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 alwayslargely 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
