86 



NATURE 



[September 23, 1915 



cisms of the necessary literature, and specifically 

 indicate the problems which are now awaiting- 

 solution. If only because they discharge these 

 functions in two departments of mathematics 

 which, in the opinion of the present writer, are 

 the most difficult of approach by the student, the 

 two volumes now under review have made a most 

 welcome and timely appearance. 



(i) The subject dealt with by Dr. Bateman is 

 of very recent growth as such, and is of nearly 

 equal interest to the pure mathematician and the 

 physicist. The references to the workers in this 

 department are sufficient to indicate this fact, and 

 yet the author has been singularly successful in 

 his effort to make the work intelligible to readers 

 who lack a somewhat extended mathematical 

 equipment. Several other branches of modern 

 electrical theory are large enough to need a text- 

 book of their own, such as the Principle of Rela- 

 tivity, and these branches, already provided for, 

 the author does not touch in detail. In fact, the 

 scope of the work is limited to such developments 

 of the electro-magnetic theory as are directly con- 

 cerned with the solution of the equation of wave 

 motion — the " boundary problems " of the 

 physicist. The theories of potential problems, the 

 scattering and diffraction of waves by obstacles 

 of various forms, the pressure of radiation, elec- 

 trical vibrations on variously shaped bodies, are 

 typical examples of problems which are admirably 

 reviewed and discussed, not only from the strictly 

 mathematical viewpoint, but with reference to 

 the physical implications of the results. For ex- 

 ample, such problems as the tails of comets, wire- 

 less teleg-raphy, and the residual blue of the sky, 

 come within the scope of the volume, although 

 the author is sometimes compelled by considera- 

 tions of space to give only the general outlines, 

 with the necessary references to enable the 

 reader to pursue the subject. An excellent 

 account of the theory of the various mathe- 

 matical functions which are necessary is given 

 when occasion arises, without, however, the exist- 

 ence theorems which would render the work 

 unwieldy. 



Perhaps the most interesting part of the book 

 is at the end, where the question of possible 

 structure in the aether is discussed. It is sufficient 

 to say that the greater part of the work is 

 Dr. Bateman 's own, and the reader is assured 

 of an authoritative treatment of a very fascinating- 

 problem. References are exhaustive throughout 

 the book, which fills all the three needs specified 

 in our first paragraph, 



(2) The subject of Prof. Hilton's work has a 

 more restricted appeal, and is still to a g-reat 

 extent a matter for the pure mathematician. But 

 NO. 2395, VOL. 96] 



recently it has come into prominence in connec- 

 tion with electrical theory, and the appeal is 

 widening. Prof, Hilton has in any case rendered 

 signal service by collecting together, into a con- 

 nected body of attractive theory, the branch of 

 mathematics which has hitherto been most scat- 

 tered, or only treated casually as a particular case 

 of more general theory. He confines his treat- 

 ment to those properties of substitutions which do 

 not depend on "groups," and all such properties 

 which are vital to the Theory of Groups, and the 

 theories of Bilinear Forms and Invariant-Factors 

 receive a satisfactory and connected account in 

 this volume. The discussion of substitutions the 

 determinant of which does not vanish is very com- 

 plete, and those with a non-zero determinant are 

 sufficiently introduced to the student. Important 

 applications of the work are indicated, the 

 examples are numerous, well graded, and very 

 suitably selected, and there are very few misprints. 

 A student who uses Prof. Hilton's book will find 

 the subject very attractive and easy to assimilate. 

 It should find a place in every college library. 



The book is issued by the Clarendon Press, and 

 Dr. Bateman 's by the Cambridge University 

 Press. Each volume maintains the best traditions 

 of the series to which it belongs. 



OUR BOOKSHELF. 

 La Science Allemande. By Prof. P. Duhem. 



Pp. 143. (Paris : A. Hermann et Fils, 1915.) 

 Prof. Pierre Duhem, member of the Institute, 

 has printed four lectures which he gave to stu- 

 dents at the University of Bordeaux. Their 

 general aim was to deliver the minds of his hearers 

 from subservience to German thought, and they 

 certainly make for fresh air. The first deals with 

 mathematics and philosophy, and its conclusion 

 is that an excessive fondness for deductive method 

 and a contempt for common sense have made 

 intellectual Germany like the house of Chrysale, 

 whence reasoning drove forth reason. Hegel's 

 axiom as to the identity of contradictories is com- 

 pared to that of Nicolas of Cues, often called the 

 cardinalis teutonicus , that the maximum is always 

 identical with the minimum. But this is surely 

 playing to the gallery. The second lecture is de- 

 voted to the experimental sciences, and includes 

 an interesting sketch of Pasteur's trial and error 

 methods, a denunciation of Haeckel, and an assur- 

 ance that Fabre's fine work has left us with scarcely 

 more than the debris of the Darwinian theory. 

 The exaggeration and misunderstanding involved 

 in the last statement show us that the illustrious 

 author has his obfuscating prejudices like the rest 

 of us. 



The best part of this lecture is its illustration 

 of "V esprit de finesse" in French science. The 

 third lecture is on the historical sciences, and 

 is full of righteous anger. It illustrates the 



