57-1 Notices respecting Neiv Books. 



which has Deeded to be made rigorous by the touch of a master- 

 hand well skilled in the refinements of modern mathematical 

 analysis. That the strengthening of the base of the structure was 

 a problem well worthy of the powers of a fully equipped analy&t 

 -was realised by Picard, and the chapters of his Traite <T Analyse 

 devoted to the exposition of the theory show the conspicuous 

 success which attended his efforts. The author of the work 

 before us has been no less successful in the independent treatment 

 of fundamental questions, and his discussion of the convergence 

 of the integrals which represent the components of attraction at 

 or near a point on a charged surface will especially repay careful 

 reading. 



The author devotes the first three Chapters of his book to what 

 would in this country be regarded as the elements of the theory, 

 sparing no pains to establish them in a thoroughly rigorous 

 fashion. In the fourth Chapter he introduces Green's function 

 and explains its importance; and in the following chapter he 

 utilises Green's function for a sphere and a point to construct a 

 function satisfying Laplace's equation within a sphere and taking 

 a given value at all points of the surface : this is Dirichlet's 

 problem for a sphere. The sixth Chapter is devoted to the 

 properties of magnetic shells or sheets of doublets. In the 

 seventh Chapter the author presents his own proof of the existence- 

 theorem : this proof is founded on a theory of the equivalence of 

 certain surface and volume distributions ; it is shown that the 

 equivalent distributions can be determined for a sphere, and that 

 for any surface without conical points the solution of Dirichlet's 

 problem can be arrived at by constructing such distributions and 

 the related potential functions for a definite series of spheres. 

 As is well known, the author has extended this proof to a more 

 general class of surfaces, and the construction of it must be 

 regarded as one of his most brilliant achievements in analysis; in 

 his book he wisely refrains from presenting the proof for the 

 more complicated case, and refers the reader to his memoir in the 

 ' American Journal of Mathematics,' vol. xii. The eighth Chapter 

 contains an account of the method by which Neumann sought to 

 construct the solution of Dirichlet's problem for a surface by 

 means of a sheet of doublets, or a magnetic shell, coinciding 

 with the surface ; and the ninth and last Chapter contains indi- 

 cations concerning a number of extensions of Neumann's method. 

 The account of the method is most lucid and suggestive, and, as 

 the author points out, the method is of especial importance on 

 account of the identity which it establishes between the functions 

 which satisfy Laplace's equation in limited regions of space and 

 the potentials of distributions of density. 



British readers will not fail to notice the use that is made, 

 throughout the last six Chapters, of Gauss's theorem that the mean 

 value of a potential function over the surface of a sphere is the 

 same as the value at the centre ; in English books on the same 

 subject this theorem appears as a curious incidental result. It 



