THE purpose of this book is to prescnt a straightforward introduction to complex numbers and their properties. Complex numbers, like other kinds of numbers, are essen tially objects with which to perform calculations a: cording to certain rules, and when this principle is borne in mind, the nature of complex numbers is no more mysterious than that of the more familiar types of numbers. This formal approach has recently been recommended in a Reportt prepared for the Mathematical Association. We believe that it has distinct advantages in teaching and that it is more in line with modern algebraical ideas than the alternative geometrical or kinematical definitions of v -1 that used to be proposed. On the other hand, an elementary textbook is clearly not the place to enter into a full discussion of such questions as logical consistency, which would have to be included in a rigorous axiomatic treatment. However, the steps that had to be omitted (with due warning) can easily be filled in by the methods of abstract algebra, which do not conflict with the 'naive' attitude adopted here. I should like to thank my friend and colleague Dr. J. A. Green for a number of valuable suggestions, especially in connection with the chapter on convergence, which is a sequel to his volume Sequences and Series in this Library."

The aim of this book is to give an elementary treatment of multiple integrals. The notions of integrals extended over a curve, a plane region, a surface and a solid are introduced in tum, and methods for evaluating these integrals are presented in detail. Especial reference is made to the results required in Physics and other mathematical sciences, in which multiple integrals are an indispensable tool. A full theoretical discussion of this topic would involve deep problems of analysis and topology, which are outside the scope of this volume, and concessions had to be made in respect of completeness without, it is hoped, impairing precision and a reasonable standard of rigour. As in the author's Integral Calculus (in this series), the main existence theorems are first explained informally and then stated exactly, but not proved. Topological difficulties are circumvented by imposing some what stringent, though no unrealistic, restrictions on the regions of integration. Numerous examples are worked out in the text, and each chapter is followed by a set of exercises. My thanks are due to my colleague Dr. S. Swierczkowski, who read the manuscript and made valuable suggestions. w. LEDERMANN The University of Sussex, Brighton."

THE purpose of this book is to present a straightforward introduction to complex numbers and their properties. Complex numbers, like other kinds of numbers, are essen tially objects with which to perform calculations according to certain rules, and when this principle is borne in mind, the nature of complex numbers is no more mysterious than that of the more familiar types of numbers. This formal approach has recently been recommended in a Reportt prepared for the Mathematical Association. We believe that it has distinct advantages in teaching and that it is more in line with modern algebraical ideas than the alternative geometrical or kinematical definitions of ..; - 1 that used to be proposed. On the other hand, an elementary textbook is clearly not the place to enter into a full discussion of such questions as logical consistency, which would have to be included in a rigorous axiomatic treatment. However, the steps that had to be omitted (with due warning) can easily be filled in by the methods of abstract algebra, which do not conflict with the 'naive' attitude adopted here. I should like to thank my friend and colleague Dr. J. A. Green for a number of valuable suggestions, especially in connection with the chapter on convergence, which is a sequel to his volume Sequences and Series in this Library."

Die elementaren Operationen in der Arithmetik bestehen darin, daB man zwei ZaWen a und b in Ubereinstimmung mit einigen wohldefinierten Regeln verkniipft und so eine neue eindeutig bestimmte zaW c erMlt. Nehmen wir zum Beispiel als Verkniipfungsregel die Multiplikation, so schreiben wir c = ab. Wenn a und b gegeben sind, dann kann die zaW c in jedem Fall gefunden werden. Es ist bekannt, daB die Multiplikation von zwei oder mehreren Zahlen gewissen for- malen Regeln gehorcht, welche fur aile Produkte gelten, unabhiingig yom spezieilen nume- rischen Wert: (Ll) ab = ba; Kommutativgesetz (1. 2) (ab)c = a(bc) Assoziativgesetz (1. 3) la=al=a Die letzte Gleichung hat die Einftihrung eines spezieilen Elementes, des Einselementes, zur Folge. Das zweite Gesetz lautet ausftihrlicher: wenn wir ab = s und bc = t setzen, dann gilt immer sc = at. In der axiomatischen Behandlung der Arithmetik ist es iiblich, zuerst die Axiome oder Postulate etwa solche wie (1. 1), (1. 2) und (1. 3) festzulegen, sowie auch gewisse andere Ver- fahrensregeln beziiglich der Addition oder der Multiplikation einzuftihren, und man leitet davon dann die logischen Folgerungen abo Es ist dabei am Anfang unwesentlich, ob die Symbole a, b, . . . ZaWen, wie wir sie im iiblichen Sinne verstehen darstellen, oder etwa an- dere mathematische Gr6Ben, ja man verzichtet oft auf eine konkrete Interpretation. Es sind auch zaWreiche axiomatische Systeme im logischen Sinne m6glich, jedoch sind diese nicht alle in gleicher Weise interessant oder wichtig.

Professor Walter Ledermann is one of the great algebraists of the twentieth century. His memoirs begin with life in pre-war Germany, the murder of several members of his family, and of the joy he found in mathematics and music. As the story of his remarkable life unfolds, we are entranced by tales of Scotland during the war and of academic life in Manchester and Sussex. His memoirs contain numerous entertaining, and often hilarious anecdotes of his encounters with famous mathematicians and physicists, such as Issai Schur, Heinz Hopf, Max Plank, Erwin Schroedinger, Edmund Whittaker, Alec Aitkin, Max Born and Alan Turing.

To an algebraist the theory of group characters presents one of those fascinating situations, where the structure of an abstract system is elucidated by a unique set of numbers inherent in the system. But the subject also has a practical aspect, since group characters have gained importance in several branches of science, in which considerations of symmetry play a decisive part. This is an introductory text, suitable for final-year undergraduates or postgraduate students. The only prerequisites are a standard knowledge of linear algebra and a modest acquaintance with group theory. Especial care has been taken to explain how group characters are computed. The character tables of most of the familiar accessible groups are either constructed in the text or included amongst the exercise, all of which are supplied with solutions. The chapter on permutation groups contains a detailed account of the characters of the symmetric group based on the generating function of Frobenius and on the Schur functions. The exposition has been made self-sufficient by the inclusion of auxiliary material on skew-symmetric polynomials, determinants and symmetric functions.