NA TURE 



6oi 



THURSDAY, APRIL 27, 1905. 



THREE CAMBRIDGE MATHEMATICAL 



WORKS. 



The Algebra 0/ Invariants. By J. H. Grace, M.A., 



and A. Younij, M.A. Pp. vii + 384. (Cambridge: 



The University Press, 1903.) Price los. net. 

 The Dynamical Theory of Gases. By J. H. Jeans, 



M.A. Pp. xvi + 352. (Cambridge: The University 



Press.) Price 155. net. 

 A Treatise on the Analytical Dynamics of Particles and 



Rigid Bodies. By E. T. VVhittaker, M.A. Pp. 



xiii + 414. (Cambridge: The University Press, 



1904.) Price 12s. 6d. net. 



WHATEVER opinions may be felt as to the 

 desirability of University Presses competing 

 with private firms in swelling the already too large 

 flood of school geometries or issuing cram books 

 for compulsory Greek examinations, there can only be 

 one opinion as to the series of standard treatises on 

 higher mathematics emanating at the present time 

 from Cambridge. In a country which, in its lack 

 of national interest in higher scientific research, par- 

 ticularly mathematical research, stands far behind 

 most other important civilised countries, it necessarily 

 devolves on a University Press to publish advanced 

 mathematical works. We may take it as certain that 

 the present volumes will be keenly read in Germany 

 and .America, and will be taken as proofs that England 

 contains good mathematicians, though Englishmen 

 as a nation may be unaware of their existence, with the 

 exception of the senior wrangler of one year, who is 

 forgotten the next. 



For years Salmon's " Higher .\lgebra " has been the 

 treatise which has done most to interest English 

 i>tudents in invariants. .At the present time a good 

 deal more is wanted in order to bring our knowledge 

 up to date. Messrs. Grace and Young have en- 

 deavoured to meet present requirements in a well 

 defined direction. .'\s they state in their preface, the 

 book 



" was started as an attempt to meet the need e.xpressed 

 by Elliott in the preface to ' The Algebra of Quantics ' 

 — ' a whole book which shall present to the English 

 reader in his own language a worthy exposition of 

 the method of the great German masters remains a 

 desideratum.' " 



While no book, unless it were written in four 

 languages, could satisfy the patriotic aspirations of 

 every native of our country by appealing to him " in 

 his own language," the production of an English 

 book on a subject largely developed in Germany meets 

 a distinct want. 



The subject is practically started ab initio. The 

 treatment does not strike us as very hard to follow, 

 although it is difficult for a beginner at first to 

 master the symbolical notation, especially in the 

 definition of transvectants (chapter iii.). In chapter 

 vi. the authors introduce Gordan's theorem, according 

 to which the number of covariants of a binary form is 

 always finite, and in the next chapter they employ his 

 method of proof to obtain the complete irreducible 

 NO. 1«52, VOL. 71] 



set of covariants of the quintic. .\ short chapter on 

 simultaneous systems brings us to Hilbert's theorem, 

 with which the algebra of binary forms may be said 

 to end. Chapters x. and xi. deal with geometrical 

 interpretations, and in particular vfith apolarity. The 

 sections dealing with ternary forms are less complete, 

 as the authors have considered that " with the methods 

 known up to the present the treatment of ternary forms 

 is too tedious for a text-book." 



Mr. Grace has previously been associated with th' 

 production of several mathematical text-books of a 

 quite elementary character, and the present book bears 

 many unmistakable traces of his experience as a 

 writer in making a somewhat difficult subject appear 

 relatively easy. 



We say " somewhat " difficult, because the subject 

 of Mr. Jeans's new book is incomparably harder than 

 the "Algebra of Invariants." This difficulty arises 

 largely from the fact that the kinetic theory of gases 

 is closely associated with the representation of physical 

 phenomena as they actually exist, and with all the 

 difficulties connected with irreversibility and the exist- 

 ence of temperature. It is only by statistical methods 

 that these phenomena are amenable to the equations 

 of reversible dynamics, and with every method of 

 attack some assumption must be made, since if any 

 motion of a molecular system exists it is equally 

 conceivable that the opposite motion should exist. 



Even Willard Gibbs's appeal to experience quoted 

 on p. 167 does not get over the difficulty. If we put 

 red and blue ink together into a vessel and stir them 

 up, it is true that if the inks differ in nothing more 

 than colour the result is a uniform violet ink. But 

 this is because the inks are viscous liquids the 

 motions of which are irreversible. If they were perfect 

 liquids perfect mixing would not take place, and the 

 effect of stirring would merely be to produce vortex 

 motions in which the vortex lines always contained 

 the same particles and remained constant in strength. 

 If we mix counters in a bag, the motions of the 

 counters are retarded by friction; if the counters 

 correctly represented perfectly reversible systems they 

 would never come to rest. 



Mr. Jeans in his preface considers that the dis- 

 crepancy between theory and experiment in connection 

 with the ratio of the specific heats of a gas " is of 

 greater importance than all the others together," and 

 he has endeavoured to emphasise the fact that when 

 account is taken of the interaction between matter 

 and the ether, theory and experiment harmonise ; 

 well as could be desired. But as soorr as this ether 

 is taken into account we have a simple means of 

 obviating the irreversibility difficulty by saddling the 

 ether with the whole responsibility. So long as 

 physicists are contented, in solving the differential 

 equations of wave motion in a medium, to omit the 

 terms which represent waves converging from an 

 infinite distance towards a centre of disturbance, so 

 long will there be an easy way out of the puzzling 

 contradictions arising out of Boltzmann's H-theorem. 



But there is really no reason why the presence of 

 a molecule in an indefinitely extended ether which 

 undoubtedly possesses some energy should not bring 

 about the convergence of waves coming in from an 



D D 



