ON THE THEORY OF NUMBERS. 365 



divisors of n, which occur in the right-hand members of his formulas, are ren- 

 dered somewhat more complicated than those of M. Kronecker. It is always 

 possible to pass from one of M. Joubert's formula) to the corresponding for- 

 mula of M. Kronecker, by an elementary process, of which M. Joubert has 

 himself given an example (at p. 25 of his memoir). 



One formula, however, has been obtained by M. Herinitc from his investi- 

 gation of the discriminant of the modular equation, which is entirely distinct 

 in form, and as it would seem in substance, from those of M. Kronecker. 

 Taking a modular equation of a prime order n, M. Hermite shows that its 

 discriminant is of the form 



where 0(m) is a reciprocal polynomial, prime to u and to 1 — u s , containing no 

 equal factors, and of order |(>r — 1)— i n + (-\ . From the nature of a 



discriminant, if w renders two of the quantities l-)<p ( y , - ) equal to 



one another, <p(u)) is a root of the equation fl(«)=0, and conversely. It is 

 thus possible, by a method of which we have already given examples, to assign 

 a system of quadratic equations (or quadratic forms) having integral coeffi- 

 cients, which shall correspond, one by one, to the roots of the equation 0(w) = O. 

 Equating the number of these quadratic forms to the index of the polynomial 

 d(u), M. Hermite obtains a formula which is essentially limited to the case 

 when n is a prime, and which, translated into the notation of M. Kronecker, 

 is as follows, 



2^F(A) + 22 2 F(A) + 62 3 G(A) 



-K*-i)H[-+@]. 



the summations 2„ 2 2 , 2 3 extending respectively to all values of S which 

 give positive values to "the numbers 



A=(8S-3») (n-2o), A=83(«-8a), A=S(»— 162). 



The difference between these series of determinants, and those which occur 

 in M. Kronecker's formula?, is very remarkable. 



136. Arithmetical Demonstrations of the Formula? of M. Kronecker. — M. Kro- 

 necker informs us that, when he had connected his formula?, in the manner 

 already described, with the expansions of certain elliptic functions, he directed 

 his attention to the process (art. 127) by which Jacobi transformed the ana- 

 lytical proof of the 'theorem of four squares' into an arithmetical one*. 

 Applying a similar transformation to the analytical proof of the equation of 



O 2 f* TT2TT GO 



he succeeded, after many reductions, in obtaining a purely arithmetical proof 

 of the formula? I. II. and V. which are included in XL This important 

 investigation has not yet been published : instead, M. Kronecker has given a 

 remarkable theorem which appears (as he observes) to contain the germ of 

 another, and very different, arithmetical demonstration of his formula?. He 

 has enunciated the theorem for prime numbers only, remarking, however, 

 that it admits of extension to composite numbers also. The result is simplest 

 in the case of a prime number p of the form 4»i + 3. 



* Monatsberichte for 18G2, p. 307. 



