ON THE THEORY OF NUMBERS. 247 



has been given by Dirichlet in his memoir, " Application de l'Analyse In- 

 finitesimale a la Theorie des Nombres" (Crelle, vol. xxi. p. 135). 



The same formulae have also been deduced by Cauchy from the equation 



Or ! + e -a 2 +e-4« 2 + e -9« 2 +..=— (i + e -4= +e -44 2 + e -9J 3 + ..), 



in which ab=&, a and b denoting real positive quantities, or imaginary 

 quantities the real parts of which are positive. This equation Cauchy 

 obtained, as early as 1817, by the principles of his theory of reciprocal 

 functions; but it is also deducible from known elliptic formulas. (See a 

 note by M. Lebesgue in Liouville's Journal, vol. v. p. 186.) If in it we write 



o 2 — — ■ for a 2 , and /3 2 H for b 2 , a and /3 being two evanescent quantities 



n 2 



connected by the relation «a=2/3, the two series 



wa(i + e- a2 + e-"« 2 + c -9* 2 + ...) 



and 2/3(i + e- j2 +e-"* 2 + e -9* 2 +...) 



become respectively \|/ (1, n) X I c~* dx, and (1 + e 2 ) x I «-*'" dx\ 



whence, dividing by the definite integral, and observing that a=\ / — e 4 , 



V n 



we obtain finally, in accordance with the formulas of Gauss, 



^ (1, n ) = \ */n(\ + 0(1+ e~^)*. 



For the case in which n is a prime number, the equality (B) has been 

 established in a very simple manner by M. Cauchy f and M. KroneckerJ. 

 But, as these latter methods have not been extended to the case in which n 

 is a composite number, they cannot be used to replace Gauss's analysis 

 in this demonstration of the law of reciprocity. 



From the formula (A) combined with the equation \p (k,p)=( -J ^ (l>p), 

 p denoting a prime number, we may infer 



-|vb=2 cos* 2 sin* U; 



s =0 P s=0 P 



° r \v) V P=2 sin* 2 —-; 2 cos* 2 — - =0, 



^P / *=0 P *=0 p 



according asj9 = l, or = 3, mod 4. 



These formulas serve to express the value of the symbol ( - \ by means of 

 a finite trigonometrical series, and are, therefore, of very great importance. 



* See M. Cauchy's Memoire sur la Theorie des Nombres in the Memoires de l'Academie 

 de France, vol. xvii, notes ix. x. and xi. See also the Comptes Bendus for April 1840, or 

 Liouville's Journal, vol. v. p. 154 ; and compare (beside the note of M. Lebesgue quoted in 

 the text) a memoir by the same author in Liouville, vol. v. p. 42. 



t In the Memoire sur la Theorie des Nombres, Note xi., or Liouville, vol. v. p. 161. 



j Liouville, New Series, vol. i. p. 392. 



(?) 



