Supplement to '■'Nature" March 5, 1908 



shape, it is oiil)- now becoming possible to construct 

 a corresponding theory for the hyperelliptic fimctions. 



Towards this Dr. Baker, in the first part of his 

 treatise, has made a really valuable contribution. The 

 first chapter contains an extremely clear account of 

 the hyperelliptic integrals, and in particular gives the 

 standard ones in their explicit algebraic form. The 

 corresponding Theta-functions are defined, and their 

 properties investigated; the solution of Jacobi's inver- 

 sion problem is given in an unusually clear form, and 

 art. 10 contains an instructive discussion of the vanish- 

 ing of a double Theta-functicn — perhaps one of the 

 jnost troublesome points in the whole theory. 



Chapter ii. contains the differential equations for the 

 Sigma-functions which are afterwards used to find 

 their e.-cpansions. By means of Aronhold's symbolical 

 notation they are expressed in a compact invariantive 

 form ; and the way in which they are obtained is an 

 elementary one. At the same time, as the author 

 would probably admit, the process is that of leading 

 up to a known result, and not a heUris'tic one; this is 

 not said by way of disparagement, because it often 

 happens that tedious methods of discovery are properly 

 replaced by others of a more artificial kind. Dr. 

 Baker, in a note at the end, directs attention to the 

 desirability of re-casting the detnonstration so as to 

 make it more strictlv analogous to the method used 

 for the elliptic Sigma-function. 



Chapter iii. deals with the properties of Rummer's 

 surface and \Veddle's surface in connection with the 

 properties of the hyperelliptic functions. Here the 

 author's powers of dealing with algebraical analysis 

 appear to great advantage. He has expressed the 

 principal results in a form that is both explicit and 

 elegant ; and the English reader who has this book 

 and Hudson's " Rummer's Suiface " will be able to 

 attack, if he likes, a very mteresting and unusually 

 definite field of research. Chapter v. is of a similar 

 character, and contains, among other things, Mr. 

 Bateman's proof of the differential equation of the 

 asymptotic lines on W'eddle's surface, and a geo- 

 metrical interpretation of the addition theorem. 

 Chapter iv. deals with the expansions of the Sigma- 

 functions, and gives a great number of explicit terms; 

 the invariantive character of the coefficients should be 

 specially noticed. 



The second part of the book, " on the reduction of 

 the theory of multiply-periodic functions to the theory 

 of algebraic functions," is of a much more recon- 

 dite and difficult character. One of its main objects is 

 to prove the theorem that the most general single- 

 valued multiply-periodic meromorphic function is ex- 

 pressible by Theta-functions. The proof given partly 

 depends upon Rronecker's theory of the definition of 

 algebraic constructs (Gi'bilde) by means of systems of 

 equations, partly upon the consideration of a set of 

 " defective " integrals. Dr. Baker is admirably 

 honest, and on p. 207 makes the remark : — " It seems 

 cert.'iin that the values of k,., can be taken so that 

 the determinant \c,.g\ is not zero "; the temptation to 

 make this a positive statement instead of a conjecture 

 would have been considerable to many writers. 

 Whether or not Dr. Baker's proof will stand minute 



NO. 2001, VOL. yj] 



examination in all its parts remains to be seen; it is 

 at any rate an original and very interesting discussion 

 of an extremely difticult and important problem. It 

 is not easy at the present time to foresee what will be 

 the ultimate shape assumed by the general theory of 

 Abelian functions. So far as mathematical rigour is 

 concerned, as well as in its definiteness and attention 

 to detail, the work of Weierstrass is preeminent, 

 and its influence may be continually noted, and is 

 frequently acknowledged in the present treatise. On 

 the other hand, the more intuitive methods of 

 Riemann and his followers are extremely illuminating 

 and fruitful in suggestions and results ; while as 

 reg-ards algebraic functions, the method of Dedekind 

 and Weber is very hard to improve upon. One main 

 difficulty, of course, is the increase in the number of 

 independent variables in the associated Theta-functions ; 

 to get a " geometrical " field for the variables we 

 must either plunge into unknown spaces or take new 

 elements (e.g. straight lines) in our own. 



Much light on the general theory and its difficulties 

 is afforded by some special examples which Dr. Baker 

 gives here and there, for instance, on pp. 255-72. In 

 fact, an accumulation of such examples would greatly 

 help beginners to grasp the arguments of the general 

 theory. 



In conclusion, attention may be directed to the 

 great economy of space which the author obtains by 

 abbreviated notation for matrices. The only draw- 

 back is that matrices are continually denoted by letters 

 of the same type as those indicating quantities. More- 

 over, double Theta-functions are expressed in the form 

 0{u), which stands for 0(!j,,tt,); consequently, the be- 

 ginner must be careful to realise the full meaning of 

 the symbols, and he must at once make himself fami- 

 liar with the elementary theory of matrices. Perhaps, 

 in anot'ner edition, matrices might be indicated by 

 letters of a special type. G. B. M. 



REINFORCED CONCRETE. 

 Principles of Reinforced Concrete Construction. By 

 F. E. Turneaure and E. R. Maurer. Pp. viii + 317. 

 (New York : John Wiley and Sons, 1907.) 



THIS is the latest text-book on a branch of engineer- 

 ing construction which during the past ten years 

 has developed from its first small beginnings to such 

 an important position that not only is it essential for 

 civil engineers and architects to be familiar with its 

 various applications, but they should also have a 

 sound knowledge of the principles which underlie the 

 design of reinforced concrete structures. The authors 

 have therefore practically divided the book into two 

 sections, the first part dealing with the theory of the 

 subject, the results of tests, and such questions as 

 working stresses and economical proportions, while 

 the second part is devoted to the application of rein- 

 forced concrete to building construction, arches, 

 retaining walls, &c. 



After discussing fullv the properties of the two 

 materials, concrete and steel, both when used in- 

 dependently and when used in combination, the 

 authors proceed to obtain working formulae for the 



