1.1. From the Preface.

The following attempt to set forth in a systematic and connected form the present state of the theory of the Motion of Fluids, had its origin in a course of lectures delivered in Trinity College, Cambridge, in 1874, when the need for a treatise on the subject was strongly impressed on my mind. Various circumstances have retarded the completion of the work in a form fit for the press; but as the delay has enabled me to incorporate the results of several important recent investigations, and altogether to render the volume less inadequate to its purpose than it would otherwise have been, this is hardly matter for regret.

I have endeavoured, throughout the book, to attribute to their proper authors the various steps in the development of the subject. The list of Memoirs and Treatises at the end of the book has no pretensions to completeness, and as it is to a great extent based on notes which I have no present means of verifying, some of the references may possibly be inexact. I trust however that the list may, in spite of these drawbacks, be of service to the student who wishes to consult the original authorities.

Horace Lamb, Adelaide, 16 May 1879.

2.1. From the Preface.

This book may be regarded as a second edition of a "Treatise on the Mathematical Theory of the Motion of Fluids," published in 1879, but the additions and alterations are so extensive that it has been thought proper to make a change in the title.

I have attempted to frame a connected account of the principal theorems and methods of the science, and of such of the more important applications as admit of being presented within a moderate compass. It is hoped that all investigations of fundamental importance will be found to have been given with sufficient detail, but in matters of secondary or illustrative interest I have often condensed the argument, or merely stated results, leaving the full working out to the reader.

In making a selection of the subjects to be treated I have been guided by considerations of physical interest. Long analytical investigations, leading to results which cannot be interpreted, have as far as possible been avoided. Considerable but, it is hoped, not excessive space has been devoted to the theory of waves of various kinds, and to the subject of viscosity. On the other hand, some readers may be disappointed to find that the theory of isolated vortices is still given much in the form in which it was. left by the earlier researches of von Helmholtz and Lord Kelvin, and that little reference is made to the subsequent investigations of J J Thomson, W M Hicks, and others, in this field. The omission has been made with reluctance, and can be justified only on the ground that the investigations in question derive most of their interest from their bearing on kinetic theories of matter, which seem to lie outside the province of a treatise like the present.

I have ventured, in one important particular, to make a serious innovation in the established notation of the subject, by reversing the sign of the velocity-potential. This step has been taken not without hesitation, and was only finally decided upon when I found that it had the countenance of friends whose judgment I could trust; but the physical interpretation of the function, and the far-reaching analogy with the magnetic potential, are both so much improved by the change that its adoption appeared to be, sooner or later, inevitable.

I have endeavoured, throughout the book, to attribute to their proper authors the more important steps in the development of the subject. That this is not always an easy matter is shown by the fact that it has occasionally been found necessary to modify references given in the former treatise, and generally accepted as correct. I trust, therefore, that any errors of ascription which remain will be viewed with indulgence. It may be well, moreover, to warn the reader, once for all, that I have allowed myself a free hand in dealing with the materials at my disposal, and that the reference in the footnote must not always be taken to imply that the method of the original author has been closely followed in the text. I will confess, indeed, that my ambition has been not merely to produce a textbook giving a faithful record of the present state of the science, with its achievements and its imperfections, but, if possible, to carry it a step further here and there, and at all events by the due coordination of results already obtained to lighten in some degree the labours of future investigators. I shall be glad if I have at least succeeded in conveying to my readers some of the fascination which the subject has exerted on so long a line of distinguished writers.

In the present subject, perhaps more than in any other department of mathematical physics, there is room for Poinsot's warning "Gardons nous de croire qu'une science soit faite quand on l'a reduite à des formules analytiques." I have endeavoured to make the analytical results as intelligible as possible, by numerical illustrations, which it is hoped will be found correct, and by the insertion of a number of diagrams of stream-lines and other curves, drawn to scale, and reduced by photography. Some of these cases have, of course, been figured by previous writers, but many are new, and in every instance the curves have been calculated and drawn independently for the purposes of this work.

Horace Lamb, May, 1895.

2.2. Review by: Ernest W Brown.
Bull. Amer. Math. Soc. 4 (1897), 73-80.

The appearance of a new treatise on any branch of higher mathematics rarely calls for anything else than congratulations to the author, and the volume before us is no exception to the rule. The problems of hydrodynamics present so many difficulties and the opportunities for students to obtain a connected view of them are so rare that any additional help is valuable. Professor Lamb, however, has gone much further than merely producing a continuous account of the subject as it stands at the present time. He has given us a treatise which will easily rank first amongst those in the English and perhaps in any language. The only other English treatise of the same scope, that by Basset published in 1888, although an advance on those which had previously appeared, rather suffers by comparison, both in its plan and the manner in which it is carried out. ...

The author is to be congratulated on the completion of a task which will earn him the gratitude of all those who are now or may in the future be interested in hydrodynamics. The manner in which his materials are put together and the fact that he never loses sight of *the practical applications make the book unusually interesting ; the large number of references will enable anyone to find out all that has been done in any branch. In fact, although the volume is a bulky one, we cannot but regret that it has not been divided into two and extended by including the investigations noted above as omitted, and by giving a much fuller index of subjects.

2.3. Review by: R D Carmichael (of 5th edition (1924)).
Bull. Amer. Math. Soc. 31 (8) (1925), 460-461.

This work reached its fifth edition in the forty-fifth year after its original publication in 1879, the successive editions having appeared at intervals of sixteen, eleven, ten, and eight years. In passing from the first edition to the second it was largely remodelled and extended, but since then there has been no change in general plan and arrangement. As it now appears it has again been carefully revised, several passages having been rewritten and some considerable additions having been made. It has, however, not undergone extensive enlargement. In his preface the author observes that "the work has less pretensions than ever to be regarded as a complete account of the science with which it deals," owing partly to the difficulty of doing justice to the growing literature. He adds: "Some memoirs deal chiefly with questions of mathematical method and so fall outside the scope of this book; others though physically important hardly admit of a condensed analysis; others, again, owing to the multiplicity of publications, may unfortunately have been overlooked. And there is, I am afraid, the inevitable personal equation of the author, which leads him to take a greater interest in some branches of the subject than in others." Though the book is extensive and important it is clear from these remarks that there are important aspects of the subject which it does not adequately treat - a fact which is stated without any implication of criticism, since it did not fall within the author's purpose to treat them and since the book probably serves better a large majority of its readers on account of the omission of certain subjects and the consequent fuller treatment of others.

The principal single increase in the scope of this work occurred in passing from the first to the second edition. The interpolations and additions incorporated in the third edition amounted to about one-fifth of the whole and thus required a renumbering of the sections. There was considerable increase in size in the fourth edition even though a few investigations of secondary interest were condensed or omitted. There are no omissions of consequence in passing from the fourth edition to the fifth; and there are several additions. Of the latter some are of the nature of interpolations (mostly short) inserted in the sections which are reproduced from the earlier edition, while the others treat new topics in the form of additional articles of which there are fourteen scattered through the volume. Throughout the whole work there is repeated evidence of careful revision both in the addition of new matter and even in the smaller details of phraseology. The printing in the fifth edition, at least in the case of formulas, is a little more compact than in the fourth, and there is a considerable increase in the amount of matter in smaller type so that there is a reduction of twenty-one pages in size of the volume while there is an actual increase in matter.

As is to be expected in a fifth edition, the author adheres closely to the terminology which he had previously adopted. The most considerable change which came to the reviewer's attention is in the usage of the words "stream-line" and "stream-tube" as set forth in Sections 19 and 21; and even this is not of great importance.

In its new edition the book will continue to hold, and a little more effectively than ever before, the important place which it has now held for a generation.

2.4. Review by: Sydney Goldstein (of 6th edition (1932)).
The Mathematical Gazette 17 (224) (1933), 215-217.

It is fifty-four years ago since Sir Horace Lamb first published his Treatise on the 'Mathematical Theory of the Motion of Fluids' as a smallish volume of some 250 pages. The second edition, published sixteen years afterwards, was more than twice the size of the first, and bore the altered title 'Hydrodynamics'. By the third edition, published in 1906, both the outward appearance and the general character of the book were determined. The classical theory of the motion of an ideal, inviscid, incompressible fluid, and the theory of waves, were being treated by a master. It is noteworthy that in each of the last three editions over 300 pages out of a total of some 700 have been devoted to waves. The excellent discussion of vortex motion has been given without explanation of the existence of vortices; one long chapter, of something well over a hundred pages, has been devoted to the motion of real, viscous fluids, incapable of slipping freely over their solid boundaries, and this chapter has included mention of some of the difficult questions of turbulent motion; whilst a final chapter has dealt with rotating masses of liquid.

In this, the sixth, edition the character of the book is, then, maintained. It remains the leading treatise on classical hydrodynamics, and serves as a corrective to some of the trends in recent compilations, such that, on the one hand, a report on hydrodynamics can be published without mention of tides or waves and very little mention of classical hydrodynamics at all, and, on the other hand, the theory of Oseen and the extensions of it made last year and the year before can be included in a section on classical hydrodynamics. Moreover, Sir Horace Lamb can rightly claim in his preface to the new edition that classical hydrodynamics has been found to have a widening field of practical applications. Nevertheless, these applications mostly require the inclusion of circulation and vorticity, for the explanation of whose existence recourse must be made to viscosity and boundary layers; and Sir Horace Lamb has nowhere included an adequate discussion of the production of circulation and vorticity in fluids of small viscosity.

3.1. From the Preface.

This book attempts to teach those portions of the Calculus which are of primary importance in the application to such subjects as Physics and Engineering. Purely analytical developments, and processes, however ingenious, which are seldom useful in practice, are for the most part omitted. Stress is laid on fundamental principles; and an endeavour has been made to cultivate the power of applying these in simple cases; but dexterity in manipulating complicated expressions has been deemed less essential.

Although the author has had in view the needs of a special class of students, he has not thought it right to construe his task too narrowly; and he has not scrupled to discuss questions of theoretical interest when they appeared to him to be at once simple, attractive, and relevant. He trusts therefore that the book may also be found accept able to mathematical students who may desire to take a general survey of the subject before involving themselves deeply in any of its more elaborate developments.

The arrangement is substantially that which has been adopted for a long series of years in lectures at the Owens College. As far as possible, priority is given to the simplest and most generally useful parts of the subject. In particular, the Integral Calculus is introduced at an early period; the propriety of this, in an elementary course, is indeed now becoming widely recognised. An attempt is made, as each stage in the subject is reached, to bring in applications of the theory to questions of interest. Thus, after the rules for differentiation come the applications of the derived function to problems of maxima and minima and to geometry; the rules for integration are followed by the application to areas, volumes, etc.; and so on. In the exposition of the theory, dynamical and physical, as well as geometrical, illustrations have been freely employed. It would have been possible to go much further in this respect, and so add greatly to the practical aspect of the book; but it was thought best to refrain from illustrations which, lying as yet too far out of the student's track, might present obscurities of their own greater than those which they were intended to banish. It is to be clearly understood, indeed, that the object aimed at in this work is not to teach Dynamics or Physics or Engineering, but to exercise the reader in the kind of Mathematics which he will find most useful for the study of those subjects.

Except for a few indications, the book deals only with functions of a single independent variable. Owing to this limitation, and to some further omissions to be noticed, it has been possible to include an elementary account of Differential Equations, as well as of the Differential and Integral Calculus, as ordinarily understood.

The omissions referred to include the general theory of Indeterminate Forms. This is somewhat tedious to establish rigorously; a good deal of it is very artificial; and, practically, the rules are not used by mathematicians, who have recourse, when the occasion arises, to more direct methods of evaluation. Again, the general theory of Algebraic Curves, with the determination of asymptotes and singularities, is properly a branch of Analytical Geometry, and requires for its successful prosecution something beyond the purely quantitative faculty which it is the office of a work like the present to develop. A few articles are however devoted to the graphical study of simple curves of the types y = f(x), y2 = f(x), such as not infrequently present themselves in the practical applications of the subject. The accounts given of methods of integration, and of solution of differential equations, are designed to include only the cases which are most useful in practice, and have no pretensions to be exhaustive.

The demands made on the previous knowledge of the student are not very extensive. The reader is of course assumed to be familiar with the elements of Geometry, Algebra, and Trigonometry, and with the fundamental notions of Analytical Geometry, but not necessarily with such things as the theory of Infinite Series, or the detailed theory of the Conic Sections. Imaginary quantities are nowhere employed in the book. The introductory Chapter will be found to supply what is required outside the above list; but its main object is to establish carefully the fundamental notions of continuity, and of limiting values, without which no real mastery of the subject can be attained. This Chapter is necessarily somewhat lengthy; and many students, especially those whose previous reading is more extensive, may be tempted to hasten over it, and proceed rapidly to Chapter II, where the specific study of the Calculus begins. It is hoped, however, that all will find it profitable to return occasionally to the introduction and revise their knowledge of the various matters there treated.

Considerable attention has been paid to the logic of the subject. Writers of textbooks, however elementary, cannot remain permanently indifferent to the investigations of the modern Theory of Functions (of a real variable), although opinions may differ widely as to the character and extent of the influence which these should exert. It is not claimed that the proofs of fundamental propositions which are here offered have the formal precision of statement which is de rigueur in the theory referred to; but it is hoped that in substance they will be found to be correct. Occasionally, where a rigorous proof of a theorem in its full generality would be long or intricate, it has been found possible, by introducing some additional condition into the statement, to simplify the argument, without really impairing the practical value of the theorem.

After the first seven chapters considerable discretion may be used as to what shall be read or omitted. Chapter VIII is occupied with applications of the process of integration to the determination of centres of mass and moments of inertia. Chapter IX investigates the properties of various special classes of curves; it need not be read consecutively, but can be referred to as occasion arises. The earlier portions of Chapter X, on Curvature, should not be passed over; but the investigations which follow are of a more technical character. The chapters on Differential Equations will probably be found to contain little that is superfluous.

The last two chapters, dealing with the theory of Infinite Series in their relation to the Calculus, and with Taylor's Theorem, require a more sustained attention than the pre ceding parts of the book. The position to which the latter theorem has been relegated may perhaps cause some sur prise. In the author's opinion, the place usually assigned to it in the exposition of the subject makes it a grievous and unnecessary obstacle to the beginner, and is a misrepresentation of its real importance. When barely over the threshold the student is confronted with a theorem which can only be established by a long and somewhat difficult investigation; and yet, from the constant appeal to it in the sequel, he is led to imagine that it is indispensable for further progress in the subject. A reference to the table of Contents of this book will show what a long array of important matters can be treated without having recourse to it. In its proper connection, the theorem is of course most important, and considerable space has been devoted in Chapter XIV to its discussion, and to the development of some of its consequences. This is preceded, in Chapter XIII, by a somewhat careful study of power - series in general, as subjects for the operations of the Calculus. The matters here treated are not essential to the understanding of Taylor's Theorem, but they cannot safely be neglected by those who wish to acquire the power of dealing intelligently with such series, and are not satisfied with the mere manipulation of symbols.

The Examples for practice have been chosen and arranged with some care. Having regard to the main purpose of the book, it is hoped that the great majority of them will be found to be easy. At all events, difficult examples have not as a rule been admitted unless they appeared to involve some result of interest. A few of the examples are original, and some have been derived from various recent sources, but many will be recognised by experts as having figured (rightly) in successive generations of textbooks.

The diagrams which profess to represent specific curves have for the most part been drawn to scale. The author is greatly indebted to the staff of the University Press for the trouble that has been taken to reproduce them accurately, as well as for much courteous assistance in other ways during the printing.

Although a few historical notes have been inserted here and there, the systematic citation of authorities has not been attempted. In a subject which has received gradual improvement at so many hands it would, indeed, often be difficult to discover the rightful originator of a particular theorem or mode of demonstration. But so far as the author is aware, no proof appearing to possess points of originality has been appropriated from a recent work without acknowledgment. And without seeking to minimise the obligations, conscious and unconscious, to various standard works, it may be right to state that the book has been written for the most part without minute reference to such works, and represents the author's own way of looking at the matters treated. It has not been thought worth while, however, to call attention to cases where the mode of presentation has, in his eyes, some degree of novelty. Further research might prove such claims to be unsubstantial; and in any event they are from the nature of the case hardly worth preferring.

H L, September, 1897.

3.2. Review by: Christian Juel.
Nyt tidsskrift for matematik 9 Afdeling B (1898), 70-72.

The first thing to say about Mr Lamb's infinitesimal calculus is the order in which the objects follow one another, an order different from the usual and in each case of educational significance. The reviewer expresses his joy at seeing that the author's ordering coincides with that which he uses himself teaching the infinitesimal calculus. First, Differential Calculus for Functions of One Variable with Geometric and Mechanical Applications, then Integral Calculus in a similar way and finally series expansions with the Taylor Series at the end.

3.3. Review by: Edmund T Whittaker.
The Mathematical Gazette 1 (13) (1898), 171-173.

The Differential Calculus in England seems to have a fatal habit of losing sight of the Calculus on the continent. The work of the Cambridge Analytical Society is a curious chapter in the history of Mathematics; by its efforts, in 1817-20, our insularity was done away with for a time. But the seventy years since then have seen another relapse; and we have now the pleasure of recording the symptoms of another recovery.

A sharp criticism of our backwardness in this subject was written in 1892 by Miss C A Scott for the Bulletin of the New York Mathematical Society, on the appearance of a fresh edition of Edward's Differential Calculus. ...

Doubtless he would be a bold man who would introduce the elements of the theory of functions of a complex variable into a treatise on the Infinitesimal Calculus; but it is easier than much of the matter now needlessly included; and it is hard to see how a reasonable account of expansion theorems can be written without it.

Interest in questions of this nature may be said to date from the appearance of Professor Forsyth's Theory of Functions in 1893. The effect is now being distinctly felt. In some recent English books, we find alterations in the order of the chapters, omission of old paragraphs, insertion of new ones, and long apologetic prefaces - the signs of a general state of transition. The Calculus is even now in the melting-pot.

Professor Lamb's new book is not an exhaustive treatise, and is not even designed for the professional mathematician at all. But its publication is nevertheless a notable event in the history of our mathematical teaching. It is not a large book; it meets the beginner at the beginning, and leaves him after considering Taylor's theorem. But importance is not to be measured by size alone. ...

We think that, if anything, Professor Lamb is too indefinite as to the class of readers he is writing for. If the book is intended for engineers, they will be chilled to find in the first chapter a discussion on the limits of assemblages; if it is meant for mathematicians, the work on moments of inertia and homogenous strain might well be left to other books. Reviewing the book as a whole, we heartily wish it success. It includes seven chapters of what would commonly be called Differential Calculus, four of Integral Calculus, two of Differential Equations, and a good chapter on the much-neglected but enormously important theory of Infinite Series. There are plenty of good figures and good examples; and teachers will probably find it the best guide to give to those who are entering for the first time the temple of the higher mathematics.

3.4. Review by: William F Osgood.
Science, New Series 7 (176) (1898), 678-680.

The English text-books on the Infinitesimal Calculus in common use afford a formal treatment of the calculus in common use afford treatment of the calculus that is all that can be desired. A student who has worked all the examples under important topics in one of these books has been through a course of shop-work that prepares him adequately for the manipulation of calculus formulas - and for the tripos examination. But he has done only shop-work. He has learned to differentiate explicit functions and to integrate (some) explicit functions, and to prove all sorts of things by Taylor's Series. He has not been trained to examine carefully the reasoning he employs or to consider even the broadest limitations in the statement of theorems. Teachers of elementary calculus are only too prone to leave the consideration of all such matters to the indefinite future; but a wise system of instruction will strive not to hide from the student, but to point out to him those difficulties that are inherent in the fundamental conceptions and methods of the science, and to provide him with the simplest means known at the present time for dealing with them. Professor Lamb has produced a text-book the distinctive feature of which, to our mind, is that a serious and successful attempt has been made to meet these latter demands. ... the book is the first of its kind on the subject of Calculus to appear in the English language. May future writers on Calculus emulate the example of Mr Lamb in trying to make their presentation rigorous according to the highest standards of their day, and at the same time not beyond the comprehension of the students whom they would instruct!

3.5. Review by Dorothy Wrinch (of the 3rd edition (1919)).
Science Progress in the Twentieth Century (1919-1933) 14 (56) (1920), 678.

The new edition of Prof Lamb's Treatise is extremely welcome. This work has for a long time past been regarded, by the majority of teachers of Mathematics, as the best account in our language of the Calculus as designed for students whose aim is less the study of its logical principles than the cultivation of a facility for applying its methods to problems which arise in the application of mathematics to other branches of Science. In this regard the book has never had a serious rival. We do not wish to imply that logical proofs have been neglected. For Prof Lamb's treatise, embracing as it does the Differential and Integral Calculus and the more important types of Differential Equations, is quite unusually logical in its procedure, avoiding, at the same time, the repellent atmosphere suggested too often to a physicist or an engineer, by the attempt to give really rigorous statements of the underlying assumptions in theorems which he would often prefer to take for granted. We regard it as the only work on this subject which has as yet succeeded in striking the proper note, and the call for a new edition is a sign of the fact that those whose work involves applications of mathematics are beginning to a greater extent to be really interested in the subject itself, and to appreciate some of its beauty and generality without too close a study.

4.1. From the Preface.

A complete survey of the theory of sound would lead into many fields, physical, physiological, psychological, aesthetic. The present treatise has a more modest aim, in that it is devoted mainly to the dynamical aspect of the subject. It is accordingly to a great extent mathematical, but I have tried to restrict myself to methods and processes which shall be as simple and direct as is possible, regard being had to the nature of the questions treated. I hope therefore that the book may fairly be described as elementary, and that it may serve as a stepping stone to the study of the writings of Helmholtz and Lord Rayleigh, to which I am myself indebted for almost all that I know of the subject.

The limitation of methods has involved some sacrifices. Various topics of interest have had to be omitted, whilst others are treated only in outline, but I trust that enough remains to afford a connected view of the subject in at all events its more important branches. In the latter part of the book a number of questions arise which it is hardly possible to deal with according to the stricter canons even of mathematical physics. Some recourse to intuitional assumptions is inevitable, and if in order to bring such questions within the scope of this treatise I have occasionally carried this license a little further than is customary, I would plead that this is not altogether a defect, since attention is thereby concentrated on those features which are most important from the physical point of view.

Although a few historical notes are inserted here and there, there is no attempt at systematic citation of authorities. The reader who wishes to carry the matter further will naturally turn in the first instance to Lord Rayleigh's treatise, where full references, together with valuable critical discussions, will be found. I may perhaps be allowed to refer also to the article entitled "Schwingungen elastischer Systeme, insbesondere Akustik," in the fourth volume of the Encyclopädie der mathematischen Wissenschaften (Leipzig, 1906).

I have regarded the detailed description of experimental methods as lying outside my province. I trust, however, that no one will approach the study of the subject as here treated without some first-hand acquaintance with the leading phenomena. Fortunately, a good deal can be accomplished in this way with very simple and easily accessible appliances; and there is, moreover, no want of excellent practical manuals.

H L, January 1910.

4.2. Review by: F J W Whipple.
The Mathematical Gazette 6 (94) (1911), 161-162.

This new book by Prof Lamb will add to his high reputation as a writer on applied mathematics. The ground covered is very nearly the same as that of Lord Rayleigh's treatise, but the investigations are not worked out in quite so much detail. For example, symmetrical spherical waves in air are discussed in full, but the unsymmetrical waves, the theory of which would involve spherical harmonics and Bessel functions of order n + 1/2 are dismissed with a reference to Lord Rayleigh. As a general rule all questions of practical interest are treated at sufficient length, although we note that singing flames are dismissed very briefly. This is less to be regretted as the theory is to be found in Thomson and Poynting's 'Physics'. In discussing scales Prof Lamb states that "the requirements of most keys can be fairly well met" by the system of equal temperament. We should have thought that all keys were on the same footing, and that the prejudice that some musicians have in favour of certain keys had no more mathematical justification than the prejudice of some whist players in favour of certain suits; but the physiological side of the subject of absolute pitch is so obscure that it would be unwise to dogmatise on this point. Of course failures in consonance are more important in the slow music of the organ with its sustained notes than in the rapid music of the piano. The explanation of equal temperament should surely be accompanied by a reference to the fact that it is attained in practice with sufficient accuracy by tuning the fifths and octaves correctly.

The subject of sound has been worked out with such thoroughness by Helmholtz and Lord Rayleigh that the new work is naturally confined to details. Amongst the references to recent work quoted by Prof Lamb we note the experiments of the Krigar-Menzel and Raps, who found that a string plucked at one point actually passes through the shapes predicted by theory, and the investigations by Lord Rayleigh which show that our perception of the direction of a sound depends on the difference of phase between the waves which reach our two ears. It is hardly necessary to add that in our opinion this treatise will prove a most valuable addition to the library of every student of mathematical physics.

4.3. Review by: Edwin Bidwell Wilson.
Bull. Amer. Math. Soc. 19 (1913), 260-264.

Mathematicians are offered a number of interesting problems in the theory of sound somewhat more advanced than elementary harmonic analysis. For example, there is the theory of finite waves, where the differential equations are no longer linear. Riemann, Hugoniot, Hadamard is the sequence of names which should be mentioned in this connection. Lamb merely discusses the matter briefly with a reference to Riemann. ...

As for matters of detail, the book, after a short introduction, takes up the study of vibrations. The pace is moderate, passing successively the simple pendulum, the general system of one degree of freedom, forced vibrations, resonance, friction and damping, systems of several degrees of freedom, and the transition to continuous systems. ...

... the author has exhibited excellent taste and balance in his selection and treatment of topics, that he has accomplished just what he intended, and that it was worth accomplishing, - a text that by easy stages fits the reader for the more advanced treatises and, we may add, a text that by its graceful composition can hardly fail to lure the reader on to the further study of the subject. All this would have been predicted in advance of reading the Sound by anybody at all familiar with the author's Hydrodynamics.

4.4. Review by: D O W (of 2nd edition (1925)).
Science Progress in the Twentieth Century (1919-1933) 21 (81) (1926), 148-149.

Prof Lamb's treatise on Sound is almost entirely mathematical in its scope, and is intended to serve as a stepping-stone to the writings of Lord Rayleigh. It forms an outstanding example of the manner in which the mathematics of a science may be presented so as to be intelligible to the physicist, who is chiefly concerned with its experimental side. The book may be read in comfort (if not in an armchair) by many who can only admire Rayleigh's classic. The new edition remains substantially the same as the last. A few errors have been corrected and a few paragraphs dealing with matters which have come into prominence during the last few years have been inserted. Such additions include brief references to the acoustic properties of buildings, to the propagation of explosion waves, to double resonators, and to the hot-wire microphone. A companion textbook dealing with the experimental side of the subject is now needed very badly, for there is no connected account of modern experimental work available to the English student.

5.1. From the Preface.

This book contains, with some modifications, the substance of lectures which have been given here far a number of years. It is intended for students who have already some knowledge of elementary Mechanics, and who have arrived at the stage at which they may usefully begin to apply the methods of the Calculus. It deals mainly with two-dimensional problems, but occasionally, where the extension to three dimensions is easy, theorems are stated and proved in their more general form.

The present volume differs from many academical manuals in the prominence given to geometrical methods, and in particular to those of Graphical Statics. These methods, especially in relation to the theory of frames, have imported a new interest into a subject which was in danger of becoming fossilised. I have not attempted, however, to enter into details which are best learned from technical treatises, or in engineering practice.

It seemed natural and convenient to treat of Hydrostatics, to a similar degree of development, and I have also, for reasons stated at the beginning of Chap. XV, included the rudiments of the theory of Elasticity.

A companion volume on Dynamics is in contemplation. Some investigations in the Chapter on Mass-Systems and elsewhere have been inserted with a view to this.

I have derived much assistance in the way of references from several of the articles in Bd. IV of the Encyclopädie der mathematischen Wissenschaften. I am also indebted for some valuable suggestions to Prof A Foppl's excellent Vorlesungen über technische Mechanik, Bd. II.

When writing out the work for the press, I found that it was hardly possible to avoid repeating, with little alteration, considerable passages of the article on Mechanics which I had recently contributed to the eleventh edition of the Encyclopaedia Britannica. The proprietors of that work, on being consulted, at once intimated that they would take no exception to this course. I beg to tender them my best thanks for their ready courtesy.

The examples for practice have been selected (or devised) with some care, and it is hoped that most of them will serve as genuine illustrations of statical principles rather than as exercises in Algebra or Trigonometry. Problems of a mainly mathematical character have been excluded, unless there appeared to be some special interest or elegance in the results.

H L, The University, Manchester, August 1912.

5.2. Review by: Anon.
The Mathematics Teacher 5 (4) (1913), 248.

This book presupposes some knowledge of elementary Mechanics on the part of the student, and while the calculus is freely used much use is made of geometrical methods and of those of graphic statics. Besides the statics of solids, five chapters are given on the statics of liquids as well as the elements of the theory of elasticity. It would, seem to be a very teachable book.

5.3. Review by: Edwin B Wilson.
Bull. Amer. Math. Soc. 27 (9-10) (1921), 475-477.

To show the breadth of treatment the titles of the chapters may be quoted: Theory of Vectors, Statics of a Particle, Plane Kinematics of a Rigid Body, Plane Statics, Graphical Statics, Theory of Frames, Work and Energy, Analytical Statics, Theory of Mass-Systems (centres of mass and moments of inertia), Flexible Chains, Laws of Fluid Pressure, Equilibrium of Floating Bodies, General Conditions of Equilibrium of a Fluid, Equilibrium of Gaseous Fluids, Capillarity, Strains and Stresses, Extension of Bars, Flexure and Torsion of Bars, Stresses in Cylindrical and Spherical Shells. This is a considerable program; it is well and consistently carried through - as should be expected by all who have known his other writings and particularly his companion volume on Dynamics ...

These books, Statics and Dynamics, are not written for the writing; they are products of teaching, for they are based on lectures delivered at the University of Manchester. If the matter were taken slowly enough, satisfactory results would attend their use in our American institutions, provided our teachers had an all round interest in the elements of mathematics, of physics, and of engineering, and a fine contempt for superficial ground-covering in any of the three.

5.4. Review by: V (of 1960 reprint).
Current Science 30 (5) (1961), 197.

Lamb's books on Dynamics and Statics are well known to students of mathematics and have been in popular demand ever since their publication nearly fifty years ago. The present volumes are the ninth reprints of the revised editions of the 1920's. These books are not classics like the author's 'Hydrodynamics' but have served as useful textbooks to graduate and honours students in mathematics. That reprints of the books were found to be desirable as well as profitable to the publishers, even in this decade, is a clear indication of both the popularity of the book as well as the sad plight of the syllabus in universities, that refuses to introduce newer developments or a more modern presentation.

6.1. From the Preface.

This book is a sequel to a treatise on Statics published a little more than a year ago, and has a similar scope. To avoid repetitions, numerous references to the former volume are made,

A writer who undertakes to explain the elements of Dynamics has the choice, either to follow one or other of the traditional methods which, however effectual from a practical point of view, are open to criticism on logical grounds, or else to adopt a treatment so abstract that it is likely to bewilder rather than to assist the student who looks to learn something about the behaviour of actual bodies which he can see and handle. There is no doubt as to which is the proper course in a work like the present; and I have not hesitated to follow the method adopted by Maxwell, in his Matter and Motion, which forms, I think, the best elementary introduction to the 'absolute' system of Dynamics. Some account of the more abstract, if more logical, way of looking at dynamical questions is, however, given in its proper place, which is at the end, rather than at the beginning of the book.

There is some latitude of judgment as to the order in which the different parts of the subject should be taken. To many students it is more important that they should gain, as soon as possible, some power of dealing with the simpler questions of 'rigid' Dynamics, than that they should master the more intricate problems of 'central forces,' or of motion under various laws of resistance. This consideration has dictated the arrangement here adopted, but as the later chapters are largely independent of one another, they may be read in a different order without inconvenience.

Some pains have been taken in the matter of examples for practice. The standard collections, and the text-books of several generations, supply at first sight abundant material for appropriation, but they do not always reward the search for problems which are really exercises on dynamical theory, and not merely algebraical or trigonometrical puzzles in disguise. In the present treatise, preference has been given to examples which are simple rather than elaborate from the analytical point of view. Most of those which are in any degree original have been framed with this intention.

H L, The University, Manchester. January 1914.

6.2. Review by: V (of 1960 reprint).
Current Science 30 (5) (1961), 197.

Lamb's books on Dynamics and Statics are well known to students of mathematics and have been in popular demand ever since their publication nearly fifty years ago. The present volumes are the ninth reprints of the revised editions of the 1920's. These books are not classics like the author's 'Hydrodynamics' but have served as useful textbooks to graduate and honours students in mathematics. That reprints of the books were found to be desirable as well as profitable to the publishers, even in this decade, is a clear indication of both the popularity of the book as well as the sad plight of the syllabus in universities, that refuses to introduce newer developments or a more modern presentation.

7.1. From the Preface.

This book treats of three-dimensional Kinematics, Statics, and Dynamics in what is I think a natural, as I have found it to be a convenient, order. It may be regarded as a sequel to two former treatises [Statics, Cambridge, 1912, and Dynamics, Cambridge, 1914] to which occasional reference is made; but it is not dependent on these, and will I trust be readily followed by students who are conversant with ordinary two-dimensional Mechanics.

The subject is of course a very wide one, and some principle of selection is necessary. I have tried to confine myself to matters of genuine kinematical or dynamical importance, avoiding developments whose interest, often considerable, is purely mathematical or now mainly historical. It is owing to such considerations that whilst some account is given of the Theory of Screws, of Null-Systems, and of Least Action, on the other hand brachistochrone problems, and the general theory of the Differential Equations of Dynamics, are left untouched.

The book does not claim to be more than an elementary one, regard being had to the nature of the subject. The reader who wishes to carry his studies further will find ample assistance in Thomson and Tait, in Rayleigh's Theory of Sound, and in Whittaker's Analytical Dynamics. And in common with other recent writers I must mention with a special sense of obligation the works of Routh, which in their later forms are an almost inexhaustible store-house of theorems and results, and abound in interesting historical references.

H L, July 1920.

7.2. Review by: Anon.
Science Progress in the Twentieth Century (1919-1933) 16 (61) (1921), 145.

Prof Lamb's book, which treats of three-dimensional Kinematics, Statics, and Dynamics, is very welcome. It is to some extent a sequel to Statics (Cam- bridge, 1912) and Dynamics (Cambridge, 1914), but it is not dependent on these. Prof Lamb has adopted, as his principle of selection in this book, that matters of genuine or dynamical importance be included, and developments whose interest is purely mathematical or mainly historical be omitted. Thus he treats the Theory of Screws, of Null Systems, and of Least Action, and omits brachistochrone problems, and the general theory of the Differential Equations of Dynamics.

7.3. Review by: George Greenhill.
The Mathematical Gazette 10 (153) (1921), 309-319.

Interesting exercises in elliptic function theory are provided by the various questions of a pendulum, clock, or governor, fixed in a whirling arm, and the solution should be carried out to the fullest extent, with no approximation. Then there is the application to the effect on the compass of the vibration of a flying machine. And the flight of the machine, taking into account the inertia of the surrounding air, will lead to the equations of Hydrodynamics, when turbulence is ignored; as intractable in the general case as the un-symmetrical top, although reduced to an elliptic function solution with uniaxial symmetry. A general discussion of vibration on variational methods tends to become a procession of algebraical equations, leading to no definite result; a concrete case requires to be added to fix the ideas.

7.4. Review by: Anon.
The American Mathematical Monthly 29 (2) (1922), 72-73.

Contents-Chapter I: Kinematics of a rigid body. Finite displacements, 1-13; II: Infinitesimal displacements, 14-33; III: Statics, 34-65; IV: Moments of inertia, 66-73; V: Instantaneous motion of a body (kinematics), 74-88; VI: Dynamical equations, 89-111; VII: Free rotation of a rigid body, 112-128; VIII: Gyrostatic problems, 129-150; IX: Moving axes, 151- 176; X: Generalized equations of motion, 177-207; XI: Theory of vibrations, 208-248; XII: Variational methods, 249-270; Index, 271-272.

8.1. Note.

This is Horace Lamb's Rouse Ball Lecture of 1924.