;iS 



NA TURE 



[Septembek 27, 1S94 



he instanced the use made by Poincar^ of the names 

 of Fuchs and Kleia in regard to theories, priority 

 ia respect of which those eminent men would be the 

 first to repudiate. In the present instance we find that 

 the problem of the " Pellian Equation " was proposed b\- 

 Fermat and solved by Lord Bronncker, and these facts 

 need not detract in the least from the reputation earned 

 bv Pell by his skill in the Diophantine analysis. 



In the solution of the problem of the " Composition 

 of Quadratic Forms," Prof. Smith introduces the im- 

 portant notion of fundamental sets of solutions of in- 

 determinate systems of equations, and thus replaces 

 Gauss' purely synthetical solution by analysis. In the 

 arrangement of the genera of quadratic forms into 

 classes he extends to irregular determinants the prin- 

 ciples employed by Gauss for the case of regular 

 determinants. 



Gauss' geometrical representation of forms of a negative 

 determinant is given at length. Klein has recently, in 

 the lectures on mathematics delivered before the 

 Evanston Colloquium in the autumn of 1893, given a 

 remarkably simple statement of the method, and has 

 introduced the expressions "line lattice'' and "point 

 lattice ' to describe the diagrams. He also has extended 

 the method to forms of positive determinant in the 

 Gittlin^cn Xachrhhten for January 1S93. To this the 

 reader's attention may be directed as elucidating and 

 amplifying Prof. Smith's statement of the work of Gauss. 

 Klein's lecture VIII. (Evanston) should also be referred 

 to in connection with the theory of complex primes and 

 the ideal numbers of Kummer. 



On the completion of the report, his attention was 

 directed to the subject of ternary quadratic forms. At 

 the time an important memoir by Eisenstein had ap- 

 peared, in which were defined the ordinal and generic 

 characters of ternary quadratic forms of uneven deter- 

 minant ; but several of the results were left undemon- 

 strated. Prof. Smith supplied the omissions, and ex- 

 tended the results to the more difiicult and complicated 

 case of the even determinant. By giving a table for 

 forming the complete generic character of any form, he 

 accomplished for the ternary theory that which had been 

 already carried out by Lejeune-Dirichlet for the binary 

 theory. He gave, moreover, a demonstration of the 

 criterion for distinguishing between possible and im- 

 possible generic characters. This he was enabled to do 

 by the important new notion of a certain particular 

 generic character, termed by him "the simultaneous 

 character of a form and its contravariant," which had 

 not been regarded by Eisenstein. He gave a more 

 complete definition of a " genus " of forms as dependent 

 upon transformation by substitutions, and showed that 

 two form; are or are not transformable into one another 

 according as their complete generic characters do or do 

 not coincide. Me proved the formal.-e which assign the 

 weight of a given genus or order both for even and un- 

 even discriminants. This he accomplished by a com- 

 parison of two expressions, obtained by different methods, 

 for the limiting ratio of the sum of the weights of the 

 representations, by a system of forms representing the 

 classes of any proposed genus, of all the numbers con- 

 tained in certain arithmetical progressions and not sur- 

 passing a given number, to the sesquiplicate power of 

 NO. 1300, VOL. 50] 



the given number when that number is indefinitely large. 

 This paper is one of great power, constituting one of his 

 most important contributions to science. 



The above was followed by another great work " On 

 the Orders and Genera of Quadratic Forms ''cnntaming 

 more than three indeterminates. This paper will always 

 be a celebrated one in the history of mathematics. It 

 contains under date 1S67, implicitly, the solution of the 

 problem proposed fifteen years later for the Grand Pri\ 

 des Sciences Mathifmatiques by the French Academv. 

 The problem referred to was given as " Tht5orie de la 

 decomposition des nombres entiers en une somme de 

 cinq carroi." In the paper of 1S67 it was indicated that 

 the four, six, and eight-square theorems of Jacobi, 

 ICisenstein and Liouville were deducible from the 

 principles set forth. He then completed Eisenstein's 

 " enunciation " of the five-square theorem by bringing 

 under view the numbers which contain a square divisor, 

 and added the corresponding seven-square theorem. The 

 demonstrations were not given, but a general theory, 

 which includes these theorems as corollaries, was given 

 in detail. On these facts being pointed out to Hermite, a 

 correspondence ensued, which the reader wiUfind given, 

 with comments, in the introduction by Dr. Glaisher 

 The result was that I'rof. Smith sent in his demonstr.i 

 tions, and that ultimately the prize was divided betwec:i 

 him and M. Hermann Minkowski, of Konigsberg. Tho 

 latter memoir followed closely the lines of the paper o: 

 1S67, a fact which gave rise at the time to much dis- 

 cussion concerning the action that was taken by the 

 French Academy. The prize memoir is the concluding- 

 paper of vol. ii. 



Passing over, for want of space, other arithmetical 

 work of much value, a few words may be said concerning 

 the papers on elliptic functions, which constitute the bulk 

 of the second volume. 



The paper, " .Memoire sur les equations modulaires, ' 

 i contains a theory of singular beauty. Mathematicians 

 were aware, thanks to profound researches of Kronecker 

 and Hermilc, of the intimate relations that exist between 

 the theory of binary quadratic forms of negative deter 

 minant and the transformation of elliptic functions, bir. 

 beyond Kronecker's elliptic function solution of the 

 '■ Pellian Equation," no association had been discovert. 1 

 between the binary quadratic forms of positive dctei- 

 minant and the elliptic functions. 



In this paper it is shown that if 



¥{k-', n -. o 



be the modular equation for the transformation of order 

 N, the Cartesian equation 



F(i + X + /Y, \ + X- /Y) = o 



is a curve which gives an exact image of the complete 

 system of forms of positive determinant N. Hy the 

 simple process of enumerating the spirals and the con 

 volutions of each spiral, he determines the number of 

 non-equivalent classes and the complete system of 

 "reduced" forms in each class. 



In " Notes on the Theory of Elliptic Transformation " 



will be found a complete discussion of the case in which 



the modular equation has equal roots ; it is shown tha; 



the squares of the corresponding multipliers are alway-. 



I different, and that this is consistent with Kccnigsberger's 



