Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree

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Courier Corporation, Jan 1, 2003 - Mathematics - 289 pages
This 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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