REVIEWS 



MATHEMATICS 



Introduction to the Theory of Fourier's Series and Integrals. By Prof. 



H. S. Carslaw, Sc.D., D.Sc, F.R.S.E. [Pp. xi + 323.] (London: 

 Macmillan & Co., 1921. Price 30s. net.) 



The task of the teacher of applied mathematics is to present mathematics 

 to his students as a powerful instrument for the solution of the problems 

 of nature. This is the spirit in which Prof. Carslaw has written his book, 

 and it is a happy circumstance that he should have chosen as his theme the 

 work of Fourier which, growing out of the needs of mathematical physics, 

 marked a new stage in the development of pure mathematics. The beginner 

 in applied mathematics is sometimes apt to be impatient with the niceties 

 of pure mathematics. Of course, you can construct series which do not 

 converge and integrals which cannot be differentiated, but these curiosities 

 never appear in the solution of a physical problem. Sooner or later he learns 

 that this is not the case ; that Nature is more subtle than even the purest 

 mathematician. Prof. Carslaw has succeeded in giving us a logical treatment 

 of his subject without that air of unreality which is so often the price of 

 modern mathematical rigour. When he is careful of a logical point he 

 carries the conviction that, if this point is neglected, sooner or later it will 

 turn up in our physical investigations to our undoing. When he says, 

 " However, this method does not give a rigorous proof of these very im- 

 portant expansions for the following reasons : (i) . . . ; (ii) . . . ; (iii) . . . ," 

 the reader is in a position to appreciate the more elaborate proof which 

 follows. This method is not only sound exposition ; it is calculated to foster 

 the true genius of physical science, which consists in inspired guessing followed 

 by a critical analysis of the guess. 



The present book is a second edition, and new editions are not popular 

 with mathematicians. The old book is an old friend, and we know our way 

 among its pages ; we hesitate to set it aside in favour of a new, even if a better 

 book. But in this case the recent advances in the subject have conspired 

 with fifteen years of further thought on the part of the author to make 

 a considerable revision inevitable. One result has been the division of the 

 book into two volumes. The present volume is devoted to a fuller treatment 

 of the pure mathematics of the subject, while the application to the theory 

 of the conduction of heat is reserved for a second volume to be published 

 later. We find ourselves hoping that when this volume appears it may not 

 be too closely confined to the theory of heat, but that Professor Carslaw 

 will use his great powers of exposition to give us some account of the applica- 

 tions which the work of Fourier has found throughout the whole realm of 

 mathematical physics. 



For the present we have a clear and adequate discussion of infinite series 

 and integrals, leading up to a theorj'^ of Fourier series and integrals of a 

 generality sufficient to cover all the present needs of mathematical physics, 

 while abundant references will assist the reader who wishes to follow the 

 subject further in its modern developments. 



The two appendices, the first giving some account of practical harmonic 

 analysis, and the second an exhaustive bibliography, provide a useful and 

 appropriate conclusion to a valuable work. 



G. B. J. 

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