February 21, 1901] 



NA TURE 



391 



edition. There were three possible ways in which this 

 task could have been fulfilled. One way was to re- 

 publish the edition of 1882, with trifling additions and 

 alterations. The second way was to retain the existing 

 text, but to add copious notes together with references 

 to recent developments bordering on the subject of 

 Riemann's lectures. The third way was to write an 

 entirely new book, based, indeed, on the earlier editions, 

 but completely brought up to date by the embodiment of 

 the new methods and problems that have come into exist- 

 ence in connection with discoveries in mathematics and 

 physics extending over nearly twenty years from the 

 date of the last edition, and nearly forty years from the 

 time when the lectures were given by Riemann. 



Prof. Weber has adopted the last of these alternatives, 

 and by so doing has produced a treatise which will be in- 

 valuable to the modern mathematical physicist. How far 

 the present treatise is to be regarded as a new work 

 written by Prof. Weber may be inferred from the fact 

 that this, the first volume only, covers 506 pages, as com- 

 pared with a total of 325 in HattendorfFs edition, and all 

 the last 350 pages are new. 



The first part, dealing with analytical methods, corre- 

 sponds more or less closely with the first three sections 

 of HattendorfiPs edition. It deals with definite integrals, 

 infinite series and the differential equations of common 

 occurrence in physics, especially linear equations with 

 constant coefficients. In this portion we are indebted to 

 Prof. Weber for an amplification of the treatment of 

 Fourier's series and Fourier's double integral theorem, 

 for a more precise treatment of continuity and for 

 entirely new sections dealing with surface and volume 

 integrals, functions of complex variables and conformal 

 representation, and Bessel's functions, the last named 

 addition occupying forty pages. 



The second part is entirely new. In it Prof Weber 

 discusses linear infinitesimal deformations and then gives 

 us a chapter on vectors, in which the modern notions of 

 *'curl" and "divergence" are fully explained, and 

 expressions for the curl of a vector given in orthogonal 

 coordinates. Thisjis followed by sections on theory of 

 the potential, including Green's theorem and potentials 

 of ellipsoids. The next section deals with spherical 

 harmonics, and this is followed by a short summary of 

 the principles of dynamics, including the Hamiltonian 

 equations and least action. 



The only branches of physics treated in Hattendorfif's 

 edition were conduction of heat, elasticity (including 

 vibrations) and hydrodynamics. The absence of any 

 reference to electricity and magnetism is accounted for 

 by the fact that these subjects, together with gravitation, 

 were treated by Riemann in a separate course, of which 

 an edition was also prepared for press by HattendorfF in 

 1876. The third part of the present volume forms a 

 treatise on the mathematical theory of electricity and 

 magnetism, for which Prof. Weber is thus solely re- 

 sponsible. The fundamental principles of electrostatics 

 and magnetism are based on the hypothesis of a con- 

 tinuous medium, the electrical and magnetic properties of 

 which depend on the existence at every point of space 

 of certain vector quantities satisfying stated laws ; and 

 the subject is thus introduced much after the manner 

 NO. 1634, VOL. 63] 



adopted by Hertz. Among the problems depending for 

 their solution upon the method of conformal represen- 

 tation, we notice an application of the transformation of 

 Schwarz and Christoffel to the distribution of electricity 

 on a prism, an example which practically amounts to an 

 exposition of this transformation. 



The subject of contact electricity, too, receives ample 

 mathematical treatment. Perhaps, however, the most 

 interesting sections are those dealing with electrolysis ; 

 and this interest is largely due to the important part 

 which Prof. Weber himself has played in advancing our 

 theories of this difficult subject. A comparison of these 

 sections, in which the problem of electrolysis is made to 

 depend on the solution of differential equations which 

 Weber integrates in certain special cases, with the frag- 

 mentary information contained in text-books of forty 

 years ago, is sufficient indication of the progress which 

 has been made during the past half century in developing 

 new fields of study in applied mathematics, and in co- 

 ordinating and perfecting the mathematical treatment of 

 electricity. 



Steadyflow of electricity, and the fundamental principles 

 of "electrodynamics" (as it used to be and still some- 

 times is called), occur in their proper places in the present 

 volume. No mention, however, is made of Hertzian 

 oscillations, which are to be dealt with in the forthcoming 

 second volume in connection with the theory of oscilla- 

 tions in general. The remaining subjects to be treated 

 in the latter volume include conduction of heat, hydro- 

 dynamics and elasticity. 



Mathematicians will, of course, not be satisfied with 

 the present treatment of such matters as convergence 

 of series and of integrals, and on the other hand physic- 

 ists will require to supplement the volume with other 

 works containing a fuller consideration of "the experi- 

 mental aspect of the various theories. It was no purpose 

 of Prof Weber's to aim at completeness in either of 

 these respects. The object of the book is rather to fur- 

 nish a statement of results both in pure mathematics and 

 in physics, and to indicate the methods by which the 

 former results, used in conjunction with the latter, lead to 

 the mathematical solution of physical problems. As an 

 illustration of the spirit of the book, we may notice the 

 article on semi-convergent series, where the use of these 

 series is explained mainly by the consideration of an illus- 

 trative example. Again, as Prof Weber points out, there 

 are many physical problems which can only be solved by 

 approximate methods of Httle or no mathematical interest, 

 and these again are omitted. 



Now a book of this character appeals to a considerable 

 class of present-day physicists. Forty years ago physical 

 laboratories hardly existed, and the pioneers of physics 

 in this country were Cambridge wranglers who approached 

 the subject from its mathematical side exclusively. Now 

 that physical laboratories are scattered all over the 

 country, and that the working man can attend science 

 classes close to his own door, we are running to the 

 opposite extreme, and there is an ever-increasing class of 

 student who requires to master the mathematics required 

 for his physical studies, but who starts his mathematical 

 reading too late in the day to work up step by step from 

 the very beginning. As was pointed out by Riemann 



