A Treatise on Algebraic Plane Curves

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Courier Corporation, 2004 - Mathematics - 513 pages
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Students and teachers will welcome the return of this unabridged reprint of one of the first English-language texts to offer full coverage of algebraic plane curves. It offers advanced students a detailed, thorough introduction and background to the theory of algebraic plane curves and their relations to various fields of geometry and analysis.
The text treats such topics as the topological properties of curves, the Riemann-Roch theorem, and all aspects of a wide variety of curves including real, covariant, polar, containing series of a given sort, elliptic, hyperelliptic, polygonal, reducible, rational, the pencil, two-parameter nets, the Laguerre net, and nonlinear systems of curves. It is almost entirely confined to the properties of the general curve rather than a detailed study of curves of the third or fourth order. The text chiefly employs algebraic procedure, with large portions written according to the spirit and methods of the Italian geometers. Geometric methods are much employed, however, especially those involving the projective geometry of hyperspace.
Readers will find this volume ample preparation for the symbolic notation of Aronhold and Clebsch.

  

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Contents

BOOK I
1
Representation in hyper space
3
Resultant of two polynomials
7
CHAPTER II
14
Tangential equations
20
Nothers Fundamental Theorem
29
Studys theorem and its consequences
36
Contacts with asymptotes
42
The RiemannRoch theorem
260
Canonical series and canonical curves
267
Sufficiency of these conditions
273
Integrals of other sorts
275
Groups common to a g and a gj
281
CHAPTER IV
289
Limiting values
295
CHAPTER V
301

Real singular points
47
Generation of curves by small variation
57
Nesting circuits
62
Cross ratios
69
Polar operator
75
Discriminant of general quadratic form
81
Ternary forms
86
The effect of singular points
92
Determination of the number of inflexions and cusps
98
Singularities of a rational curve
104
Genus
108
Projection of a real curve on an imaginary plane
111
CHAPTER VIII
119
Correspondences of value 0
126
Deduction of the general formula Riemanns theorem
128
Correspondences on different curves
135
Line polars
142
Pencils of curves
148
Pliicker characteristics of Hessian Steinerian and Cayleyan
154
Satellite curve
160
Warings theorem
166
Products of distances
174
Sums of angles determined by tangents and foci
180
Metrical covariants associated with polars
188
Conditions for an algebraic involute Humberts theorem
194
Reduction of singularities
196
Limiting cases
200
Nothers transformation theorem
207
DEVELOPMENT IN SERIES
213
Order and class of a branch
219
Number of intersections in general case
225
CHAPTER III
232
Satellite points
239
Definition of genus of a general curve
245
Adjunction theorem
254
Transformation of hyperelliptic curve to canonical form
305
Transformation to canonical form
311
Residuation theorems 245
313
Series of index 1
317
Extension of RiemannRoch theorem
323
CHAPTER VII
329
Linear dependence of correspondences
335
Relation of circuits to rational points in hyperspace
336
p p correspondences
342
Curves with only simple branches Clebschs transformation
348
PARAMETRIC REPRESENTATION OF THE GENERAL
354
Applications of uniformizalion
360
CHAPTER IX
368
Determination of the equation of a rational curve
370
Cuspidal and undulational conditions
376
Postulation bymeans of singular points
383
Situation of singular points
392
Necessary and sufficient conditions for the reduction to a curve
399
Reduction of curves lacking adjoint systems of high index
406
Apolarity
412
The inflexional locus
421
NONLINEAR SYSTEMS OF CURVES
425
The inflexions
433
Number of curves in a fcparameter system which touch k curves
439
Properties of the Jacobian 424
446
Montesanos theorem
453
Transformations in one plane
459
Fixed points of two sorts
466
Transformations with curves of fixed points
474
Involutory transformations of lowest class
482
CHAPTER VIII
489
Finite groups
496
SUBJECT INDEX
511
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