## Lectures on the Icosahedron and the Solution of Equations of the Fifth DegreeThis well-known work covers the solution of quintics in terms of the rotations of a regular icosahedron around the axes of its symmetry. Its two-part presentation begins with discussions of the theory of the icosahedron itself; regular solids and theory of groups; introductions of (x + iy); a statement and examination of the fundamental problem, with a view of its algebraic character; and general theorems and a survey of the subject. The second part explores the theory of equations of the fifth degree and their historical development; introduces geometrical material; and covers canonical equations of the fifth degree, the problem of A's and Jacobian equations of the sixth degree, and the general equation of the fifth degree. Second revised edition with additional corrections. |

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### Contents

THE REGULAR SoLIDS AND THE THEORY of GROUPs SEQ PAGE 1 Statement of the question | 3 |

Preliminary notions of the grouptheory | 6 |

The cyclic rotation groups | 8 |

The group of the dihedral rotations | 10 |

The quadratic group | 13 |

The group of the tetrahedral rotations | 14 |

The group of the octahedral rotations | 16 |

The group of the icosahedral rotations | 17 |

Computation of the forms t and W | 111 |

On the products of differences for the us and the Ys | 117 |

GENERAL THEOREMs AND SURVEY of THE SUBJECT | 124 |

Algebraically integrable linear homogeneous differential equa | 132 |

90 | 135 |

THEORY OF EQUATIONS OF THE FIFTH DEGREE | 137 |

Infinite groups of linear substitutions of a variable | 139 |

Formulae for the direct solution of the simplest resolvent | 147 |

On the planes of symmetry in our configuration | 21 |

General groups of pointsFundamental domains | 23 |

The extended groups | 24 |

Generation of the icosahedral group | 26 |

Generation of the other groups of rotations | 29 |

CHAPTER II | 31 |

On those linear transformations of ciy which correspond to rotations round the centre | 34 |

14 | 36 |

Homogeneous linear substitutionsTheir composition | 37 |

Return to the groups of substitutionsThe cyclic and dihedral groups | 39 |

The groups of the tetrahedron and octahedron | 40 |

CHAPTER III | 58 |

Plan of the following investigations | 72 |

On the conformable representation by means of the function zZ | 79 |

Linear differential equations of the second order for 2 and | 87 |

General remarks on resolvents | 94 |

Marshalling of our fundamental equations | 100 |

sec Page | 104 |

91 | 150 |

CHAPTER I | 153 |

Data concerning elliptic functions | 159 |

Kroneckers method for the solution of equations of the fifth | 168 |

Object of our further developments | 175 |

The simplest special cases of equations of the fifth degree | 181 |

A resolvent of the twentieth degree of equations of the fifth | 195 |

10 | 222 |

THE PROBLEM of THE As AND THE JAcoBIAN EQUATIONS | 234 |

The problem of the As and its reduction | 243 |

Brioschis resolvent | 250 |

10 | 256 |

THE GENERAL EQUATION of THE FIFTH DEGREE | 265 |

Of the substitutions of the A AsDefinite formulation | 274 |

Comparison of our two methods | 280 |

94 | |

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accessory square root algebraical arbitrary auxiliary equation binary Brioschi canonical equation canonical resolvent canonical surface co-ordinates coefficients collineations compute connection consider consideration construct corresponding covariant curve cyclic groups denote determine developments diagonal differential equation dihedral group dihedron discriminant edition elliptic functions elliptic modular functions equa establish exposition expression fact factor fifth degree form-problem formulae func further Galois geometrical given Gleichungen groups of linear hedral Hence Hermite Herr Gordan Herr Kronecker homogeneous homogeneous functions icosa icosahedral equation icosahedral substitutions icosahedron identical integral functions introduce invariant forms Jacobian equation linear complex linear substitutions magnitudes Math Mathematische Annalen means method modular equation obtain octahedron paragraph parameter permutations plane preceding chapter problem quadratic group rational functions regard relation remain unaltered remark respectively rotations simplest sixth solution square root sub-group tetrahedron theorem theory of equations tion Tschirnhausian transformation Ueber Unabridged republication values variables