Agnes Scott College

Hilda P. Hudson

Cremona Transformations in Plane and Space
Cambridge University Press, 1927

Cremona Transformations Cover Page


Cremona transformations are powerful tools in many lines of research; the aim of this book is to bring together all that has so far been published on their construction and use, as regards points and loci in two and three dimensions. The most important application is to the resolution of singularities of curves and surfaces, which is treated fully in Chapters VII and XVI. The useful and interesting part of a transformation is the way in which it changes the neighborhood of its fundamental elements, and the fundamental systems are first studied, both in general and in the examples which occur most often in the literature. Special attention is drawn to elements of contact.

A historical account of the subject is given in Chapter XVII; Cayley and cremona are still leaders, though we British have fallen behind the rest of the world in their track. Pure and analytical methods are here used together; a curve or surface, a function and an equation are treated as the same thing; under different aspects, and certain liberties of language are taken in this connection, which make the sentences shorter but not less clear.

The bibliography shows how varied is the mass of material; yet the space theory is far from complete, and it ishoped that the list of outstanding problems on p.394 may attract more workers. The most promising line of advance is probably from space of higher dimensions: this I must leave to other writers.

By the untimely death of Miss Grace Sadd, a mathematician of promise, and my friend and fellow-worker, the book has suffered loss and delay. Chapter III is mainly her work, also the collection of material for Chapters V and VI, and much helpful criticism of all the MS. which then existed.

I offer grateful thanks to Mr. Arthur Berry, who first introduced me to the subject; to Mr. T. L. Wren, for his skilled and tireless work on the proofs; to many correspondents who have helped with the bibliography; and to the Cambridge University Press for their care in the printing.

London, 1927

Table of Contents

Part I. Cremona Plane Transformations
  1. Outline of the General Plane Theory
  2. Clebsch's Theorem
  3. The Quadratic Plane Transformation
    I. Planes Distinct
    II. Planes Superposed
    III. Involutions
  4. Composition and Resolution of Plane Transformations
    I. The Problem of Composition and Resolution
    II. Construction of Tables
    III. Properties of the Characteristic Numbers
  5. Transformation in One Plane
    I. Plance Superposed
    II. Involutions
  6. Special Plane Transformations
    I. The De Jonquières Transformations
    II. Other Special Transformations
  7. Resolution of Singularities of Plane Curves
  8. Noether's Theorem
Part II. Cremona Space Transformations
  1. Outline of the General Space Theory
  2. The Quadro-Quadric Transformation
    I. Spaces Distinct
    II Spaces Superposed
  3. Postulation and Equivalence
  4. Contact Conditions
    I. Points of Total Contact
    II. Points of Partial Contact
    III. Curves of Contact
  5. The Principal System
  6. Special Space Transformations
    I. Transformations of Low Degree
    II. The Bilinear T3-3
    III. Monoidal Transformations
    IV. Other Special Types
  7. A Cubo-Quartic Transformation
  8. Resolution of Singularities of Surfaces
    I. Composition of Space Transformations
    II. Resolution of Singularities of Surfaces
    III. Second Method of Resolution
    IV. Classification of Transformations
  9. History and Literature
  10. Bibliography