Theory of Lie Groups I

854 NATURE December 20. 1947 Vol. t60 Theory of lie Groups, I deficiency is especially well treated, and the chapters By Claude Chevalley. (Princeton Mathematical on feeding experiments contain one of the best Series.) Pp. xii + 217. (Princeton, N.J.: Princeton reasoned discussions on the pros and cons of the University Press; London: Oxford University paired-feeding technique. In the author's view this Press, 1946.) 20s. net. method would appear to find its largest usefulness N recent years great advances have been made in in comparisons in which food consumption is not I our knowledge of the fundamental structures of markedly restricted by the conditions imposed, and analysis, particularly of algebra and topology, and in which the measure is in terms of the specific an exposition of Lie groups from the modern point effect of the nutrient under study, instead of the more of view is timely. This is given admirably in the general measure of increase in weight. It is considered book under review, which could well have been called that there are many problems concerning which the "Lie Groups in the Large". To make such a treatment use of separate experiments of both ad libitum and intelligible it is necessary to re-examine from the new controlled feeding will give much more information point of view such subjects as the classical linear than either procedure alone. groups, topological groups and their underlying The chapters on growth, reproduction, lactation topological spaces, and analytic manifolds. Excellent and work production are likewise excellent. In its accounts of these, with the study of integral manifolds clarity and brevity this work has few equals. and their constructions in the large, are given in the D. P. CUTHBERTSON earlier chapters. In Chapter 4 we come to the defini• Photoelectric Cells tion of an analytic group and its underlying manifold, By Dr. A. Sommer. (Methuen's Monographs on and to the introduction of the important concept of Physical Subjects.) Pp. viii+ 104. (London: a Lie algebra associated with an analytic group. A Methuen and Co., Ltd., 1946.) 5s. net. Lie group is then defined as a locally connected R. SOMMER is well known for his work on topological group related in a certain way to some D photo-electric cells, and in this monograph he analytic group ; from this it follows that the concept collects the basic work done by himself and others of Lie algebra can be used freely when dealing with between 1931 and 1945. While there have been Lie groups. Chapter 5 contains an account ofCartan's excellent texts on the industrial applications of calculus of exterior differential forms and its applica• photo-electric devices, there appears to be no review tion to the theory of Lie groups. The final chapter is of the theoretical and research work since about concerned with compact Lie groups, and after a 1934, so we are grateful to Dr. Sommer for his work brief consideration of the general theory of representa• and his bibliography. He is here concerned solely tions it is proved, among other things, that every with photo-emissive cells. Commencing with Hertz, representation of a compact Lie group is semi-simple. Planck and Einstein, he outlines the theory of The style of writing, though very condensed, is emission, and shows how the number of materials clear, and the printing is excellent. The notation is useful as photo-emitters is strictly limited by ioniza• that of the modern school, with 'union' (U) and tion energy and work function, as well as physical 'intersection' (n) in place of the older 'sum' and properties. He then deals with the relationship of 'product' of sets. (A misprint has unfortunately the wave-length of the incident light, the resulting interchanged the symbols u and n in their definition emission, and the effect of gas in the enclosure. The on p. xii.) The author promises a second volume, to manufacture of photo-cathodes is then outlined, and be mainly concerned with semi-simple Lie groups. also tho way in which they can be used for practical A. G. WALKER applications. Of great value is his critical table outlining the suitable types of cells for established Animal Nutrition technology, such as photometry, sound-fihn repro• By Prof. Leonard A. Maynard. (McGraw-Hill Pub• duction, picture-telegraphy, and television. lications in the Agricultural Sciences.) Second edition. L. E. c. HUGHES Pp. xviii+494. (New York and London: McGraw• Hill Book Co., Inc., 1947.) 25s. City Region and Regionalism HIS is one of the very few books which deal A Geographical Contribution to Human Ecology. T adequately with the fundamentals of the bio• By Robert E. Dickinson. (International Library of chemistry and physiology of nutrition and with the Sociology and Social Reconstruction.) Pp. xv+327. applications of this knowledge to the feeding of farm (London: Kogan Paul and Co., Ltd., 1947.) 21s. net. animals. Since the first edition was published some HILE this book is not directly about planning, nine years ago, much of the newer knowledge of Wit deals with matters and conceptions that are nutrition has emerged, and this has meant extensive essential to any far-sighted planning schemes. alterations in the sections on amino-acid, vitamin and Human ecology, as the relation of society to the area mineral nutrition. In his preface, the author points in which it lives, is a difficult but most important out that owing to the limited opportunity which was study and one that is too often overlooked in schemes enforced by war and its aftermath, it has been of administration. The spatial relations of human impossible to take account of the results of recent society is a study that has received too little attention research in foreign countries, because the data have in Great Britain. Tho city is a regional centre of a either remained unpublished or the journals contain• larger or smaller area, and its function as such is an ing them have not reached his desk. The student important condition in delimiting regions within is warned by the author that he should not thereby any country. Some cities weld together larger areas gain a distorted picture of the progress of our know• than others and size is not a necessary criterion of ledge of this very important field of physiology and this function. Mr. Dickinson deals mainly with the chemistry·. The present volume takes us up to the town or city as a regional capital, but has also a good end of March 1946. deal to say on the possible delimitation of regions in It is difficult to single out any particular parts the United States, France and England and Wales. for special comment, but the chapter on mineral The book merits close attention.