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In our introductory chapter on Kinematics, the consideration of Harmonic Motion naturally leads us to Fourier's Theorem, one of the most important of all analytical results as regards usefulness in physical science. In the Appendices to that chapter we have introduced an extension of Green's Theorem, and a short treatise on the remarkable functions known as Laplace's Coefficients. There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students. In the simplified and symmetrical form in which we give it, it will be found quite within the reach of readers moderately familiar with modern mathematical methods.

In the second chapter we give Newton's Laws of Motion in his own words, and with some of his own commentaries—every attempt that has yet been made to supersede them having ended in utter failure. Perhaps nothing so simple, and at the same time so comprehensive, has ever been given as the foundation of a system in any of the sciences. The dynamical use of the Generalized Coordinates of Lagrange, and the Varying Action of Hamilton, with kindred matter, complete the chapter.

The third chapter, "Experience," briefly treats of Observation and Experiment as the basis of Natural Philosophy.

The fourth chapter deals with the fundamental Units, and the chief Instruments used for the measurement of Time, Space, and Force

Thus closes the First Division of the work, which is strictly preliminary.

The Second Division is devoted to Abstract Dynamics (commonly of late years, but not well, called Mechanics). Its object is briefly explained in the introductory (fifth) chapter, and the rest of the present volume is devoted to Statics. Chapter vi., after a short notice of the Statics of a Particle, enters into considerable detail on the important subject of Attraction. In Chapter vn. the Statics of Solids and Fluids is treated with special detail in various important branches; such as the Deformation of Elastic Solids, the Statical Theory of the Tides, and the Figure and Rigidity of the Earth.

In the next volume, Division II. will be completed by chapters on the Kinetics of a Particle, and the Kinetics of Solids and Fluids. The Vibrations of Solids, and Wave-motion in general, will be fully treated. This volume will probably also contain Division m., which is to deal with Properties of Matter.

We believe that the mathematical reader will especially profit by a perusal of the large type portion of this volume; as he will thus be forced to think out for himself what he has been too often accustomed to reach by a mere mechanical application of analysis. Nothing can be more fatal to progress than a too confident reliance on mathematical symbols; for the student is only too apt to take the easier course, and consider the formula and not the fact as the physical reality.

A great deal of apparently purposeless matter has been introduced into the present volume, but it will be found to have a direct bearing on portions of the remaining three. The necessity of thus anticipating the wants of future volumes has been one of the main reasons for the delay in the publication of the present instalment, the printing having gone on at irregular intervals since November 1862.

The present volume has been printed by T. Constable, Esq., Printer to the Queen, and to the University of Edinburgh; and, as a specimen of mathematical printing, can scarcely be surpassed. The volume was not far from completion when we were informed that the Delegates of the Clarendon Press were desirous of publishing the work as one of their educational series. This gave us much pleasure, as it appeared likely to assist us in one of our main objects, the introduction into University.study and examinations of something like a complete course of Natural Philosophy. The three remaining volumes will, of course, be printed at Oxford.

Those illustrations in which accuracy was considered essential have been photographed on the wood-block by E. W. Dallas, Esq., F.R.S.E., from large drawings carefully executed by ourselves, or by Mr. D. Macfablaxe, Assistant to the Professor of Natural Philosophy in the University of Glasgow; and all have been engraved by Mr. J. Adam with a skill which leaves nothing to be desired.

W. THOMSON.

P. G. TAIT.

July 1867

ERRATA.

ITbe following are an we hare notice!, but there can hardly fail to Be a good many more. We shall be ranch obliged by being informed of any that may happen to be detected.)

Paoa

41, line 7, after tide, insert in the equilibrium theory (§ 811). „ line 8, for at open coast stations, read of this theory. 115, line 5 from foot, for Cup. Ir., read Vol. n. 131, upper marginal, for Freedon, read Freedom. 143, line 7 from foot, for equations, read equation. .. line 2 from foot, for degrees, read orders. 150, equation (40), in denominators, after (s+l) and («+2), insert .1 and .1.2.

line 7 from foot, after — , insert , if i s is odd.

152, line 5 from foot, for valuation, read evaluation.
150, end of line 2, for dif, readd£'; and end of line 11, for r, readr'.
„ equation (60), same correction as for page 150, equation (40).

lines 10 and 11 from foot, for previous notation, read notation of ()0). ., equation (61), in numerator and denominator of first factor, for (i — k) and i, read (« — J) and: 160, equation (G5), before Is -f- is, insert (. SI2, line 3 from foot, after momentum, insert as. ,. last line, transpose comma from motion to than.

Page

213, lines 10 and 25, for gravity, read inertia.

228, line 1, for (c), read (e).

236, equation (15), for fcty, read tdd.

(16), for tf, read W.

237, in first member of equation (22), delete vincula.

„„„ i- „„ r dx , dx, , . di j dt,

238, line 23, for -=- , read j—; and for -=— , read 3— • ''J dxi dx,' J dxt dxt

„ last line, for 'x,dt, read m&dt.

239, line after (26), for \p and 4, read yjr and 4,

240, in equation (31), for dT, read o~T.

243, line 3 from foot, after expression, intent , as distinguished from those obtainable by differentiation of (34), which are now denoted simply be

dA M etc
W If'

286, second line of § 352, for rectineal, read rectilineal.

304, 7th line of small type, for a, read the.

323, line 12, after 720 B.C., insert : but (§ 830) an error has been found in this calculation, and the corrected result renders it probable that the time of the earth's rotation is longer by —-— now than at that date.

Wa1 Wa*

335, line 7 from foot, for Q= - _- , etc., read <>= —j-> etc.

D1357, line 3, for +

ax

372, line 7 of case 1, for case, read cone.
377, line 7, for to, read to be.
401, line 8 from foot, for gravity, read inertia.
443, line 6, for axes, read axis.
582, line 2 from foot, for vanishes, read vanish.

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