Cambridge University Press, 1927

**Preface**

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**

- Outline of the General Plane Theory
- Clebsch's Theorem
- The Quadratic Plane Transformation

I. Planes Distinct

II. Planes Superposed

III. Involutions - Composition and Resolution of Plane Transformations

I. The Problem of Composition and Resolution

II. Construction of Tables

III. Properties of the Characteristic Numbers - Transformation in One Plane

I. Plance Superposed

II. Involutions - Special Plane Transformations

I. The De Jonquières Transformations

II. Other Special Transformations - Resolution of Singularities of Plane Curves
- Noether's Theorem

- Outline of the General Space Theory
- The Quadro-Quadric Transformation

I. Spaces Distinct

II Spaces Superposed - Postulation and Equivalence
- Contact Conditions

I. Points of Total Contact

II. Points of Partial Contact

III. Curves of Contact - The Principal System
- Special Space Transformations

I. Transformations of Low Degree

II. The Bilinear T_{3-3}

III. Monoidal Transformations

IV. Other Special Types - A Cubo-Quartic Transformation
- 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 - History and Literature
- Bibliography