ledge of Determinants and their applications. We have therefore given a brief elementary discussion of Determinants in Chapter XXXIII., in the hope that it may provide the student with a useful introductory course, and prepare him for a more complete study of the subject. The last chapter contains all the most useful propositions in the Theory of Equations suitable for a first reading. The Theory of Equations follows so naturally on the study of Algebra that no apology is needed for here introducing propositions which usually find place in a separate treatise. In fact, a considerable part of Chapter XXXV. may be read with advantage at a much earlier stage, and may conveniently be studied before some of the harder sections of previous chapters. It will be found that each chapter is as nearly as possible complete in itself, so that the order of their succession can be varied at the discretion of the teacher; but it is recommended that all sections marked with an asterisk should be reserved for a second reading. In enumerating the sources from which we have derived assistance in the preparation of this work, there is one book to which it is difficult to say how far we are indebted. Todhunter's Algebra for Schools and Colleges has been the recognised English text-book for so long that it is hardly possible that any one writing a text-book on Algebra at the present day should not be largely influenced by it. At the same time, though for many years Todhunter's Algebra has been in constant use among our pupils, we have rarely adopted the order and arrangement there laid down; in many chapters we have found it expedient to make frequent use of alternative proofs; and we have always largely supplemented the text by manuscript notes. These notes, which now appear scattered throughout the present work, have been collected at different times during the last twenty Н. Н. А. b years, so that it is impossible to make definite acknowledgement in every case where assistance has been obtained from other writers. But speaking generally, our acknowledgements are chiefly due to the treatises of Schlömilch, Serret, and Laurent; and among English writers, besides Todhunter's Algebra, we have occasionally consulted the works of De Morgan, Colenso, Gross, and Chrystal. To the Rev. J. Wolstenholme, D.Sc., Professor of Mathematics at the Royal Indian Engineering College, our thanks are due for his kindness in allowing us to select questions from his unique collection of problems; and the consequent gain to our later chapters we gratefully acknowledge. It remains for us to express our thanks to our colleagues and friends who have so largely assisted us in reading and correcting the proof sheets; in particular we are indebted to the Rev. H. C. Watson of Clifton College for his kindness in revising the whole work, and for many valuable suggestions in every part of it. May, 1887. H. S. HALL, S. R. KNIGHT. PREFACE TO THE THIRD EDITION. IN this edition the text and examples are substantially the same as in previous editions, but a few articles have been recast, and all the examples have been verified again. We have also added a collection of three hundred Miscellaneous Examples which will be found useful for advanced students. These examples have been selected mainly but not exclusively from Scholarship or Senate House papers; much care has been taken to illustrate every part of the subject, and to fairly represent the principal University and Civil Service Examinations. March, 1889. CONTENTS. Inverse variation Joint variation If Ax B when C is constant, and AC when B is constant, then Formation of equations when the roots are given Conditions that the roots of a quadratic should be (1) equal in magni- Definitions of function, variable, rational integral function Condition that ax2+2hxy + by2+2gx+2fy+c may be resolved into two |