ru2 



SCIENCE. 



[N. S. Vol. XV. No. :H78. 



states. This inadequacy of presentation is 

 offset by the addition of an important supple- 

 ment which Borel himself contributes to the 

 theory of Stieltjes. We regret the omission 

 of the method of Lindelof. Its dismissal with 

 a half dozen lines and without even a refer- 

 ence to his article in the Acta Societatis Fen- 

 iiicae is possibly due to a certain haste in 

 preparation which we have fancied we have 

 detected in several places. While the method 

 of conformal representation (or transforma- 

 tion of the variable) which Lindelof employs 

 has been applied only to a restricted class of 

 divergent series, it seems probable that it could 

 be developed so as to give a more general 

 theory. 



On account of its somewhat abstract char- 

 acter Borel's treatise will probably be of 

 greater interest to the pure mathema- 

 tician than to the astronomer or student of 

 applied mathematics. Few applications of the 

 various theories have been given, probably 

 because but few applications have yet been 

 made, except in the case of the asymptotic 

 theory of Poincare. The author leaves us in 

 some uncertainty as to how far his own 

 theory has been carried and applied to differ- 

 ential equations. We hope that in a subse- 

 quent edition' the important applications will 

 be more fully developed. 



We turn now to the little book of Hada- 

 mard. This is one of a series of short mono- 

 graphs published under the general title 

 'Scientia' and devoted to the 'Expose et 

 developpement des questions scientifiques a 

 I'ordre du jour.' The special topic taken up 

 by Hadamard, as has already been stated, is 

 the analytic continuation of a power-series, 

 a^-\-a^z-\-a,z'-\-. . .. In the consideration of 

 this question two problems of the greatest 

 importance and difficulty present themselves. 

 These are: (1) The determination of the 

 nature and position of the singular points of 

 the analytic function defined by the series, 

 and (2) the calculation of the value of the 

 function at points exterior to the circle of con- 

 vergence. 



Hadamard has had the extremely difficult 

 task of compressing into a few pages what 

 has been done on these problems. In this he 



has succeeded admirably. It is extraordinary 

 what an amount of information is packed 

 away in the space of one hundred pages. Yet 

 the work is no dry compilation of facts. 

 Nowhere is the skill of the author more fully 

 shown than in the manner in which he has 

 woven his materials together. The theorems 

 are analyzed, their significance is pointed out, 

 and their demonstrations are outlined suffi- 

 ciently to show the manner in which the sub- 

 ject is treated. Attention should also be 

 called to the excellent bibliography with 

 which the book opens and to which reference 

 is constantly made. In correlating the one 

 hundred and fifty memoirs here included 

 Hadamard has performed a very important 

 service. His admirable report is not suited 

 to the reader who has little acquaintance with 

 the general subject, but to the specialist and 

 investigator it will be invaluable. 



E. B. Van Vleck. 



The Elementary Principles of Chemistry. By 

 A. V. E. Young, Professor of Chemistry in 

 North western University. New York, D. 

 Appleton & Company. 1901. 

 This book differs so radically from those in 

 general use that if reviewed at all, it must 

 be at some length. The author has used this 

 method successfully for thirteen years; his 

 object being to instruct the student during 

 the first year by this method, which he calls 

 the quantitative method. He says that its 

 inception is due to Professor Josiah P. Cooke, 

 of Harvard; he believes it 'both scientifically 

 and pedagogicallj' an improvement on prevail- 

 ing methods.' The presentation of a topic in 

 the text is to be studied by the student after 

 jierforming the laboratory experiment illus- 

 rating the same. 



The first 97 pages of the book are devoted 

 to the physical and chemical properties of 

 substances and to simple theoretic chemistry, 

 including the fundamental quantitative laws 

 of chemical action, the gas laws, atomic and 

 molecular theory, kinetic theory of gases, 

 structure and stereoisomerism. The author 

 lays particular stress on the quantitative laws, 

 and also on the laws of Gay Lussac, Dulong 

 and Petit, Mitscherlich and Eaoult, as illus- 



