Notices respecting New Books. 573 



faces and Curves of the second degree, Electric Images, and Systems 

 of Conductors. In addition to the clear textual exposition of the 

 matter, there is an extensive collection of examples, very many of 

 which are worked out with full detail : these will be of great use 

 to the student. There are several elegant aud instructive proofs 

 which have been furnished by Prof. E. Purser. 



With a Table of Contents and an Index, this text-book, which 

 is very accurately printed, is likely to be appreciated by junior 

 students, while at the same lime more advanced ones may consult 

 it with advantage. 



Theorie du Potentiel Neivtonien. Lecons professees a la Sorbonne 

 par H. Poincaee redigees par Edouard Leeot et G-eokges 

 Vincent. Pp. 36b*. 8vo ; Paris, 1899. 



This book forms part of Poincare's Cours de Physique Maihema- 

 tique, and it may be said at once that it is the best of the series. 

 The author is, as everyone knows, a master of mathematical 

 analysis; he is besides the possessor of a most luminous style of 

 exposition, and, as he is here occupied with a subject in which he 

 has himself made important advances, great hopes must have been 

 aroused among mathematicians interested in the Theory of the 

 Potential by the announcement of the publication of these lectures. 

 Such hopes will not be disappointed. 



The theory under consideration is the most striking example of 

 the assistance rendered to pure mathematics by applied mathe- 

 matics. The importance of a thorough study of what we now call 

 " potential functions " was seen by Laplace in his researches on 

 gravitation, and especially on the figure of the earth; it was 

 emphasised by Green in connexion with electrostatics and mag- 

 netism. Such a study has been found to be no less requisite 

 as a preliminary to the mathematical theories of hydrodynamics 

 and elasticity ; in the formulation of each of these branches of 

 applied mathematics it has been found that, if the theories are not 

 to be self-contradictory, there ought to exist functions which 

 satisfy Laplace's equation at all points in a particular region of 

 space, and also satisfy certain conditions at the boundaries of the 

 region. The determination of a function from such conditions 

 becomes a problem of profound interest to mathematicians, and 

 the principle of the existence of such functions led, in the hands 

 of Eiemann, to nothing less than a revolution in the theory 

 of functions of a complex variable. In the time of Eiemann the 

 proof of this principle which had been given by Dirichlet was 

 accepted as valid; but since the fallacy that underlies this proof 

 has been exposed by "Weierstrass, mathematicians have had to 

 seek for proof in other directions. Thus has arisen the famous 

 "existence-theorem," to the construction of the rigid proof of 

 which the efforts of many analysts have been directed. 



But if the existence-theorem is the roof and crown of the theory 

 of the potential, it is by no means the only part of the theory 



