November 7, 1901] 



NA TURE 



■of summability is shown to comprise an area bounded by 

 a (finite or infinite) number of straight Hues, each of 

 which goes through a critical point. This new region 

 M. Borel calls the polygon of siiinmability. An obvious 

 • question arises here ; does the continuation of f\x) 

 obtained by exponential summation necessarily coincide 

 with one obtained by other methods, for example 

 Weierstrass's ? In some cases it certainly does ; for 

 instance, when jlx) is a rational function of x, or one 

 Ibranch of an algebraic function. 



So far it has been supposed that the object of inquiry 

 is in the first instance a series given by the law of con- 

 struction of its terms ; and the main result has been to 

 show how, in certain cases where the series is divergent 

 in the ordinary sense, it may be associated with a finite 

 function, called its sum i in an extended sense', which the 

 series so far represents that relations satisfied formally 

 'by the series are actually and arithmetically satisfied by 

 its sum. But there is another side of the question which 

 is of equal importance, especially from the practical point 

 of view. We may have a function explicitly or implicitly 

 defined by certain properties, and try to obtain a series 

 which for purposes of computation or otherwise may be 

 regarded as its equivalent. A typical illustration is 

 afforded by the ordinary process of solving differential 

 equations by series ; here we have a uniform method 

 which, if it does not fail altogether, leads us to a power- 

 series, formally satisfying the equation, but not necessarily 

 convergent. Exponential summation, when it is applic- 

 able, enables us to obtain a solution from the merely 

 formal equivalent In this connection we have Poincare's 

 theory of asymptotic series, which is expounded by M. 

 Borel in Chapter i. Independently of its convergence, 

 ithe expansion 



<•«, + '-J- + -S + . • + ^ + . ■ 



X A- .V" 



lis said to representy"(x) asymptotically if 



L X X- X'' J • 



vanishes when x is infinite. Asymptotic expansions may 

 be combined by the ordinary formal rules of rational 

 operations, and the result is asymptotically equivalent to 

 the corresponding combination of the functions repre- 

 sented. These considerations justify the use of semi- 

 convergent series in computation ; the classical e.xample 

 occurs in the theory of the gamma function. It must be 

 carefully observed, however, that although the asymptotic 

 expansion (if it exist) of a definite function is itself 

 definite, sve cannot infer the existence of a definite 

 function corresponding to a given e.xpansion 2c„^~" : the 

 reason of this rather paradoxical result is that innumer- 

 able functions (for instance e~'') lead to an asymptotic 

 expansion with zero coefficients throughout. 



In Chapter ii. M. Borel discusses the results contained 

 in Stieltjes's memoir {Aiinales de la Faculie des Sciences 

 de Toulouse, tt. viii. ix.), and in Chapter v. deals with the 

 polynomial expansions due to Mittag-Leffler. Interest- 

 ing as they are, it seems hopeless to try to analyse these 

 chapters within the compass of a review ; they are, 

 indeed, themselves of the nature of summaries, and will 

 be best appreciated by those readers who accept M. 

 Borel's invitation to consult the original memoirs. 

 Attention may, however, be called to the author's 

 NO. 1 67 I, VOL. 65] 



estimation] of these researches. It is, in eftect, that 

 the memoir of Stieltjes, though of great originality and 

 suggestiveness, isjof restricted application and not likely 

 to lead to a general theory ; and that, on the other hand, 

 while iMittag-Lefflers theory does not immediately afford 

 a calculus of divergent series, in the proper sense of the 

 term, it may 'very probably lead to one. It should be 

 added that M. Borel himself has made substantial con- 

 tributions to:; this theory of polynomial expansions ; 

 some of them [appear for the first time in the present 

 volume. 



The fact is that most of the field traversed in this very 

 attractive course is of recent discovery, and we cannot 

 expect to beipresented with a complete and symmetrical 

 doctrine all at once. Let us be thankful that M. Borel, 

 himself one of the pioneers on this novel route, has so 

 clearly and impartially indicated the progress that has 

 hitherto been made. G. B. M. 



067? BOOK SHELF. 



The Chemical Essays of Charles- William Scheele. Pp. 

 XXX -I- 294. (London : Scott, Greenwood and Co., 

 1901.) Price 5^. net. 



This is a reprint of Dr. Beddoes' translation of Scheele's 

 essays, which was published in 1786 by John Murray 

 and may still be picked up occasionally in second-hand 

 book shops. The reproduction is faithful even to the 

 mis-spelling of Priestley's name in Beddoes' preface. 

 Between this preface and the essays, however, there now 

 appears a memoir of the life and work of Scheele, 

 written for the reissue by Mr. John Geddes Mcintosh. 

 Mr. Mcintosh presumably has inspired the reissue of 

 the essays, and if this will be the means of getting them 

 more generally read by students of chemistry, he may so 

 far prove a benefactor. 



Of the essays themselves it is hardly necessary to say 

 anything. The facts they establish belong for the most 

 part to what is now very elementary chemistry and the 

 phlogistic hypothesis with which the explanations are 

 involved did not long outlive Scheele ; but the spirit 

 which breathes in these essays and the method they in- 

 culcate can never grow commonplace or antiquated. 



The strict fidelity to experiment, the rare sagacity, the 

 scrupulous and minute observation and the extraordinary 

 experimental skill combine to make Scheele a model for 

 all time. When we add to this the pathos of his early 

 struggles, the simplicity of his blameless life and the 

 nobility of his untimely death, there can be no wonder 

 that Scheele is reckoned a hero among chemists. 



It cannot be said that the memoir which accompanies 

 these essays is worthy of the subject. Mr. Mcintosh has 

 apparently considerable enthusiasm for the solid virtues 

 of Scheele and for the material outcome of Scheele's 

 discoveries, but he shows little critical insight or literary 

 taste. Speaking, for example, of the discovery of 

 chlorine, he says : " Let us now glance at the radical 

 errors of the French school, the chief of whom was 

 BerthoUet, the man who was the first to make practical 

 application of Scheele's discovery, and, as is usually the 

 case with such men, they propound a theory of their 

 own, so that some at least of the merit, if not all of the 

 original discovery, may descend upon their own mantle." 



The violence here done to BerthoUet, to the rules of 

 English composition, and to a time-honoured metaphor 

 is very remarkable. 



On the following page it is stated to be " a well-known 

 fact at the present day " that the product of distilling 

 fluorspar and sulphuric acid in a glass retort is gaseous 

 hydrofluosilicic acid. 



