February 29, 191 2] 



NATURE 



581 



present reviewer encountered a slight difficulty at the 

 outset. 



Dr. Kneser starts with the problem of linear con- 

 duction of heat, but the reduction to an integral 

 ■equation depends essentially on the introduction of 

 Green's function. Now we have always regarded 

 Green's function as connected with the problem of 

 the potential; and, further, most elementary English 

 books on analytical statics get no further than explain- 

 ing Green's theorem, and possibly stating Green's 

 problem, but not defining Green's function. The 

 necessary light was throw^n on the subject by referring 

 to p. 241 of Weber's treatise reviewed below, which 

 not only gives, for the problem of the potential, the 

 definition of Green's function, but shows, further, how 

 this leads at once to an integral equation for the 

 potential of a given distribution subject to given 

 boundary conditions. In the case of the potential due 

 to fixed charges, this integral equation, of course, 

 degenerates into an ordinary integration formula. 



In heat conduction and allied problems Green's 

 function between two points, P and Q, might be de- 

 fined as the temperature or potential function at 

 one point due to a unit source, at the other sub- 

 ject to the given boundary conditions. This function 

 IS the nucleus or kernel (Kern) of the integral 

 equation, i.e. the factor which multiplies the un- 

 known variable under the sign of integration. 



After heat conduction in one dimension. Dr. Kneser 

 applies the method to the problem of small oscilla- 

 tions, Sturm-Liouville's functions, and problems in 

 two or three dimensions, the remaining sections being 

 devoted to the existence theorem, Dirichlet's problem, 

 and Fredholm's series. 



(3) A closely allied line of investigation is de- 

 veloped in Dr. Haupt's pamphlet on " Oscillation- 

 theorems," which is divided into two parts, the first 

 dealing with the general linear homogeneous differ- 

 ential equation of the second order containing an 

 arbitrary parameter in its undifferentiated coefficient, 

 while the second deals with the special differential 

 equation of the fourth order, in which the second 

 differential coefficient of a multiple of the second 

 differential coeflfiicient of the dependent variable is pro- 

 portional to a multiple of the variable itself, the multi- 

 pliers being functions of the independent variable. It 

 is largely a development of Hilbert's work on integral 

 equations, and deals in particular largely with the 

 conditions under which such differential equations 

 may lead to a "Green's system," and the correspond- 

 ing forms of the boundary conditions. 



(4) That integral equations are destined to play an 

 important part in the formal treatment of mathe- 

 matical physics is now evident. It does not appear, 

 however, probable that they will at present supersede 

 the use of harmonic analysis for purposes of calcula- 

 tion. In bringing out a fifth edition of his " Partial 

 Differential Equations based on Riemann's lec- 

 tures," Prof. Weber plainly states at the outset that 

 he has not been able to rewrite the book on the lines 

 of recent researches, partly owing to want of time 

 and energy, and also partly on the ground that the 

 iibject is now in a transient stage, in which further 

 livelopments may be expected every day, so that if 



NO. 2209, VOL. 88] 



an attempt were made to start afresh, the whole book 

 would soon be out of date. In addition to the theory 

 of integral equations, a second line of recent develop- 

 ment has grown up in the study of the principle of 

 relativity. 



In these circumstances Prof. Weber has adopted 

 the most desirable course, namely, to introduce refer- 

 ences to this recent work into the text at suitable 

 places, and his book still constitutes as good an intro- 

 duction to the study of mathematical physics as could 

 well be written. Although the book has more and 

 more become the work of W'eber himself, he still 

 desires to pepetuate the name of Riemann as having 

 sown the seed from which this large tree has grown 

 up. 



The second volume has only just appeared. It 

 deals with the theory of certain differential equations, 

 heat-conduction, elasticity, theory of vibrations of 

 strings, and membranes, electric oscillations, and 

 hydrod3mamics. 



We have referred to the unsuitability, for English 

 readers, of the German hyperbolic notation, but a 

 greater difficulty is introduced into these German 

 treatises by the use of a single integral sign to denote 

 surface and volume integrals. It certainly adds con- 

 siderably to clearness of exposition to use double and 

 triple integral signs in these cases. Of course, this 

 method is illogical as practised in this country, where 

 a triple sign of integration is often followed by a 

 single differential. The correct plan in such cases 

 is to denote surface and volume differentials by d^S 

 and dW instead of dS and dV, and it is much to be 

 wished that this were always done. 



(5) The somewhat brief discussion of vibrations in 

 the last-named treatise is a reminder that, unfor- 

 tunately, a review of Prof. Horace Lamb's 

 " Dynamical Theory of Sound " has been unavoidably 

 delayed for an inordinate time. The subject is one 

 which lends itself to treatment in three ways; by 

 the publication of memoirs on specialised researches, 

 by the production of a treatise even larger and more 

 exhaustive than Lord Rayleigh's three volumes, and 

 by the compilation of an introductory treatise in which 

 the most important fundamental principles are dealt 

 with concisely, over-elaboration being avoided. Prof. 

 Lamb has chosen the last alternative, and has thus 

 produced a book which should be of great use to 

 students of applied mathematics, physics, and 

 acoustics. 



This aim at brevity necessitates the omission of 

 details of long analytical investigations, the results 

 of which are stated without proof; for example, under 

 Fourier's series no attempt is made to reproduce the 

 existing literature relating to its convergency, and the 

 theory of vibrating systems in general is discussed 

 briefly without reference to more than a statement of 

 results in connection with transformation to normal 

 coordinates. The various chapters following the in- 

 troduction deal with general theories of vibration, 

 strings, Fourier's theorem, bars, membranes and 

 plates, plane and spherical sound waves, generation 

 and diffraction of waves, pipes and resonators, and 

 physiological acoustics, the last-named chapter being 

 a summary of a branch of acoustics falling outside 



