q8i ratioli'" triiJdfor~mation 2 & 9 '


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886

I

GENERAL INDEX

Quadric Inversionanalytical treatment of, 259applications of, 267

Quadrio:transformation 274special 269

Quartic, trinodal 267tricuspidal 268

Kadius of curvature 0 aber-rancy curve 360of aberrancy 356

Ramphoid cusp 322Rational ourves 65

special olass of, 298transformation lU9

Reoiprocalpolars 19, 135, 144in homogeneous co-ordinates 137

w.r.t. focus 224singularities 74

Beeiprocation, theory of, 19, 135skew, 253

Reduotion of the order oftransformed curve 279with a multiple pt. 281

Relation bet. line and point 4oharacteristics of afamily 373

Residuation, theory of, 36ReBidual 37

theorem 43extension of, 44

Beaiduationaddition theorem on, 38subtraction 39multiplication 40

Riemann transformation 276, 277Beco,dary caustic 157, 165Semicubioalparabola 152, 235Sextactic points 320

Singular focus 211Singular points on curves 47at infinity .71on unioursal curves 290

Singularities, equivalent 325reciprocal 74higher, on curves 321analysis of, 326method of expansionapplied to 384

Skew reciprocatiou 253Speoiesof cusps 322

of doublepoints 56Spimode 57St. Laurent'scaustic equation of 159

49, 74, 170tationary tangentBteinerian 107, 110, 190

characteristics of, 192class of, 116order of, 119inflexionaltangent of, 119

Bteinerian of a net of curves 269Sucoessive transformations 328Superlinear branches

order of,class of,

333, 334331332

Tacnodal branch 35iTacnode 144, 247, 323, 326Tact-invariant of two curves 3MTangential co-ordinates 4.

equation derived frompoint equation 138

equation of caustic 160Tangents to a cubic 85Traoing of curves

in Cartesian" homogeneous

227, 245245

. 245


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GENERAL IND~X 387Transcendental functions 23 Triple point 51,82, 12/\ ,161, 133

curves 21 classification of, 324Transformation, rational 249 Undulation, points of 59

birational 249linear 250 Unicursal curves 65,69.287dualistic 253 order of, 289special quadric 269 class of, 290byadjoints 285 singular pts. on, 292successive 328 bitangents of, 295Noetber's 271 circuit of, 298Riemann's 276 Unipartite curves 298

Transon's cheory of not necessarily uni-aberrancy 355 cursal 299

Triangular symmetric curves United points 31084, 298 Vanishing line 12

Tricuspidal quartic Vertex of projection 12268Trilinear co-ordinates 1 Weierstrass's elliptic funo-

equation 6, 138 tions 302of caustics 160

Witch 248Trinodal quartic 70, 2 67 Zero residual 37

Ganguli, Theory of Plane Curves, Index

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