64: 



NATURE 



[February 13, 19 13 



tributes some interesting data on the thermo- 

 cliemistry of silicon. F. Weigert lias an important 

 paper on the effect of oxygen in retarding and 

 inhibiting photochemical reactions. It will be 

 evident from this very brief summary that the 

 Nernst Festschrift is full of interesting material 

 and is in every sense worthy of the distinguished 

 physical chemist in whose honour it was published. 



F. G. DONNAN. 



FUNCTION THEORY. 

 Lectures on the Theory of Functions of Real 

 Variables. Vol. ii. By Prof. J. Pierpoinl. 

 Pp. xiii + 64S. (Boston, New York, Chicago 

 and London: Ginn and Co., 191 2.) Price 

 20^. net. 



THE main topics of this volume are proper 

 and improper integrals, series and pro- 

 ducts, point-sets and aggregates, continuity and 

 discontinuity, and the geometrical notions derived 

 from intuition. In style and method it follows 

 the same lines as vol. i. 



The first thing to notice is the substitution of 

 " metric set " instead of " measurable set " for a 

 set of points the upper and lower contents of 

 which are equal. In chapter xi. we have the 

 theory of the upper and lower measure of any set 

 which can be enclosed by a countable aggregate 

 of metric sets. The treatment is quasi- 

 geometrical, after the manner of Minkowski, but, 

 of course, all the arguments used are purely 

 arithmetical. In this, as in all other parts of the 

 subject, we are struck by the variety of cases 

 presented by the strict arithmetical theory, where 

 intuition suggests one definite conclusion. There 

 are so many new symbols used that it is difficult 

 to give an account of this valuable and original 

 discussion of measure ; it concludes with a 

 theorem which may be regarded as an extension 

 of the statement that, if a finite segment is divided 

 into any finite number of parts, the length of the 

 segment is the sum of the lengths of the parts. 

 It is extremely interesting to compare the analyti- 

 cal theorem, and those which lead up to it, with 

 the geometrical theorem, taken as obvious by all 

 previous generations of mathematicians. 



Another novel feature of the work is the author's 

 definition of improper integrals. This is given 

 on p. 32, after a statement of two other current 

 definitions. It leads (pp. 402 sqq.) to a discussion 

 of improper Lebesgue integrals, and an extended 

 theory of the change of order of integration in a 

 multiple integral. The theory of Fourier series 

 is discussed from the same point of view, and we 

 are thus led to see that the Fourier expansions 

 are valid for cases which do not satisfy the con- 

 NO. 2259, VOL. 90] 



ditions of Riemann and Dirichlet. Lebesgue, in 

 fact, has made an addition to the theory of trigono- 

 metrical series so great that it ought not to be 

 ignored in any treatise dealing with them with 

 any attempt at completeness. 



The sections which deal with geometrical con- 

 ceptions are those which are likely to have the 

 greatest educational effect upon the mathematical 

 student. If we define an analytical curve by the 

 equations .\- = <^(f), y = i/'(i), where <l>{t), ip(t) are 

 one-valued continuous functions of a real variable 

 t in a certain interval, then the curve is continu- 

 ous, and, if closed, bounds a region in the plane 

 [x, y). But it need not have a tangent at every, 

 or any, point : it may fill up a plane area, such 

 as a square, and no arc of it need have a finite 

 length. Anything more remote from the conclu- 

 sions of ordinary intuition it is hard to conceive. 

 At the same time, all these statements have been 

 proved with the utmost degree of rigour at present 

 attainable, and seem to be proof against all 

 possible objections. 



In a similar way, the definition of the area of 

 a surface as the limit of that of an inscribed poly- 

 hedron was shown to be fallacious by Schwarz 

 (whose proof is reproduced on p. 626). The author 

 gives a definition of the area, based upon the 

 assumption x, y, s = ^(u, v), ^{u, v), x ('*» 'v)i where 

 <j>, ij/, x are functions of the independent real 

 variables u, v, which range over a certain field. 

 Thus we are brought back once more to Gauss's 

 classical memoir as the first analytical treatment 

 of surfaces destined to be of permanent value in 

 the widest sense. 



Practically, a good deal of the success of a 

 mathematical treatise depends upon its symbolism. 

 Dr. Pierpoint seems to have fairly hit the mark; 

 his new symbols are not too many to remember, 

 and each of them crystallises an important concep- 

 tion. Undoubtedly we shall have, before very 

 long, a new and generally accepted system of 

 symbols. At present we are in a state of com- 

 parative chaos, just as at the time of the invention 

 of the infinitesimal calculus. The sooner this is 

 ended the better ; and it might be appropriately 

 discussed by the next mathematical congress, if 

 there is any prospect of agreement. 



An Englishman naturally compares this work 

 with that of Dr. Hobson. Dr. Pierpoint has the 

 advantage of writing at a later date, and is thus 

 able to include more recent discoveries ; in other 

 respects there is a contrast, which does not detract 

 from the merits of either work, but is more or 

 less typical of the nationafities of their authors. 

 One abbreviates when he can, the other when he 

 feels that he must ; one tries to avoid meta- 

 physics, but scarcely succeeds, while the other 



