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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
