INTRODUCTORY TREATISE ON LIES THEORY OF FINITE CONTINUOUS TRANSFORMATION GROUPS BY JOHN EDWARD CAMPBELL, M. A. FELLOW AND TUTOR OF HERTFORD COLLEGE, OXFORD AND MATHEMATICAL LECTURER AT UNIVERSITY COLLEGE, OXFORD OXFORD AT THE CLARENDON PRESS 3903 HENRY FKOWDE M. A. ISHER TO THE UNIVERSITY OF OXFORD LONDON, EDINBURGH NEW YORK PREFACE TN this treatise an attempt is made to give, in as elementary a form as possible, the main outlines of Lies theory of Continuous Groups. I desire to acknowledge my great indebtedness to Engels three standard volumes on this subject they If ve been constantly before rne, and but for their aid the present work could hardly have been undertaken. His Con tinuierliche Gruppen, written as it was under Lies own supervision, must always be referred to for the authoritative exposition of the theory in the form in which Lie left it. During the preparation of this volume I have consulted the several accounts which Scheffers has given of L s work in the books entitled Differential-gleichungen, Continuierliche Gruppen, and the Beriihrungs-Transformationen and also the inte resting sketch of the subject given by Klein in his lectures on Higher Geometry. In addition to these I have read a number of original memoirs, and would specially refer to the writings of Schur in the Mathe matische Annalen and in the Leipziger Berichte. Yet, great as are my obligations to others, I am not with out hope that even those familiar with the theory of Continuous Groups may find something new in the form in , which the theory is here presented. Within the limits of a volume of moderate size the reader will not expect to find an account of all parts of the subject. Thus the theory of the possible types of group-structure has been omitted. This branch of iv PREFACE group-theory has been considerably advanced by the labours of others than Lie especially by W. Killing, whose work is explained and extended by Cartan in his These sur la structure des groupes de transforma tions jinis et continus 1 . A justification of the omission of this part of the subject from an elementary treatise may perhaps also be found in the fact that it does not seem to have yet arrived at the completeness which characterizes other parts of the theory. The following statement as to the plan of the book may be convenient. The first chapter is in troductory, and aims at giving a general idea of the theory of groups. The second chapter contains elementary illustrations of the principle of extended point transformation. Chapters III-V establish the fundamental theorems of group-theory. Chapters VI and VII deal with the application of the theory to complete systems of linear partial differential equa tions of the first order. Chapter VIII discusses the invariant theories associated with groups. Chapter IX considers the division of groups into certain great classes. Chapter X considers when two groups are transformable, the one into the other. Chapter XI deals with isomorphism. Chapters XII and XIII show how groups are to be constructed when the structure constants are given. Chapter XIV discusses PfafFs equation and the integrals of non-linear partial differential equations of the first order. Chapter XV considers the theory of complete systems of homo geneous functions. Chapters XVI-XIX explain the theory of contact transformations. Chapter XX deals 1 See the article on Groups by Burnside in the Encyclopaedia Bri tannica. PREFACE v with the theory of Differential Invariants. Chapters XXI XXIV show how all possible types of groups can be obtained when the number of variables does not exceed three. Chapter XXV considers the relation subsisting between the systems of higher complex numbers and certain lineai groups. I have added a fairly full table of contents, a reference to which will, I think, make the general drift of the theory more easily grasped by the reader to whom the sub ject is new...