LXVI. Notices respecting New Books. A Course of Modern Analysis. By E. T. WHITTAKER, M.A. Cambridge University Press. 1902. THE HE extended title of this exceptionally excellent work is "An Introduction to the General Theory of Infinite Series and of Analytic Functions; with an account of the principal Transcendental Functions." When it is stated further that the functions discussed are the Gamma, the Legendre, the Hypergeometric, the Bessel, and the Elliptic Functions, the general scope of the book will be understood. The author has shown great discrimination and reserve in his treatment; for it is only too easy in a subject of this kind to deviate into side issues so that the student has the vaguest ideas as to the general aim of his wanderings. Here, however, we find in the 170 pages which constitute Part I. a lucid, compact, and yet sufficiently detailed development of the theory of functions in a form necessary and sufficient-to use the familiar phrase-for a sound discussion of the important special types of functions treated of in Part II. Each chapter is enriched with an appropriate set of examples or exercises, many of which are important theorems associated with the names of the discoverers. Had the majority of these been treated at length as part of the text--as is the custom with some authors--the book could easily have been made of formidable dimensions. In this connexion especially the author has shown great wisdom. The working student will probably find it profitable to read the text carefully through so as to get a general view of the subject, and then turn back and familiarize himself with the methods by working a selection of the exercises out in detail. The value of the book is further enhanced by a series of short historic notes and by constant references to important memoirs by the many distinguished mathematicians who have helped to develop the modern theory of functions. Mathematical Crystallography and the Theory of Groups of Movements. By HAROLD HILTON, M.A. Oxford: At the Clarendon Press. 1903. MR. HILTON is to be congratulated on having made an important contribution to our mathematical literature. Crystallography is a science which in its practical aspects concerns the mineralogist and the chemist; but very few of those who are familiar with the forms and classification of crystals will find Mr. Hilton's pages easy reading. After a discussion of the geometry of crystals and the various laws recognized by crystallographers, the author enters upon his real work, namely, the complete mathematical discussion and classification of the various groups of movements possible under the limitations suggested by the laws of crystals. The book is in fact a treatise on the theory of a set of finite groups of a special type, involving certain operations of translation, reflexion, and rotation. The theory of the point-groups is worked out in detail in Chapters IV., V., and VI., and it is proved that there are 32, and only 32, finite groups of movements consistent with the law of rational indices. Two brief chapters on the relations between crystalline symmetry and physical properties and on the growth of crystals complete Part I.; and the rest of the book is devoted to the structure theory of crystalline matter. There is thus presented for the first time to English readers in connected form the geometrical theory of crystal structure. The 230 space groups capable of representing crystalline form are worked out in sufficient detail, and are profusely illustrated by diagrams drawn on the lines suggested by Federow. Mr. Hilton lays no claim to originality, but aims at reproducing mainly the system developed by Schoenflies. He has, however, laid other writers under contribution; and his own powers are in evidence in the clearness of exposition and compactness of demonstration. The argument is frequently very condensed, and every line demands the closest attention on the part of the reader if he wishes really to follow the demonstration. There is a steady strain upon the geometrical and kinematical imagination, a strain which comparatively few of those who are practically interested in crystallographic questions will care to undergo. But the mathematical student interested in the theory of groups will be greatly benefited by a careful study of Mr. Hilton's pages. Towards the close of the book reference is inade to dynamic possibilities of crystalline structure as distinguished from geometrical possibilities. Here, of course, we encounter questions of molecular stability which can hardly at present be stated, far less solved. Thermodynamics and Chemistry: A non-Mathematical Treatise for Chemists and Students of Chemistry. By P. DUHEM, Professor of Theoretical Physics at the University of Bordeaux. Authorized translation by George K. Burgess. New York: John Wiley & Sons. 1903. Pp. xxi+445. A LARGE number of chemists and students of chemistry find it difficult, if not impossible, to follow the modern developments of the theory of chemical equilibria, by reason of their insufficient knowledge of the mathematical processes whose aid must be invoked in a thorough discussion of the subject. To such the translation of Professor Duhem's book will be a welcome addition to their scientific library, as the author deals with the subject without the use of analysis. Such a method must necessarily have its shortcomings, and the reader must now and then be asked to take certain things for granted which could readily be demonstrated were the use of mathematical analysis not forbidden. On the other hand, the non-mathematical reader has the satisfaction of knowing that, having consented to take certain statements on the authority of the mathematicians, he will not, in this book, have to wade through pages of, to him, unintelligible symbols. The first six chapters of the book deal with work and energy, quantity of heat and internal energy, chemical calorimetry, chemical equilibrium and the reversible transformation, the principles of chemical statics and the phase rule. Chapter vii. contains a large number of applications of the phase rule to multivariant systems. The succeeding chapters, viii. to xii., are concerned with mono- and bi-variant systems. Chapters xiii. and xiv. deal with mixed crystals and metallic alloys; chapter xv. with the chemical mechanics of perfect gases; chapter xvii. with capillary actions and apparent false equilibria; chapter xviii. with genuine false equilibria; chapter xix. with unequally heated spaces; and the concluding chapter xx. with chemical dynamics and explosions. A noticeable and highly praiseworthy feature of the book is the very large number of illustrative examples. Especially is this feature valuable in the earlier chapters of the book, devoted to general theoretical considerations, as it enables the reader to form a much more accurate and vivid idea of the subject under discussion than would otherwise be possible. Considering the valuable service which Dr. Burgess has rendered to English-speaking students by translating this work, it may seem ungrateful to criticise the translation adversely. Yet no one could possibly mistake the rendering for an original work in English. French idioms abound. These may be overlooked by an indulgent reader, but when it comes to the wholesale importation of French words without any attempt at translation, the reader's patience cannot but be sorely tried. What, for example, are we to make of the following:-"it is shown in mechanics by methods which we cannot expose here..."? Or why does the author speak of a renversable change (without even italicising the term)? To pass to another matter, we consider that a modern writer on chemical theory has no more right to speak of vapour-tension, meaning pressure, than a modern writer on dynamics has to apply the term "power" to a force. The revision of the proof-sheets must have been carried out very carelessly, as there are numerous instances of missing letters. For this, however, the publishers are to blame. Fractional Distillation. By SYDNEY YOUNG, D.Sc., F.R.S., Professor of Chemistry in University College, Bristol. With 72 illustrations. London: Macmillan & Co., Ltd. 1903. Pp. xii +284. PROFESSOR SYDNEY YOUNG'S name is so well known in connexion with the numerous difficult and highly important physicochemical researches carried out by him, that the present volume, which contains a vast amount of information, some of which has never been published elsewhere, is sure to meet with a warm welcome from the increasing band of workers on the borderland of physics and chemistry. The book is remarkable alike for the logical arrangement of the subject-matter and the lucid and easy style of exposition. The first chapter is devoted to a careful account of the construction of various forms of still, and contains many useful practical hints. The next few chapters deal with the boiling-point of a pure liquid, the vapour-pressures and boiling-points of mixed liquids, and the composition of the liquid and vapour phases (considered both experimentally and theoretically). In chapter vii. we have detailed directions for carrying out a fractional distillation. The next two chapters deal with the theoretical relations between the weight and composition of the distillate, and the relation between the boiling-points of residue and distillate. Chapters x.-xii. are devoted to a very full account of modifications of the still-head; and in chapter xiii. the subject of continuous distillation is dealt with. Fractional distillation with an improved still-head is then taken up in chapter xiv., and distillation on the manufacturing scale in chapter xv. The important subject of fractional distillation as a method of quantitative analysis is next dealt with. In chapter xvii. we have an account of methods by which the composition of mixtures of constant boiling-point may be determined; and in chapter xviii. an account of the indirect method of separating the components of a mixture of constant boiling-point. The concluding chapter is devoted to general remarks on the subject. An Appendix containing tables of temperature corrections for the height of the barometer, and a very copious index form useful additions. A highly commendable feature of the book consists in the numerous bibliographical references given at the end of each chapter. LXVII. Intelligence and Miscellaneous Articles. AN INSTRUMENT FOR DRAWING CONICS. To the Editors of the Philosophical Magazine. GENTLEMEN, Trinity College, Dublin, 14th April, 1904. WITH reference to my article in the March number of the Philosophical Magazine, I wish to state that I have since received a paper in the Russian language by Prof. Prince Kougoushef of Warsaw, which was published in 1899, and is a description of a new conicograph' which is evidently the same as that which forms the subject of my paper. Although my instrument was constructed in 1895 and shown at the time to several scientific gentlemen connected with Trinity College, Dublin, I did not previously publish any account of it, and consequently Prince Kougoushef is fully entitled to claim priority. In making this acknowledgment I desire to express my regret that, at the time of writing my paper, I was unaware that it had been anticipated. I am, Gentlemen, THE LONDON, EDINBURGH, AND DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES.] JUNE 1904. LXVIII. On Deep-water Two-dimensional Waves produced by any given Initiating Disturbance. By Lord KELVIN *. § 1. CONSIDER frictionless water in a straight canal, infinitely long and infinitely deep, with vertical sides. Let it be disturbed from rest by any change of pressure on the surface, uniform in every line perpendicular to the plain sides, and left to itself under constant air pressure. It is required to find the displacement and velocity of every particle of the water at any future time. Our initial condition will be fully specified by a given normal component velocity, and a normal component displacement, at every part of the surface. § 2. Taking O, any point at a distance h above the undisturbed water level, draw OX parallel to the length of the canal, and OZ vertically downwards. Let, be the displacement-components of any particle of the water whose undisturbed position is (x, z). We suppose the disturbance infinitesimal; by which we mean that the change of distance between any two particles of water is infinitely small in comparison with their undisturbed distance; and the line joining them experiences changes of direction which are infinitely small in comparison with the radian. Water being assumed frictionless, its motion, started primarily from rest by pressure applied to the free surface, is essentially irrotational. Hence we have where (a, z, t), or 4, as we may write it for brevity when From the Proceedings of the Royal Society of Edinburgh for Feb. 1, 1904. Communicated by the Author. Phil. Mag. S. 6. Vol. 7. No. 42. June 1904. 2 T |