A Treatise on Algebraic Plane Curves Book

Book I	Elementary Theory
Chapter I	The Fundamental Properties of Polynomials
1.	Polynomials in one variable
2.	Polynomials in N variables
Chapter II	Elementary Properties of Curves
1.	Ordinary and singular points
2.	Determination of curves by points
3.	Residuation theorems
Chapter III	Real Curves
1.	Asymptotes
2.	Real singular points
Chapter IV	Real Circuits of Curves
1.	Topological properties of even and odd circuits
2.	Generation of curves by small variation
3.	Nesting circuits
4.	Apparent order and index of circuits
Chapter V	Elementary Invariant Theory
1.	Trilinear coordinates
2.	Invariants and covariants, first principles
3.	Symbolic notation
4.	Binary domain
5.	Ternary forms
Chapter VI	Projective Theory of Singular Points
1.	Polar curves and singular points
2.	The effect of singular points
Chapter VII	Plucker's Equations, and Klein's Equation
1.	Forms of Plucker's equations
2.	Existence conditions
3.	Klein's equation
Chapter VIII	The Genus
1.	Definition of genus
2.	Chasles-Cayley-Brill correspondence formula
3.	Correspondences on different curves
Chapter IX	Covariant Curves
1.	Polar curves
2.	Simplest linear systems
3.	Fundamental covariant curves
4.	Covariants of two curves or envelopes
Chapter X	Metrical Properties of Curves
1.	Centres of gravity
2.	Foci
3.	Products of distances
4.	Sums of angles
5.	Polars
6.	Transversals
7.	Evolutes
Book II	The Singular Points
Chapter I	The Reduction of Singularities
1.	Quadratic transformations
2.	The effect on singularities
Chapter II	Development in Series
1.	Development of the branches at a point
2.	Invariant numbers for branches
3.	Intersections of two branches
4.	Effect of a quadratic transformation on a branch
Chapter III	Clustering Singularities
1.	General idea of clustering singularities
2.	Singularities of a single branch
Chapter IV	Adjoint Curves and Plucker's Equations
1.	Adjoint curves in general
2.	Residuation
3.	Genus
4.	Plucker's equations
Book III	Systems of Points on a Curve
Chapter I	General Theory of Linear Series
1.	Linear systems of curves and linear series
2.	Sums and differences of series
3.	Representation in hyperspace
4.	The Riemann-Roch theorem
5.	Simple series and composite series
Chapter II	Abelian Integrals
1.	Integrals of the first sort
2.	Integrals of other sorts
3.	Abel's theorem
Chapter III	Singular Points of Correspondences
1.	The Chasles-Cayley-Brill correspondence formula
2.	The Jacobian series
3.	The De Jonquieres formula
Chapter IV	Moduli and Limiting Values
1.	The Gap Theorem of Weierstrass
2.	Moduli
3.	Limiting values
Chapter V	Curves of Special Type
1.	Curves containing series of given sort
2.	Elliptic curves
3.	Hyperelliptic curves
4.	Polygonal curves
5.	[phi]-curves
6.	Reducible curves
Chapter VI	Non-Linear Series of Groups of Points on a Curve
1.	General theorems about series
2.	Series of index 1
3.	Groups common to a g[superscript r subscript N] and a [gamma superscript rho subscript M]
4.	The defect of equivalence
Chapter VII	Higher Theory of Correspondences
1.	General theorems
2.	Application of Abelian Integrals
3.	Representation in hyperspace
4.	Generalized values
5.	(p, p) correspondences
6.	Birational correspondences
7.	Halphen's transformation
8.	Rational determination of characteristics of a curve
Chapter VIII	Parametric Representation of the General Curve. A Sketch
1.	Riemann surface for the general curve
2.	Uniformization
3.	Applications of uniformization
4.	Curves of genus 1
Chapter IX	Rational Curves
1.	Binary apolar forms
2.	Determination of the equation of a rational curve
3.	Groups of intersections
4.	Equations and conditions for singularities
Book IV	Systems of Curves
Chapter I	Postulation of Linear Systems By Points
1.	Fundamental properties of linear systems
2.	Systems defined by simple points
3.	Postulation by means of singular points
4.	Fundamental curves and special adjoints
5.	Situation of singular points
Chapter II	The Transformation of Linear Systems
1.	Invariants of linear systems
2.	Curves transformable into straight lines
3.	Reduction of singularities
4.	Reduction to linear systems of minimum order
5.	Reduction of curves of low genus
Chapter III	Ternary Apolarity
1.	Linear systems and hyperspace
2.	Apolarity
3.	Apolarity between forms of different orders
4.	Expression of forms in terms of perfect nth powers
Chapter IV	Special Curves in Linear Systems
1.	The pencil
2.	Two-parameter nets
3.	The Laguerre net
Chapter V	Non-Linear Systems of Curves
1.	General formulation
2.	Singular points and the envelope
3.	The inflexions
4.	Projective theorems
5.	Systems depending on more than one parameter
Chapter VI	The General Cremona Transformation
1.	Fundamental properties
2.	Nother's factorization theorem
3.	Applications of the factorization theorem
4.	The identities of Clebsch
5.	Transformations in one plane
Chapter VII	Types of Cremona Transformations
1.	Types of lowest order
2.	Numerical relations
3.	Transformations with curves of fixed points
4.	Involutory transformations
5.	Cremona transformations and integral collineations in hyperspace
Chapter VIII	Groups of Cremona Transformations
1.	Continuous groups
2.	Infinite discontinuous groups
3.	Finite groups
	Index of Authors Quoted	499
	Subject Index	511