二十面体和5次方程的解的讲义（英文版）

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目录
- 前辅文
- Part I Theory of the Icosahedron Itself
  - Chapter I The Regular Solids and the Theory of Groups
    - 1.Statement of the Question.
    - 2.Preliminary Notions of the Group-Theory
    - 3.The Cyclic Rotation Groups
    - 4.The Group of theDihedral Rotations.
    - 5.The Quadratic Group.
    - 6.The Group of the Tetrahedral Rotations.
    - 7.The Group of the Octahedral Rotations
    - 8.The Group of the Icosahedral Rotations
    - 9.On the Planes of Symmetry inOur Configurations
    - 10.General Groups of Points—Fundamental Domains
    - 11.The Extended Groups
    - 12.Generation of the IcosahedralGroup.
    - 13.Generation of the Other Groups of Rotations.
  - Chapter II Introduction of (x +i y)
    - 1.First Presentation and Survey of the Developments of This Chapter
    - 2.On Those Linear Transformations of (x +i y) WhichCorrespond to Rotations Round the Centre
    - 3.Homogeneous Linear Substitutions—Their Composition.
    - 4.Return to the Groups of Substitutions—the Cyclic and DihedralGroups
    - 5.The Groups of the Tetrahedron andOctahedron
    - 6.The IcosahedralGroup.
    - 7.Non-Homogeneous Substitutions—Consideration of the Extended Groups.
    - 8.Simple Isomorphism in the Case of Homogeneous Groups of Substitutions
    - 9.Invariant Forms Belonging to a Group—The Set of Forms for the Cyclic andDihedralGroups
    - 10.Preparation for the Tetrahedral andOctahedral Forms.
    - 11.The Set of Forms for the Tetrahedron.
    - 12.The Set of Forms for the Octahedron
    - 13.The Set of Forms for the Icosahedron.
    - 14.The Fundamental Rational Functions
    - 15.Remarks on the Extended Groups
  - Chapter III Statement and Discussion of the Fundamental Problem, According to the Theory of Functions
    - 1.Definition of the Fundamental Problem.
    - 2.Reduction of the Form-Problem.
    - 3.Plan of the Following Investigations
    - 4.On the Conformable Representation byMeans of the Function z(Z)
    - 5.March of the z1, z2 Function in General—Development in Series
    - 6.Transition to theDifferential Equations of the ThirdOrder.
    - 7.Connection with Linear Differential Equations of the Second Order
    - 8.Actual Establishment of the Differential Equation of the Third Order for z[Z].
    - 9.Linear Differential Equations of the Second Order for z1 and z2
    - 10.Relations to Riemann’s P-Function.
  - Chapter IV On the Algebraical Character of Our Fundamental Problem
    - 1.Problemof the Present Chapter
    - 2.On the Group of an Algebraical Equation.
    - 3.General Remarks on Resolvents.
    - 4.The Galois Resolvent in Particular
    - 5.Marshalling of our Fundamental Equations
    - 6.Consideration of the Form-Problems
    - 7.The Solution of the Equations of the Dihedron,Tetrahedron,andOctahedron.
    - 8.The Resolvents of the Fifth Degree for the Icosahedral Equation
    - 9.The Resolvent of the r ’s
    - 10.Computation of the Forms t andW
    - 11.The Resolvent of the u’s.
    - 12.The Canonical Resolvent of the Y ’s
    - 13.Connection of the New Resolvent with the Resolvent of the r ’s
    - 14.On the Products of Differences for the u’s and the Y ’s.
    - 15.The Simplest Resolvent of the SixthDegree.
    - 16.Concluding Remarks.
  - Chapter V General Theorems and Survey of the Subject.
    - 1.Estimation of our Process of Thought so far, and Generalisations Thereof
    - 2.Determination of all Finite Groups of Linear Substitutions of a Variable
    - 3.Algebraically Integrable Linear Homogeneous Differential Equations of the Second Order
    - 4.Finite Groups of Linear Substitutions for a Greater Number of Variables.
    - 5.Preliminary Glance at the Theory of Equations of the Fifth Degree,and Formulation of a General Algebraical Problem.
    - 6.InfiniteGroups of Linear Substitutions of a Variable
    - 7.Solution of the Tetrahedral, Octahedral, and Icosahedral Equations by EllipticModular Functions.
    - 8.Formulae for the Direct Solution of the Simplest Resolvent of the SixthDegree for the Icosahedron
    - 9.Significance of the Transcendental Solutions.
- Part II Theory of Equations of the Fifth Degree
  - Chapter I The HistoricalDevelopment of the Theory of Equations of the Fifth Degree
    - 1.Definition of Our First Problem.
    - 2.Elementary Remarks on the Tschirnhausian Transformation—Bring’s Form.
    - 3.Data Concerning Elliptic Functions
    - 4.On Hermite’sWork of 1858
    - 5.The Jacobian Equations of the SixthDegree
    - 6.Kronecker’sMethod for the Solution of Equations of the Fifth Degree
    - 7.On Kronecker’sWork of 1861.
    - 8.Object of our Further Developments
  - Chapter II Introduction of GeometricalMaterial
    - 1.Foundation of the Geometrical Interpretation
    - 2.Classification of the Curves and Surfaces.
    - 3.The Simplest Special Cases of Equations of the Fifth Degree.
    - 4.Equations of the Fifth DegreeWhich Appertain to the Icosahedron.
    - 5.Geometrical Conception of the Tschirnhausian Transformation.
    - 6.Special Applications of the Tschirnhausian Transformation
    - 7.Geometrical Aspect of the Formation of Resolvents
    - 8.On Line Co-ordinates in Space.
    - 9.A Resolvent of the Twentieth Degree of Equations of the Fifth Degree
    - 10.Theory of the Surface of the Second Degree.
  - Chapter III The Canonical Equations of the Fifth Degree
    - 1.Notation–The Fundamental Lemma.
    - 2.Determination of the Appropriate Parameter λ
    - 3.Determination of the Parameter μ.
    - 4.The Canonical Resolvent of the Icosahedral Equation.
    - 5.Solution of the Canonical Equations of the Fifth Degree
    - 6.Gordan’s Process
    - 7.Substitutions of the λ,μ’s—Invariant Forms.
    - 8.General Remarks on the Calculations WhichWe Have to Perform.
    - 9.Fresh Calculation of theMagnitude m1
    - 10.Geometrical Interpretation ofGordan’s Theory
    - 11.Algebraical Aspects (After Gordan)
    - 12.The Normal Equation of The rν’s
    - 13.Bring’s Transformation
    - 14.TheNormal Equation ofHermite.
  - Chapter IV The Problem of the A’s and the Jacobian Equations of the Sixth Degree
    - 1.The Object of the Following Developments
    - 2.The Substitutions of the A’s—Invariant Forms.
    - 3.Geometrical Interpretation—Regulation of the Invariant Expressions
    - 4.The Problem of the A’s and Its Reduction.
    - 5.On the Simplest Resolvents of the Problem of the A’s
    - 6.The General Jacobian Equation of the SixthDegree
    - 7.Brioschi’s Resolvent
    - 8.Preliminary Remarks on the Rational Transformation of Our Problem.
    - 9.Accomplishment of the Rational Transformation
    - 10.Group-Theory Significance of Cogredience and Contragredience
    - 11.Introductory to the Solution of Our Problem.
    - 12.Corresponding Formulae.
  - Chapter V The General Equation of the Fifth Degree
    - 1.Formulation of TwoMethods of Solution.
    - 2.Accomplishment of Our FirstMethod
    - 3.Criticismof theMethods of Bring andHermite.
    - 4.Preparation for Our SecondMethod of Solution.
    - 5.Of the Substitutions of the A,A’s—Definite Formulation
    - 6.The Formulae of Inversion ofOur SecondMethod
    - 7.Relations to Kronecker and Brioschi
    - 8.Comparison of Our TwoMethods.
    - 9.On theNecessity of the Accessory Square Root
    - 10.Special Equations of the Fifth DegreeWhich Can Be Rationally Reduced to an Icosahedral Equation.
    - 11.Kronecker’s Theorem.
- Appendix

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二十面体和5次方程的解的讲义（英文版）

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