•April 8, 1922] 



NA TURE 



437 



the more difficult postulate (4) — but it would be 

 ungracious to insist on such small criticisms of the 

 most comprehensive presentation of the theory. 



Tn chap. 5, on functions of a real variable (chap. 4 



! the first edition), there are very many important 

 additions. The ideas of absolute (p. 276) and approxi- 

 mate (p. 295) continuity are introduced. The treat- 

 ment of functions of bounded variation (we are glad to 

 find Prof. Hobson now adopting the ordinary language) 

 has been materially simplified, and there is a new 

 section (pp. 318-320) on rectifiable curves. The latter 

 part of the chapter includes an account of some of the 

 most recent work of Den joy, G. C. Young, and W. H. 

 Young concerning derivatives. Above all, there is a 

 discussion of implicit functions, omitted somewhat 

 unaccountably from the earlier edition. This is a 

 most welcome addition, but we are surprised that 

 Prof. Hobson does not state the fundamental theorem 

 (p. 407) in its most general form. No reference to 

 derivatives is necessary, as was made clear by Young, 

 and a theorem more general than Prof. Hobson's is to 

 be found in so elementary a book as the reviewer's 

 " Pure Mathematics." 



Finally, chaps. 6-8 contain the theory of integration, 

 and it is here that we find the most that is new. These 

 chapters are naturally far better than the correspond-' 

 ing parts of the first edition, both in completeness and 

 in logical arrangement, for the first edition appeared 

 at the awkward moment when Lebesgue's ideas were 

 new, and the consequences of his work had not 

 been developed to their conclusion. It may be 

 questioned whether the space (eighty pages) devoted 

 to the Riemann integral is not excessive, since so much 

 of the theory is now of historical or didactic interest 

 only ; but Prof. Hobson's object is, of course, to be 

 complete. The importance of the Stieltjes integral 

 is fully lecognised in this edition. The last chapter 

 (" Non-absolutely convergent integrals "), dealing as 

 it does with the extreme limits of generalisation of 

 which, in the hands of Denjoy and of Young, the notion 

 of an integral has so far proved to be capable, is very 

 heavy reading ; but to have given the first systematic 

 account of these generalisations is in itself a most 

 important achievement. 



It is to be hoped that we shall not wait long for the 

 appearance of the second volume, and the completion 

 of a work which has added so much, not only to the 

 personal reputation of the author, but to the status of 

 English mathematics. 



(2) Prof. Carslaw's book was conceived on a much 

 less ambitious scale than Prof. Hobson's, but he too 

 has had to rewrite it and turn one volume into two. 

 This first volume contains pure mathema^tics only, 

 and there is no reference to any physical phenomenon 

 NO. 2736, VOL. 109] 



after the introduction. It is, in short, a treatise on 

 analysis, restricted within certain limits, and written 

 with a special end in view. 



Prof. Carslaw confines himself quite rigidly and 

 consistently within the limits which he has chosen. 

 It was necessary to have definite limits, but we do 

 not agree entirely with his judgment in selecting 

 them. We think that he has made them too narrow, 

 and that he would have written a still better and 

 more attractive book if he had allowed himself a rather 

 wider scope. It is a very good book even as it is, for 

 it is accurate and scholarly, it contains a mass of most 

 interesting and important theorems which it would be 

 difficult to find collected in an equally attractive form 

 elsewhere, and it is written in an admirably clear and 

 engaging style. It also contains an excellent biblio- 

 graphy of the subject. 



Prof. Carslaw has gone too far, however, in his 

 anxiety to eliminate the refinements of the modern 

 theory of functions. For example, the notion of a 

 function of bounded variation is quite explicitly and 

 dehberately excluded (p. 207). The only functions 

 admitted — if we confine our attention, for simplicity 

 of statement, to bounded functions — are those which 

 satisfy Dirichlet's famous condition ; they have at 

 most a finite number of maxima and minima within 

 the interval considered. Now there is a serious logical 

 objection to a treatment of Fourier's series in which 

 this class of functions is taken as fundamental, an 

 objection which even a physicist might feel. It is an 

 artificial and not a natural class, since it does not form 

 a group for the elementary operations. Neither the 

 sum nor the product of two functions of the class is 

 in general a function of the class ; and it is difficult to 

 see why, if a physicist is interested in two functions, 

 he should not also be interested in their sum. 



Prof. Carslaw alludes to the notion of bounded 

 variation as " somewhat difficult," and so, no doubt, 

 it is. But the necessary analysis, as presented, for 

 example, by de la Vallee Poussin, is certainly not 

 more difficult than a good deal which Prof. Carslaw 

 includes. It is not more difficult, for example, than 

 the second mean value theorem, or the theory of 

 Poisson's integral, or Pringsheim's discussion of 

 Fourier's double integral, of all of which Prof. Carslaw 

 gives a very careful account. In any case a book may 

 be made much easier by the inclusion of a difficult 

 theorem, if it helps to elucidate the theorems which 

 the book already contains. 



It is inevitable that an analyst, reading a book like 

 this, should be longing to go further all the time. No 

 account of the theory of Fourier's series can possibly 

 satisfy the imagination if it takes no account of the 

 ideas of Lebesgue ; the loss of elegance and of simplicity 



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