ON THE THEORY OF NUMBERS. 347 



is only necessary to observe that we must choose for 0(w) and »/<(w), and 

 similarly for <p(7w) and ^(7u/), expressions having the same denominator. 



At the beginning of his memoir Jacobi says that the formulae to which it 

 relates are probably finite in number. It would seem that when he expressed 

 himself thus, he had not yet found his three general formulae, each of which 

 contains an infinite number of equations between series having their exponents 

 contained in the same quadratic form. But it is certainly very unlikely that 

 equations between series whose exponents are contained in different quadratic 

 forms, exist for any but a few of the simplest forms, or for them in infinite 

 number. 



130. The Formula of M. Kronecker. — We now come to an important series 

 of results, discovered within the last few years by M. Kronecker, which form 

 a memorable accession to our knowledge of quadratic forms, and which have 

 opened an entirely new field of arithmetical inquiry. Their demonstration 

 requires considerations of a very complicated kind ; and as they are certainly 

 among the most interesting, so also they must be reckoned among the most 

 abstruse of arithmetical truths. Unfortunately, in the brief notices* which 

 M. Kronecker has given of his investigations, his methods are indicated only 

 in a very general manner ; and, notwithstanding the light which has been 

 thrown on them in the subsequent memoirs of MM. Hermite and Joubertt, 

 it is occasionally difficult to rediscover them. Nevertheless, as a mere enu- 

 meration of formulae, unaccompanied by any explanation of the methods by 

 which they have been obtained, would be of little use to the reader, we shall 

 attempt in the next article a complete demonstration of one or two of them, 

 which may serve as specimens of the rest. 



The following (with an unimportant change in the notation) are the eight 

 equations given by M. Kronecker (Crelle, vol. lvii. p. 248 ; Liouville, New 

 Series, vol. v. p. 289). 



I. F(2»m) + 2F(2> l m-l 2 ) + 2F(2 li m—2 2 ) + .. . 

 = 2*(m) + fc^-fy) + ¥-(2"- 2 m). 



II. F(2ro) + 2F(2m- 1 2 ) + 2F(2m-2 2 ) + 2F(2m-3 2 ) + . . ," ; 

 =2<J>(m). 



III. F(2»i)-2F(2m-l 2 ) + 2F(2>n-2 2 )-2F(2m— 3 2 ) + . . . 



=0. 



IV. 3G(m) + 6G(m-l 2 ) + 6G(Hi-2 2 ) + 6G(m-3 2 )-f-... 



= $(m) + 3¥(». 



V. 2F(m) + 4F(m-l 2 ) + 4F(m-2 2 ) + 4F(m-3 2 ) + . . . 



= $(m) + V(m). 



* The following are the memoirs of M. Kronecker on the application of the theory ot 

 elliptic functions to quadratic forms. 



(1) " Ueber elliptische Functionen und Zahlen-Theorie," Monatsberichte, Oct. 29, 1857 ; 

 and translated in Liouville, New Series, vol. iii. p. 265. 



(2) "Ueber die Anzahl der verschiedenen Klassen von quadratischen Formen von 

 negativer Determinante," Crelle, vol. lvii. p. 248 ; and translated in Liouville vol v 

 p. 289. 



(3) " Ueber eine neue Eigenschaft der quadratischen Formen von negativer Determi- 

 nante," Monatsberichte, May 26, 1862. 



(4) " Ueber die complexe Multiplication der elliptischer Functionen," Ibid,June26, 1862. 



(5) "Aunosung der Pellschen Gleichung mittelst elliptischer Functionen," Ibid Jan 

 22,1863. 



t M. Hermite, " Sur la theorie des equations Modulaires ; " M. Joubert, " Sur la Theorie 

 des Fonctions Elliptiques et son application a la Theorie des Nornbres," already cited in 

 the note on art. 125. 



