The present work is an expansion of a course of lectures which I have annually delivered for some years past at Queen's College, Oxford.
Its primary object is, as was the case in the lecture room, that of explaining with all the clearness at my command the leading principles of invariant algebra, in the hope of making it evident to the junior student that the subject is attractive as well as important, and that its early difficulties are only such as he can readily surmount. Lucidity in mathematical works has often suffered from undue compression. My constant aim has been to guard against such a possibility here. In a book of moderate size dealing with a great subject much must remain unsaid, if the fundamental considerations are to be presented with the thoroughness and the perspicuity necessary to enable the student adequately to realize them, and give him the interest in them which will prepare him to pursue for himself the study to which they introduce him.
But, while the interests of the beginner have thus been given precedence, I am not without hope that the mathematician who is not new to the higher algebra will, especially in the chapters near the middle of the book, find in its pages matter of value to him as an aid to his researches. In some branches of the theory, which though of really elementary character are of comparatively recent investigation., as for instance in much of the algebra of differential operators, it is believed that a welcome supplement to previous treatises is offered.
The title 'Algebra of Quantics' is perhaps, one of my own introduction. It probably needs no defence, and can hardly fail to convey the right meaning. The mathematical world has now for half a century associated the algebra of invariants and covariants with the name of Cayley, and with his Memoirs on Quantics, so that it may perhaps be regarded as appropriate that a new work, appearing in the year which has seen the close of the labours of the renowned author of those memoirs, and dealing with their subject, should bear a name which recalls his memory.
To Salmon's Higher Algebra and his other works it is impossible to say how much I am indebted, both for direct reference and for guidance to the use of other authorities. Faà de Bruno's Formes Binaires has also been constantly before me. Of Clebsch's Binäre Formen and Gordan's Invariantentheorie less use has been made, as their symbolical method, and their successful, application of it to the great problem of the investigation of complete irreducible systems, have been reluctantly passed over with little more than an allusion in the following pages. A scanty chapter or two on this subject would have been utterly inadequate, and inconsistent with the general plan, as stated above, of an introductory treatise which prefers to omit rather than to obscure by condensation. A whole work which shall present to the English reader in his own language a worthy exposition of the method of the great German masters remains a desideratum.
The reader will not, however, find that the present work is a compilation from others which have preceded it, great as has been the help which those others have afforded Constant recourse has been had to the original authorities, particularly of course to Cayley's series of memoirs, and to Sylvester's writings in the Cambridge and Dublin Mathematical Journal, the American Journal of Mathematics, and elsewhere.
No bibliography of works and memoirs on the subject has been introduced. All mathematicians who wish to go deeply into the study of original authorities will have in their hands Dr F Meyer's Bericht über den gegenwärtigen Stand der Invariantentheorie in the Jahresbericht der Deutschen Mathematiker-Vereinigung for 1890-91, which is so full and thorough a bibliography and analysis of what has been done, especially in the later period of the history of the invariant theory, that it is hard to see how more can be desired. With regard to the originators of particular results, the difficulty continues, and has grown with the multitude of investigators, which was felt by Dr Salmon when he wrote, 'I can scarcely pretend to assign to their proper authors the merits of the several steps; and, as between Messrs Cayley and Sylvester, perhaps these gentlemen themselves, who were in constant communication with each other at the time, would now find it hard to say how much properly belongs to each.' To the difficulty with regard to Cayley and Sylvester may in particular be added that of discriminating between what in Salmon's work should be ascribed to them or others at all and what to Salmon himself. Throughout the following pages discoverers' names are very frequently attached to results; but it is too much to hope, though all care has been taken, that there are not cases in which the names given are those of authors in whose writings the results in question have certainly occurred, rather than those of the authors who first gave them.
I am indebted to several friends for suggestions and other help. Among them there is one, Mr J Hammond, M.A., one of the most distinguished of living researchers in the higher Algebra, to whom my especial thanks are due for a manuscript on the binary quintic which has been exceedingly helpful.
Some students, approaching the subject for the first time, will be advised to omit Chapters VII to XI till part of what follows them has been read.
E B ELLIOTT.
In preparing this new edition my aim has been, as before, to express principles clearly, with a directly didactic purpose, and to exhibit salient conclusions which have been derived from the principles by the use of those methods which are often characterized as distinctively English. The old arrangement, and the old numbering of articles, have been nearly always retained. A good deal has been added; and the additional has sometimes been stated with a brevity which in the original matter was deliberately avoided in the effort never to be obscure. Pains have been taken to insert the additions in places where they assist the argument, or at any rate do not impede its flow. When this has been found impossible or seriously difficult, a practice much resorted to in the first edition has been further adopted; and facts of interest or utility have been stated, with guidance to ways of obtaining them, in examples.
In two Chapters, V and XV, rearrangement has not been avoided. In the former I had neglected an important consideration; and in the latter stress has now been laid on facts of which the importance has only begun to be appreciated since the first edition appeared.
Altogether, fully one-eighth has been added to the book, but the skill of the staff of the University Press has kept the volume within its old bulk-without, as I trust, making it difficult to read.
A need to which I called attention in the preface to the first edition has been satisfied by Messrs Grace and Young in their Algebra of Invariants. I have therefore, with easy conscience, said little more than before about the powerful symbolic method which, in its parallelism to the English method of always keeping in view explicit forms, and bringing to bear upon them knowledge about linear differential operators and about enumerative arithmetic, has been called distinctively German. In such additions as I have made to my mention of that method I have as a rule adopted, not the usual symbolism, but a modification of it which stands in direct relationship to the calculus of differential operations.
My hope is that the work will still appeal, as I am glad to think it has in the past, to the beginner in Higher Algebra; will give him real interest in the subject, and provide him with sound knowledge in one of its departments. In endeavouring to improve this second edition, which is the last I shall live to produce, and is probably definitive, I have continued to think mainly for him, and to picture him as one better prepared for being led on from the simple - it may be the crude -to the elaborate, than for first receiving, and then applying, comprehensive theory. But I further trust that the book will retain, for some time to come, a certain value as one to be referred to by men and women who, while occupied with the continued advancement of Pure Mathematics, will need to look back upon and utilize the labours of those pioneers of modern Algebraic Theory, who adorned the latter half of the nineteenth century, and have now passed away.
E B ELLIOTT.
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