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XIII. Notices respecting New Books. 

 Die Parti ellen Differential- Glekhung en der Maihematischen Physik. 



Nach Riemann's Vorlesungen in Vierter Auflage neu bearbeitet 



von Heinrich Weber. Erster Band. Pp. xviii-f506. 



Braunschweig : Friedrich Vieweg und Sobn. 1900. 

 T\/ r ELL-NIGrH half a century has elapsed since the time 

 "" when Riemann first delivered the course of lectures on 

 partial differential equations which was subsequently" published 

 in book-form, and which is well known to all students of mathe- 

 matical physics. The first three editions of this important and, 

 at the time of its appearance, unique work were practically 

 identical in form and scope, the last edition appearing in 1882. 

 Meanwhile, the subject was making enormous strides, the rapid 

 progress of experimental investigation constantly opening up 

 new vistas of problems, and calling for fresh methods of dealing 

 with them. The gradual abandonment by leading physicists of all 

 action-at-a-distance theories, and their replacement by theories 

 based on the transmission of strains and stresses through a 

 continuous medium, have in no small degree modified the methods 

 of dealing with various classes of problems. Quite apart from 

 the stimulus received by mathematical physics from the accumu- 

 lation of new facts and the development of new theories, rapid 

 progress was also being made in various branches of pure 

 mathematics, and some of the results obtained were shown to 

 lend themselves easily to the solution of important physical 

 problems. Among such developments may be mentioned the 

 modern theory of functions of a complex variable. 



Accordingly, when the demand arose for a new edition of 

 Riemann's work, the publishers, realizing that a mere reprint of 

 the former editions would no longer meet modern requirements, 

 entrusted the task of bringing the book up to date to the able 

 hands of Professor H. Weber. The work is not yet complete, for 

 the book before us is only the first volume, and a second is 

 promised shortly. 



Some idea of the magnitude of the task undertaken by Professor 

 Weber may be gathered from the fact that the last edition of 

 Riemann's work was a small volume of only 325 pages. The 

 first volume of the new edition contains over 500. 



Practically the outcome is a new treatise on mathematical 

 physics by Professor Weber. Riemann's method has been re- 

 tained, but the scope of the book has been enormously extended. • 



The volume is divided into three sections. Section I. is devoted 

 to an account of the analytical methods used in the solution of 

 physical problems, and deals with the following subjects : — definite 

 integrals, Dirichlet's and Fourier's integrals, infinite series, 

 Fourier's expansion, multiple integrals, functions of a complex 

 variable, differential equations, and Bessel's functions. Section II. 

 contains an account of certain fundamental geometrical and 

 dynamical propositions, including the subject of strains, vector 

 analysis, the theorems of Gauss and Stokes, the theory of 



