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 Part I Theory of the Icosahedron Itself
  Chapter I The Regular Solids and the Theory of Groups
   atement of the Question.
   eliminary Notions of the Group-Theory
   e Cyclic Rotation Groups
   e Group of theDihedral Rotations.
   e Quadratic Group.
   e Group of the Tetrahedral Rotations.
   e Group of the Octahedral Rotations
   e Group of the Icosahedral Rotations
    the Planes of Symmetry inOur Configurations
   neral Groups of Points—Fundamental Domains
   e Extended Groups
   neration of the IcosahedralGroup.
   neration of the Other Groups of Rotations.
  Chapter II Introduction of (x +i y)
   rst Presentation and Survey of the Developments of This Chapter
    Those Linear Transformations of (x +i y) WhichCorrespond to Rotations Round the Centre
   mogeneous Linear Substitutions—Their Composition.
   turn to the Groups of Substitutions—the Cyclic and DihedralGroups
   e Groups of the Tetrahedron andOctahedron
   e IcosahedralGroup.
   n-Homogeneous Substitutions—Consideration of the Extended Groups.
   mple Isomorphism in the Case of Homogeneous Groups of Substitutions
   variant Forms Belonging to a Group—The Set of Forms for the Cyclic andDihedralGroups
   eparation for the Tetrahedral andOctahedral Forms.
   e Set of Forms for the Tetrahedron.
   e Set of Forms for the Octahedron
   e Set of Forms for the Icosahedron.
   e Fundamental Rational Functions
   marks on the Extended Groups
  Chapter III Statement and Discussion of the Fundamental Problem, According to the Theory of Functions
   finition of the Fundamental Problem.
   uction of the Form-Problem.
   an of the Following Investigations
    the Conformable Representation byMeans of the Function z(Z)
   rch of the z1, z2 Function in General—Development in Series
   ansition to theDifferential Equations of the ThirdOrder.
   nnection with Linear Differential Equations of the Second Order
   tual Establishment of the Differential Equation of the Third Order for z[Z].
   near Differential Equations of the Second Order for z1 and z2
   lations to Riemann’s P-Function.
  Chapter IV On the Algebraical Character of Our Fundamental Problem
   blemof the Present Chapter
    the Group of an Algebraical Equation.
   neral Remarks on Resolvents.
   e Galois Resolvent in Particular
   rshalling of our Fundamental Equations
   nsideration of the Form-Problems
   e Solution of the Equations of the Dihedron,Tetrahedron,andOctahedron.
   e Resolvents of the Fifth Degree for the Icosahedral Equation
   e Resolvent of the r ’s
   putation of the Forms t andW
   e Resolvent of the u’s.
   e Canonical Resolvent of the Y ’s
   nnection of the New Resolvent with the Resolvent of the r ’s
    the Products of Differences for the u’s and the Y ’s.
   e Simplest Resolvent of the SixthDegree.
   ncluding Remarks.
  Chapter V General Theorems and Survey of the Subject.
   timation of our Process of Thought so far, and Generalisations Thereof
   termination of all Finite Groups of Linear Substitutions of a Variable
   gebraically Integrable Linear Homogeneous Differential Equations of the Second Order
   nite Groups of Linear Substitutions for a Greater Number of Variables.
   eliminary Glance at the Theory of Equations of the Fifth Degree,and Formulation of a General Algebraical Problem.
   finiteGroups of Linear Substitutions of a Variable
   lution of the Tetrahedral, Octahedral, and Icosahedral Equations by EllipticModular Functions.
   rmulae for the Direct Solution of the Simplest Resolvent of the SixthDegree for the Icosahedron
   gnificance 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
   finition of Our First Problem.
   ementary Remarks on the Tschirnhausian Transformation—Bring’s Form.
   ta Concerning Elliptic Functions
    Hermite’sWork of 1858
   e Jacobian Equations of the SixthDegree
   onecker’sMethod for the Solution of Equations of the Fifth Degree
    Kronecker’sWork of 1861.
   ject of our Further Developments
  Chapter II Introduction of GeometricalMaterial
   undation of the Geometrical Interpretation
   assification of the Curves and Surfaces.
   e Simplest Special Cases of Equations of the Fifth Degree.
   uations of the Fifth DegreeWhich Appertain to the Icosahedron.
   ometrical Conception of the Tschirnhausian Transformation.
   ecial Applications of the Tschirnhausian Transformation
   ometrical Aspect of the Formation of Resolvents
    Line Co-ordinates in Space.
   9.A Resolvent of the Twentieth Degree of Equations of the Fifth Degree
   eory of the Surface of the Second Degree.
  Chapter III The Canonical Equations of the Fifth Degree
   tation–The Fundamental Lemma.
   termination of the Appropriate Parameter λ
   termination of the Parameter μ.
   e Canonical Resolvent of the Icosahedral Equation.
   lution of the Canonical Equations of the Fifth Degree
   rdan’s Process
   bstitutions of the λ,μ’s—Invariant Forms.
   neral Remarks on the Calculations WhichWe Have to Perform.
   esh Calculation of theMagnitude m1
   ometrical Interpretation ofGordan’s Theory
   gebraical Aspects (After Gordan)
   e Normal Equation of The rν’s
   ing’s Transformation
   eNormal Equation ofHermite.
  Chapter IV The Problem of the A’s and the Jacobian Equations of the Sixth Degree
   e Object of the Following Developments
   e Substitutions of the A’s—Invariant Forms.
   ometrical Interpretation—Regulation of the Invariant Expressions
   e Problem of the A’s and Its Reduction.
    the Simplest Resolvents of the Problem of the A’s
   e General Jacobian Equation of the SixthDegree
   ioschi’s Resolvent
   eliminary Remarks on the Rational Transformation of Our Problem.
   complishment of the Rational Transformation
   oup-Theory Significance of Cogredience and Contragredience
   roductory to the Solution of Our Problem.
   rresponding Formulae.
  Chapter V The General Equation of the Fifth Degree
   rmulation of TwoMethods of Solution.
   complishment of Our FirstMethod
   iticismof theMethods of Bring andHermite.
   eparation for Our SecondMethod of Solution.
    the Substitutions of the A,A’s—Definite Formulation
   e Formulae of Inversion ofOur SecondMethod
   lations to Kronecker and Brioschi
   parison of Our TwoMethods.
    theNecessity of the Accessory Square Root
   ecial Equations of the Fifth DegreeWhich Can Be Rationally Reduced to an Icosahedral Equation.
   onecker’s Theorem.
 Appendix