104 



NATURE 



[March 30, 19 16 



Another great modern theory is that of elliptic 

 modular functions, with its development, that of 

 automorphic functions. In a letter to Borchardt 

 ("Crelle," vol. Ixxxiii. (1877)) Uedekind pointed 

 out the importance of the function he calls the 

 Valenz ; essentially this is no other than the modu- 

 lar function ;(w), which enjoys the property that 

 j(a)) = j((oi) if, and only if, a)i = (ow + )8)/(-)'a) + 5) 

 where a, j8, y, 5 are real integers such that 

 ad - fiy—i. This introduction of / as funda- 

 mental, instead of Hermite's (f>, ^ functions, marks 

 an epoch in the theory; it should be noted, how- 

 ever, that H. J. S. Smith had practically reached 

 similar results as early as 1865 (see his report on 

 the Theory of Numbers, Arts. 125 ff.). 



We now pass on to Dedekind's work in the 

 theory of numbers. Gauss extended the theory 

 so as to include complex integers m + ni, and 

 proved that all the usual rules, especially that of 

 the unique resolution of an integer into prime 

 factors, still remained valid. Kummer investi- 

 gated algebraic integers derived from the period- 

 equations of cyclotomy, and was confronted by 

 the vexatious fact that the theorem about prime 

 factors broke down; thus we might have a^ = yd 

 with a, fi, y, 5 all integral, each irresolvable in 

 the field considered (and in that sense prime), yet 

 y essentially differing from a, j3 by having a 

 different norm. By the invention of ideal primes, 

 Kummer overcame the diflSculty, so far as these 

 cyclotomic integers were concerned. His dis- 

 coveries naturally suggested a definition of an 

 algebraic integer in general, and the problem of 

 defining its prime factors. Dedekind first gave 

 a complete solution in supplement xi. of the third 

 edition (1879) of Dirichlet's " Zahlentheorie " ; this 

 is undoubtedly one of the finest mathematical works 

 that have ever been written, and although in the 

 fourth edition (1894) the method is simplified, 

 the original exposition should always be read, and 

 in some ways is unsurpassed, not to say unsur- 

 passable. Briefly, the author establishes the 

 notions of corpus (or field), ideals and their bases, 

 discriminants, including that of the field con- 

 sidered ; he proves the general laws of divisibility 

 for every field, and in particular shows how to fac- 

 torise the real integral prime factors of the dis- 

 criminant of the corpus — one of the main difficul- 

 ties of the theory. Besides this, he discusses 

 systems of units, the copiposition and equivalence 

 of ideals, their connection with the theory of 

 forms, and the problem of finding the number of 

 non-equivalent classes for a given field. All 

 these results are of the highest generality and 

 importance ; and every arithmetician, who wishes 

 to advance the theory, must be familiar with 

 them. 



In conjunction with H. Weber, Dedekind pub- 

 lished in "Crelle," vol. xcii. (1882), a long and im- 

 portant memoir on algebraic functions of one 

 variable. The main feature is the discussion of 

 "algebraic divisors," which play much the same 

 part here as ideals do in an arithmetical field. 

 They allow us to gain a precise conception of a 

 "place" on a Riemann surface, and lead in a 



remarkably simple way to proofs of the invariance 

 of the deficiency (genre, Geschlecht) of the sur- 

 face, the Reimann-Roch theorem, and so on. 

 Consideration of expansions in a variable t is 

 reduced to a minimum, though (as pointed out 

 by Weierstrass) it cannot be avoided altogether. 

 The methods of this memoir have been developed 

 by Hensel and Landsberg in their treatise on 

 algebraic functions ; it seems to us that they form 

 a happy mean between merely heuristic methods 

 and the very dry presentation of the Weierstrass- 

 ian school. 



Another subject on which Dedekind wrote some 

 valuable notes is the theory of groups ; however, 

 this is not the place to give a list of his writings. 

 It is to be hoped that they will be published in a 

 collected form, as some of them are not easily 

 accessible; they are not voluminous, and, so far 

 as our experience goes, they are remarkably accu- 

 rate, so there is no reason for delay. G. B. M. 



4 



NOTES. 



A coNFEKENtE convened by the president and council 

 of the I'J.oyal Society was held at Burlington House 

 on Wednesday. March 22, to consider the desirability 

 of establishing a Conjoint Board of Scientific Societies 

 for the purpose of organising scientific elfort in this 

 country. Delegates from the following societies 

 attended to center with the president and council of 

 the Royal Society : — Royal Society of Edinburgh; 

 Royal Society of Arts, Royal Anthropological Institute, 

 Royal Astronomical Society, Royal College of Phy- 

 sicians, Royal College of Surgeons, Royal Geo- 

 graphical Society, Royal Institution, Institution ot 

 Civil Engineers, Institution of Electrical Engineers, 

 Institution of Mechanical Eng:ineers, Institution of 

 Mining Engineers, Institution of Naval Architects, 

 Institute of Chemistry, Society of Chemical Industry, 

 British Association, Chemical Society, Geological 

 Society, Linnean Society, London Mathematical 

 Society, Physical Society, Physiological Society, 

 Zoological Society. The following resolution was, 

 passed unanimousl)", and a committee was appointed 

 to draft a scheme for giving effect to the resolution 

 and to report thereon to a future meeting, viz. : — 

 "This meeting considers that it is desirable to estab- 

 lish a Conjoint Board of Scientific Societies for the 

 purpose of (i) promoting the co-operation of those 

 interested in pure or applied science ; (2) supplying <a 

 means by which the scientific opinion of the country 

 may, on matters relating to science, industry, and 

 education, find effective expression; (3) taking sucb 

 action as may be necessary to promote the applicatioii 

 of science to our industries and to the service of the 

 nation ; (4) discussing scientific questions in whicfc 

 international co-operation seems advisable." — We aR 

 glad that the Royal Society has taken this st^ 

 towards the organisation of scientific activities for tfy 

 promotion of national welfare. . The necessity for th< 

 unity of effort, contemplated in the principles embodiA 

 in the foregoing resolution led to the establishment 

 the British Science Guild in 1905 ; and Sir Ron? 

 Ross, in the Times of March 29, expresses the opini< 

 that the business affairs of science would be hetti 

 entrusted to such a separate body as the guild thJ 

 to a board of scientific societies, the members of whic 

 are chiefly interested in the publication and discussiol 

 of scientific papers. 



On February 23 the French Academy of Agricultt 

 held its annual meeting. There is always a touc] 



