/VA rURR 



265 



THURSDAY, JULY 19, 1906. 



SOME RECENT MATHEMATICAL WORKS. 



Correspondance d'Hermite et de Slieltjes. Edited 

 by B. Baillaud and H. Bourget. Vol. ii. Pp. 

 viii + 404. (Paris: Gauthier-Villars, 1905.) Price 

 16 francs. 



G. Lejeune Dirichlet's Vorlesungen iiber die Lehre von 

 den einfachen und mehrfachen bestimmten Inte- 

 gralen. By G. Arendt. Pp. x.xiv + 476. (Bruns- 

 wick : Friedrich Vieweg and Son, 1904.) Price 12 

 marks. 



/,(• Calcitl des R^sidus et ses Applications a la Thiorie 

 dcs Fonctions. By Ernst Lindelof. Pp. viii+144. 

 (Paris: Gautliier-Villars, 1905.) Price 3.50 francs. 



I.cs Principes des Mathematiques. By Louis Couturat. 

 Pp. viii + 342. (Paris: F^lix Alcan, 1905.) 



Mi'thodes de Calcul graphique. By Frederico Oom. 

 Pp. 26; with 4 plates. (Lisbon : Imprimerie nation- 

 rile, 1905.) 



]'oliime and Surface Integrals used in Physics. By 

 J. G. Leathern. Pp. 48. (Cambridge : University 

 Press, 1905.) Price 2S. 6d. net. 



Recherche sur les Champs de Force hydrodynamiques. 

 By V. Bjerknes. Reprinted from Acta Mathematica, 

 vol. .\xx. Pp. 146. 



Sur la Recherche des Solutions particulieres des Sys- 

 ieines difjirentiels et sur les Motivcments station- 

 naircs. By T. Levi Civita. Pp. 40. (Warsaw : J. 

 Sikorskiego, 1906.) 



ONE of the most noticeable features of recent times 

 has been the increasing interest taken in the 

 history of mathematics. That two international con- 

 gresses — the historical and the mathematical — have 

 devoted separate sections to this study is, let us hope, 

 a stepping-stone towards the realisation of the resolu- 

 tions passed at both congresses in favour of the es- 

 tablishment of chairs of mathematical history in the 

 leading universities of the Continent and America, 

 and possibly even Great Britain. 



Reference has previously been made to the first 

 volume of the interesting correspondence between 

 Hermite and Stieltjes. The second volume, covering 

 the period 1889-1894, is no less delightful reading than 

 the first. Every letter fills up some gap in the reader's 

 mathematical knowledge, either by introducing him to 

 some little-known proposition or bv presenting some 

 well-known result in a new aspect. .An appendix 

 contains four letters addressed by Stieltjes to Prof. 

 Mittag-Letfler in 1885-1S87 dealing with Riemann's 

 Zeta function. These letters afford a striking insight 

 into the difficulties experienced bv Stieltjes in his 

 efforts to master Riemann's works and his ingenuity 

 in devising alternative methods. The present volume 

 contains a portrait of Hermite and the facsimile of a 

 manuscript by Stieltjes. 



The historic spirit has further shown itself in Prof. 

 .'\rendt"s issue of the nearest possible approach to a 

 verbatim report of the lectures on definite integrals 

 as given by Dirichlet at Berlin in 1854. It is true, as 

 the author points out, that the lectures which Dirichlet 

 gave on the same subject at Gottingen four years later 

 NO. I916, VOL. 74] 



formed the basis of the well-known treatise by Gustav 

 Ferdinand Meyer, but it appears that the notes on 

 which Meyer's account was based were far from com- 

 plete, and it was necessary to spend considerable time 

 in filling up the gaps in tlie reasoning, and, moreover, 

 the object was to give a complete treatment of the 

 subject rather than an exact account of the lectures. 

 Prof. Arendt, on the other hand, has compiled the 

 present work from a set of notes mostly transcribed 

 on the actual dates of the lectures. The course covers 

 a branch of mathematics well known to the average 

 student, namely, the definition of an integral and its 

 connection with summation, the theorems on change 

 of limits and differentiation of integrals, the evalu- 

 ation of the ordinary well-known definite integrals, the 

 Beta and Gamma functions, transformation of mul- 

 tiple integrals, the attractions of ellipsoids, and ap- 

 plications to harmonic and hypergeometric series. The 

 notes at the end afford evidence of the care with which 

 the original manuscript has been followed ; where any 

 divergence has been necessary the changes are care- 

 fully pointed out; the only important innovation, how- 

 ever, is the introduction of the modern notation \a\, 

 which greatly simplifies certain formulae. At the 

 present time these lectures of Dirichlet make an 

 excellent text-book, and an interesting historical com- 

 parison may be made between the present course and 

 Kronecker's lectures delivered at the same university 

 about thirty years later. 



.\nother prominent place among the " classics " 

 must be assigned to Prof. Ernst Lindelof's charming 

 e.xposilion of Cauchy's calculus of "residues." This 

 is the eighth of a series of monographs on the theorv 

 of functions appearing under the editorship of Prof. 

 Eniile Borel. In Prof. Lindelof, Cauchy's ideas have 

 found an able exponent, and from a detailed study 

 of a number of papers, including some of Cauchy's 

 little-known writings lent for the purpose bv Prof. 

 Mittag-Leffler, the author has produced a treatise in 

 which the simplicity and perfection of this important 

 method of analysis are well shown. Of the applica- 

 tions, those in the second chapter are mainlv 

 due to Cauchy. The third shows how certain 

 formulae of summation can be immediately deduced 

 from the same principle, while in the fourth it is shown 

 how this method of treatment greatly simplifies the 

 study of the Gamma function and of Riemann's 

 function. Of the importance of the latter application 

 the difliculties of Stieltjes already referred to give 

 sufficient proof, and on the other hand the name of 

 Stieltjes figures conspicuously in the discussion of 

 Stirling's series, in connection with which Prof. 

 Lindelof contributes several new results and proofs. 

 Finally we have a general account of certain modern 

 results relating to functions defined by Taylor's series, 

 thus bringing into one small volume a general survey 

 of the recent as well as the original developments of 

 Cauchy's method. The book includes new matter for 

 which the author is himself responsible as regards 

 methods of treatment no less than as regards results. 



\ second line of modern mathematical development 

 consists in the attempt to probe ever deeper and deeper 

 into the foundations of mathematics. In France, 



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