24': 



NATURE 



[July i6, 1908 



The author is already favourably known for his 

 excellent rdsuine of observations and experiments re- 

 lating- to temperature in insects, this, with the present 

 volume, constituting the first two instalments of his 

 projected "Experimental Studies in Entomology." 

 The production of the third volume of the series will 

 be awaited with interest. F. A. D. 



INFINITE SERIES. 

 All Introduction to the Theory of Infinite Series. Bv 

 T. J. l'.\. Bromwich. Pp. xvi + 511. (London : 

 Macmillan and Co., Ltd., 1908.) Price 155. net. 



THE first impression this book is likely to produce 

 is that, considering its title, it is very big. How- 

 ever, it is not diffuseness that is to blame for this; 

 the fact is that quite a third of the volume consists 

 of matter that does not strictly come under the title, 

 but is either introductory or supplemental. Thus we 

 have ah appendix dealing with irrational numbers 

 and limits; another on logarithms and exponentials; 

 a third on infinite integrals and gamma functions. 



It is pleasant to find the author adopting Dedekind's 

 definition of an irrational number, the only one which 

 is really scientific. In the second appendix, the ex- 

 ponential function is introduced after the logarithm, 

 but as the latter is defined by an integral, this does 

 not matter much. There can be little doubt that for 

 methodical treatment, the integral definition of log c, 

 with the elements of the theory of complex integra- 

 tion, is by far the most satisfactory; and it does not 

 introduce any gratuitous difficulties. The third 

 appendix is interesting, because it introduces recent 

 results obtained by the author, Mr. Hardy, and others, 

 which illustrate very clearly how the problems of 

 series are complicated when we pass to integrals over 

 an infinite range. In passing, it may be observed 

 that Mr. Bromwich refers with due appreciation to 

 Mr. Gibson's excellent text-book on the calculus. 



Passing on to the main subject of the book, it is 

 curious to note how much there is that is com- 

 paratively recent. Of course, .\bel and Cauchy were 

 the great pioneers ; but if we take, for instance, the 

 distinction between uniform and non-uniform con- 

 vergence, this does not seem to have been fullv recog- 

 nised before Stokes's paper of 1847 (see Mr. Brom- 

 wich 's note, p. 115); and the new definitions of the 

 " sum " of a divergent series are creations of 

 yesterday. 



The subject last mentioned is discussed in chapter 

 xi., mainly after Borel and Cesiro, and is a good 

 example of the extension of mathematical terms. 

 Borel gives a process by which, from a divergent 

 series (or sequence) 2, we can in certain cases find 

 an expression S(S) which is finite. Moreover, if S is 

 a convergent series, S(i:) is the sum in the ordinary 

 sense, and if S(S), S(S') exist, then S(2 + 2') = S(2}-F 

 S(S'). Mr. Bromwich makes some very interesting 

 comparisons between this recent theory and some 

 of Euler's transformations of divergent or oscillating 

 series. Like Fourier, Euler had a wonderful instinct, 

 which led him right, even when his logic was 

 defective. 



NO. 2020, VOL. 78] 



It is fairly plain that, with the exception of con- 

 vergent series, there is no one definition of the sum 

 superior to all others; different definitions may be 

 useful for different purposes. Again, with regard to 

 ordinary series, there is no universal test for cbn- 

 vergency, except, of course, the definition; and the 

 same remark applies to integrals with infinite limits. 

 Oddly enough, one of the most useful tests for the 

 convergence of a series (p. 35) is practically due to 

 Gauss. 



One of Mr. Bromwich 's great merits is that he con- 

 structs examples to show the fallacy of various 

 plausible assumptions which have occasionally misled 

 even the elect. For instance (p. 99), we have a pro- 

 duct n(i-t-^/„) which is convergent, although 2z/„ and 

 2«„= both diverge. The discussion of double series is ' 

 also very instructive. The fact is that any actual case 

 of summation is the construction of a linear sequence 

 .Sj, s,, S3, &c. ; so-called derangements or permutations 

 of series are best regarded as constructions of new 

 series, the terms of which have a one-one correspond- 

 ence to those of the first. Two series thus related • 

 may, or may not, have the same sum. 



Attention should be directed to the discussion (pp. ■ 

 157-60) of certain Fourier series, especially as to the 

 limiting form of the curve 



j' = sin ;'£: + 3 sin 2.\- + . . .-!-«-' sin n.v, 

 when n increases indefinitely. Reference might have 

 been made to the correspondence in Nature, vol. lix. 

 (i8q8), in which W'illard Gibbs and Prof. Michelson 

 took part. The point is that the limiting form is 

 not a mere zig-zag but a zigzag with projecting 

 spines. 



Finally, a word may be said about the examples, 

 which are very numerous and diversified. It is per- 

 haps a trifling matter in itself, but to some minds it 

 will give satisfaction, that as modern analysis is 

 becoming assimilated, illustrations of it are being pro- 

 duced which have something of the elegance and in- 

 dividual beauty of the Cambridge or Oxford problem 

 of years gone by. After all, a plant must grow before 

 it flowers. G. B. M. 



STUDIES IN EDUCATION. 

 The Demonstration Sclwols Record. Being Contribu- 

 tions to the Study of Education by the Department 

 of Education in the University of Manchester. No. i. 

 Edited by Prof. J. J. Findlay. Pp. xvi+126. (Man- 

 chester : The L'niversity Press, 1908.) Price is. bd. 

 net. 



THE work before us is to be regarded rather as an 

 introduction to future issues of the " Record " 

 than as an arranged and classified record of the results 

 of educational observation and experiment. Here the 

 authors take us into their confidence ; we are told 

 what their view of a demonstration school is, what 

 questions they hope to solve, and on what principles 

 they think the answers should be sought. When the 

 future volumes are available the record will be one in 

 which full confidence can be placed ; it will not be 

 a statement of partial facts selected consciousl}' or 

 unconsciously to fit a particular theory. For this 



