Book contents
- Frontmatter
- ADVERTISEMENT
- Contents
- CLASSIFICATION
- 101 Notes on Lagrange's Theorem
- 102 On a Double Infinite Series
- 103 On Certain Definite Integrals
- 104 On the Theory of Permutants
- 105 Correction to the Postscript to the Paper on Permutants
- 106 On the Singularities of Surfaces
- 107 On the Theory of Skew Surfaces
- 108 On certain Multiple Integrals connected with the Theory of Attractions
- 109 On the Rationalisation of certain Algebraical Equations
- 110 Note on the Transformation of a Trigonometrical Expression
- 111 On a Theorem of M. Lejeune-Dirichlet's
- 112 Demonstration of a Theorem relating to the Products of Sums of Squares
- 113 On the Geometrical Representation of the Integral
- 114 Analytical Researches connected with Steiner's Extension of Malfatti's Problem
- 115 Note on the Porism of the In-and-circumscribed Polygon
- 116 Correction of two Theorems relating to the In-and-circumscribed Polygon
- 117 Note on the Integral
- 118 On the Harmonic Relation of two Lines or two Points
- 119 On a Theorem for the Development of a Factorial
- 120 Note on a Generalisation of the Binomial Theorem
- 121 Note on a Question in the Theory of Probabilities
- 122 On the Homographic Transformation of a Surface of the Second Order into Itself
- 123 On the Geometrical Representation of an Abelian Integral
- 124 On a Property of the Caustic by Refraction of the Circle
- 125 On the Theory of Groups as depending on the Symbolical Equation θn = 1
- 126 On the theory of Groups as depending on the Symbolical Equation θn = 1. Second Part
- 127 On the Homographic Transformation of a Surface of the Second Order into itself
- 128 Developments on the Porism of the In-and-circumscribed Polygon
- 129 On the Porism of the In-and-circumscribed Triangle, and on an irrational Transformation of two Ternary Quadratic Forms each into itself
- 130 Deuxième Mémoire sur les Fonctions doublement Périodiques
- 131 Nouvelles Recherches sur les Covariants
- 132 Réponse à une Question proposée par M. Steiner
- 133 Sur un Théorème de M. Schläfli
- 134 Remarques sur la Notation des Fonctions Algébriques
- 135 Note sur les Covariants d'une Fonction Quadratique, Cubique, ou Biquadratique à deux Indéterminées
- 136 Sur la Transformation d'une Fonction Quadratique en ellemême par des Substitutions linéaires
- 137 Recherches Ultérieures sur les Déterminants gauches
- 138 Recherches sur les Matrices dont les termes sont des fonctions linéaires d'une seule Indéterminée
- 139 An Introductory Memoir on Quantics
- 140 Researches on the Partition of Numbers
- 141 A Second Memoir on Quantics
- 142 Numerical Tables Supplementary to Second Memoir on Quantics
- 143 Tables of the Covariants M to W of the Binary Quintic: from the Second, Third, Fifth, Eighth, Ninth and Tenth Memoirs on Quantics
- 144 A Third Memoir on Quantics
- 145 A Memoir on Caustics
- 146 A Memoir on Curves of the Third Order
- 147 A Memoir on the Symmetric Functions of the Roots of an Equation
- 148 A Memoir on the Resultant of a System of two Equations
- 149 On the Symmetric Functions of the Roots of certain Systems of two Equations
- 150 A Memoir on the Conditions for the Existence of given Systems of Equalities among the Roots of an Equation
- 151 Tables of the Sturmian Functions for Equations of the Second, Third, Fourth, and Fifth Degrees
- 152 A Memoir on the Theory of Matrices
- 153 A Memoir on the Automorphic Linear Transformation of a Bipartite Quadric Function
- 154 Supplementary Researches on the Partition of Numbers
- 155 A Fourth Memoir on Quantics
- 156 A Fifth Memoir on Quantics
- 157 On the Tangential of a Cubic
- 158 A Sixth Memoir on Quantics
- Notes and References
127 - On the Homographic Transformation of a Surface of the Second Order into itself
Published online by Cambridge University Press: 03 May 2011
Book contents
- Frontmatter
- ADVERTISEMENT
- Contents
- CLASSIFICATION
- 101 Notes on Lagrange's Theorem
- 102 On a Double Infinite Series
- 103 On Certain Definite Integrals
- 104 On the Theory of Permutants
- 105 Correction to the Postscript to the Paper on Permutants
- 106 On the Singularities of Surfaces
- 107 On the Theory of Skew Surfaces
- 108 On certain Multiple Integrals connected with the Theory of Attractions
- 109 On the Rationalisation of certain Algebraical Equations
- 110 Note on the Transformation of a Trigonometrical Expression
- 111 On a Theorem of M. Lejeune-Dirichlet's
- 112 Demonstration of a Theorem relating to the Products of Sums of Squares
- 113 On the Geometrical Representation of the Integral
- 114 Analytical Researches connected with Steiner's Extension of Malfatti's Problem
- 115 Note on the Porism of the In-and-circumscribed Polygon
- 116 Correction of two Theorems relating to the In-and-circumscribed Polygon
- 117 Note on the Integral
- 118 On the Harmonic Relation of two Lines or two Points
- 119 On a Theorem for the Development of a Factorial
- 120 Note on a Generalisation of the Binomial Theorem
- 121 Note on a Question in the Theory of Probabilities
- 122 On the Homographic Transformation of a Surface of the Second Order into Itself
- 123 On the Geometrical Representation of an Abelian Integral
- 124 On a Property of the Caustic by Refraction of the Circle
- 125 On the Theory of Groups as depending on the Symbolical Equation θn = 1
- 126 On the theory of Groups as depending on the Symbolical Equation θn = 1. Second Part
- 127 On the Homographic Transformation of a Surface of the Second Order into itself
- 128 Developments on the Porism of the In-and-circumscribed Polygon
- 129 On the Porism of the In-and-circumscribed Triangle, and on an irrational Transformation of two Ternary Quadratic Forms each into itself
- 130 Deuxième Mémoire sur les Fonctions doublement Périodiques
- 131 Nouvelles Recherches sur les Covariants
- 132 Réponse à une Question proposée par M. Steiner
- 133 Sur un Théorème de M. Schläfli
- 134 Remarques sur la Notation des Fonctions Algébriques
- 135 Note sur les Covariants d'une Fonction Quadratique, Cubique, ou Biquadratique à deux Indéterminées
- 136 Sur la Transformation d'une Fonction Quadratique en ellemême par des Substitutions linéaires
- 137 Recherches Ultérieures sur les Déterminants gauches
- 138 Recherches sur les Matrices dont les termes sont des fonctions linéaires d'une seule Indéterminée
- 139 An Introductory Memoir on Quantics
- 140 Researches on the Partition of Numbers
- 141 A Second Memoir on Quantics
- 142 Numerical Tables Supplementary to Second Memoir on Quantics
- 143 Tables of the Covariants M to W of the Binary Quintic: from the Second, Third, Fifth, Eighth, Ninth and Tenth Memoirs on Quantics
- 144 A Third Memoir on Quantics
- 145 A Memoir on Caustics
- 146 A Memoir on Curves of the Third Order
- 147 A Memoir on the Symmetric Functions of the Roots of an Equation
- 148 A Memoir on the Resultant of a System of two Equations
- 149 On the Symmetric Functions of the Roots of certain Systems of two Equations
- 150 A Memoir on the Conditions for the Existence of given Systems of Equalities among the Roots of an Equation
- 151 Tables of the Sturmian Functions for Equations of the Second, Third, Fourth, and Fifth Degrees
- 152 A Memoir on the Theory of Matrices
- 153 A Memoir on the Automorphic Linear Transformation of a Bipartite Quadric Function
- 154 Supplementary Researches on the Partition of Numbers
- 155 A Fourth Memoir on Quantics
- 156 A Fifth Memoir on Quantics
- 157 On the Tangential of a Cubic
- 158 A Sixth Memoir on Quantics
- Notes and References
Summary
I pass to the improper transformation. Sir W. R. Hamilton has given (in the note, p. 723 of his Lectures on Quaternions [Dublin, 1853)] the following theorem:—If there be a polygon of 2m sides inscribed in a surface of the second order, and (2m − 1) of the sides pass through given points, then will the 2m-th side constantly touch two cones circumscribed about the surface of the second order. The relation between the extremities of the 2m-th side is that of two points connected by the general improper transformation; in other words, if there be on a surface of the second order two points such that the line joining them touches two cones circumscribed about the surface of the second order, then the two points are as regards the transformation in question a pair of corresponding points, or simply a pair. But the relation between the two points of a pair may be expressed in a different and much more simple form. For greater clearness call the surface of the second order U, and the sections along which it is touched by the two cones, θ, ϕ; the cones themselves may, it is clear, be spoken of as the cones θ, ϕ. And let the two points be P, Q. The line PQ touches the two cones, it is therefore the line of intersection of the tangent plane through P to the cone θ, and the tangent plane through P to the cone ϕ.
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- Information
- The Collected Mathematical Papers , pp. 133 - 137Publisher: Cambridge University PressPrint publication year: 2009First published in: 1889