SCIENCE. 



[N. S. Vol. XXI. Xo. 524. 



of some one variable. Instead of true total 

 differentials, he gets total derivatives. The du 

 in § 137 are not total differentials, but differ- 

 tials of functions of one variable. In the dif- 

 ferentiation of implicit functions the author 

 assumes merely the existence of the partial 

 derivatives. He shoidd assume also their con- 

 tinuity. The form of demonstration is bad, 

 as it requires him to assume (tacitly) the ex- 

 istence of the very thing he is seeking, viz., 

 dy/dx. 



In the treatment of envelopes, § 141, the 

 author does not as usual give sufficient condi- 

 tions for the validity of his reasoning, but 

 contents himself with the vague statement in 

 a footnote that the process is all right ' in all 

 applications made in this book.' This blemish, 

 which a few lines will remedy, should be re- 

 moved in another edition. The definition of 

 an infinite series given in § 147 is not felici- 

 tous. In avoiding the lax definition usually 

 given the author has gone to the opposite ex- 

 treme. The simplest way seems to be to con- 

 sider 



fTj -|-- "2 + "3 -f- , • • • in inf. 



as a symbol to which a meaning is attached as 

 to other symbols, as > < = , etc. The solu- 

 tion of Ex. 3, § 152, is not quite rigorous, as 

 it postulates the covergence of G. In § 160 

 undefined arithmetical operations are per- 

 formed on series. 



We can not agree with the author that the 

 remainder in Taylor's series for several vari- 

 ables is too complicated to be given. The 

 treatment of maxima and minima can be made 

 much more complete without complications or 

 difficulty. The reasoning given at the bottom 

 of page 248 can be made not only ' plausible,' 

 but entirely conclusive, using no more space 

 that that required b.y the author. 



In the reduction of indefinite integi-als the 

 author proves the trivial formula? 



J {du + dv — dw) -=r y (III + J* '/'' — I* dw, 



J* adv ^ a J' dv, 



but omits entirely the demonstration relative 

 to the transformation of the variable. This 

 is all the more surprising as this transforma- 



tion is constantly employed, even in establish- 

 ing important theorems. Two chapters, 

 XXIX. and XXX., are devoted to definite 

 integrals. In the first we arrive at the notion 

 of a definite integral by means of the notion 

 of area; in the second, by means of the limit 

 of a sum. In our opinion the first treatment 

 is not only superfluous, but should be entirely 

 omitted on several counts. 



The relatively few blemishes in this work, 

 the reviewer is glad to state, will be removed 

 in the next edition. James Pierpont. 



Yale Uxiversity. 



The Study of the Atom, or the Foundations 

 of Chemistry. By F. P. Vex able. Easton, 

 Pa., The Chemical Publishing Co. Pp. 290. 

 The history of an important scientific theory 

 is an interesting study, where it is possible, 

 as it often is, to trace the orderly development 

 of that theory from stage to stage. The evo- 

 lution of the atomic theory is a subject which 

 has claimed the attention of many writers, 

 and the story has been told so often and so 

 well in works on the history of chemistry, that 

 one wonders whether it is not familiar to most 

 chemists. A careful perusal of this book does 

 not disclose any new point of view, or any- 

 thing new in the method of treatment, though 

 the matter is generally presented in a satis- 

 factory manner, especially Chapter V., which 

 deals with the periodic system. In the last 

 chapter of the book the author considers the 

 most recent hypotheses regarding the constitu- 

 tion of matter by J. J. Thomson, Rutherford 

 and others. The book is generally clear, con- 

 servative in tone and, on the whole, well-pro- 

 portioned, though 75 pages, or one fourth of 

 tlie contents, seems rather too much to de- 

 vote to the conception of the atom before the 

 time of Dalton, especially as this material 

 must be taken entirely from secondary sources. 

 The book may be commended as a good sum- 

 mary for students. E. T. Allex. 



SOCIETIES AND ACADEMIES. 



NEW YORK ACADEMY OF SCIENCES. SECTION OF 

 GEOLOGY AND MINERALOGY. 



The section was called to order at 8:15 r.M., 

 November 21, 1904, with Vice-president Kemp 

 in the chair and forty persons present. 



