PREFACE.

To the first edition of this work, published in 1856, the following was prefixed:

"In the present Treatise will be found all the ordinary propositions, connected with the Dynamics of particles, which can be conveniently deduced without the use of D'Alembert's Principle.

"Its publication has been delayed by many unforeseen occurrences; more especially by the early and lamented death of Mr STEELE, whose portion of the work was left uncompleted, and whose assistance in its final arrangement and revision would have been invaluable. The principal portions due to him are the greater part of Chapters III., V. and VIII. together with a few pages of Chapter I.

"Considerable use has been made of Pratt's Mechanical Philosophy: indeed a large portion of Chapter XI. is reprinted verbatim from that work.

"Throughout the book will be found a number of illustrative examples introduced in the text, and for the most part completely worked out; others with occasional solutions or hints to assist the student are appended to each Chapter. For by far the greater portion of these the Cambridge SenateHouse and College Examination Papers have been applied to."

I am glad of the opportunity, presented by the call for a second edition, to make reparation for many of the faults of the first. Numerous trivial errors, and a few of a more serious character, have now been corrected; many sections and several new examples have been added; and the whole of the second Chapter has been rewritten, upon the basis of the corresponding portion of Thomson and Tait's Natural Philosophy which, though as yet unpublished, was printed off nearly two years ago.

When I wrote that Chapter, in 1855, I had not read Newton's admirable introduction to the Principia; and I endeavoured to make the best of the information I had then acquired from English and French treatises on Mechanics. These five pages, faulty and even erroneous as I have since seen them to be, cost me almost as much labour and thought as the utterly disproportionate remainder of my contributions to the volume. And I cannot but ascribe this result, in part at least, to the vicious system of the present day, which ignores Newton's Third Law of Motion, though constantly assuming it (tacitly) as an axiom; and erects Statics upon a separate basis from Kinetics, thereby necessitating several additional Physical Axioms, the splitting of Newton's Second Law into two, and the introduction of a so-called Statical measure of Force.

To be enabled to preserve the title of the work, I have added (apropos of the Second Law of Motion) a few hints about Statics of a particle.

The Examples are, for the most part, reprinted verbatim from the papers in which they were set; in a few the language has been altered, or the theorem involved has been generalized; several, however, have defied all attempts at improvement, and now stand in their unintelligibility as a

warning, to the Candidate for Mathematical Honors, of the ordeal he may have to pass through.

To several important theorems more than one demonstration has been appended: with the object of exhibiting the use of the various processes by applying them to the deduction of results of real value, instead of to the solution of "Problems" of unquestionable absurdity.

Various friends to whom I have applied for suggestions as to any important changes which they might think desirable in this second edition, and especially I. TODHUNTER, Esq. of St John's, have replied that they had none to offer, as they liked the book well enough in its original form. This has prevented me from attempting a thorough alteration of style which I had contemplated, viz. to cease breaking up the subject into detached propositions-specially fitted for "writing out." I retain my own opinion, however, that this is not the form in which such a treatise ought to be written; although there can be no doubt that it offers certain advantages to the student whose sole object in reading is to pass an examination.

The treatise is intended to be merely an analytical one: for the full discussion and experimental demonstration of the elementary principles on which the analysis is founded, the reader must be referred to works on Natural Philosophy; of which, so far as mere Abstract Dynamics is concerned, we have a most admirable example in the Principia. For the general application of modern theories to the whole range of physical phenomena, the reader is referred to the forthcoming work on Natural Philosophy by Professor W. THOMSON and myself, in which the subject will be developed from the grand basis of Conservation of Energy.

I have been dissuaded from introducing into this work the

Newtonian notation for Fluxions. It is true that in Kinetics of a particle it is not very greatly superior to the ordinary notation of differential coefficients: though, when the general equations of motion of a system have to be treated, in the beautiful manner invented by Lagrange, a partial use of it is absolutely necessary. Newton's idea of Fluxions was purely Kinematical; and, in fact, the fundamental ideas of the Differential Calculus are essentially involved in the most elementary considerations regarding velocity. It is also to be observed, that, whenever we write f'(x) for the differential coefficient of f(x), we are really employing the principal feature of Newton's. notation, though in a form somewhat more expressive than his.

It is possible that in this edition a few of the objectionable terms or methods, which the first edition contained, may have remained undetected-but I hope that in every essential respect the volume will be found to be an improvement on its predecessor.

I am encouraged in this hope by the fact that the sheets in passing through the press have been read by J. STIRLING, Esq. of Trinity, to whose care and knowledge I am indebted for many valuable suggestions.

P. GUTHRIE TAIT.

COLLEGE, EDINBURGH,

March, 1865.