250 report — 1859. 



besgue. This coefficient, which is connected with C a by the equation 

 C' a =/7? _1 + C a , represents the number of solutions of the congruence 

 x 1 2 + x 2 2 +x 3 2 + .. + Xq 2 = a, mod q. From this definition M. Lebesgue de- 

 duces the equation C' a = j /9?- 1 + (-l)* (p - 1)( *- 1) (-)P~, and the congru- 

 ence C' a = l+(-) (- )> mod q, by processes which, though different from 



those of Eisenstein, involve, like them, the consideration of integral numbers 

 only. 



22. Other proofs of the Theorem of Reciprocity have been suggested to 

 subsequent writers by a comparison of the different methods of Gauss. The 



symbol r denoting a root of the equation — — r-=0, it is very easily shown 

 that 



(r-r-i) 2 [t*-r-*y (r 2 -r 2 j=(-l) 2 P- (C) 



It is natural therefore to employ this equation to replace the equation 

 h=p-l /ZA "I 2 £=1 



methods of Gauss. It is also found that the product 



k=2 (P — I) r kq_~-kq 



n 

 fetal 



=( — 1) 2 p, which presents itself in the 4th and 6th 



»-i(f»-i) ,*_,-» ^ q m (D) 



fc=:1 ri—r-1 n \pj v ' 



This is an immediate consequence of the property of a half-system of Resi- 

 dues (see Art. 19 supra) on which Gauss's 3rd and 5th methods depend. 

 From a combination of the equations (C) and (D), the law of reciprocity is 

 immediately deducible. (See a note by M. Liouville, Compt. Rend. vol. xxiv., 

 or Liouville's Journal, vol. xii. p. 95, and especially a memoir by Eisenstein, 

 entitled " Application de l'Algebre a l'Arithmetique transcendante," Crelle, 

 vol. xxix. p. 177. The proof by the same author in vol. xxxv. p. 257, is the 

 same as that in the earlier memoir, only that the properties of the circular 

 functions, which here replace the roots of unity, are in the later memoir 

 deduced immediately from the definition of the sine as the product of an in- 

 finite number of factors.) 



23. Algorithm for the Determination of the Value of the Symbol (-gY — 

 Gauss has shown in the memoir " Demonstrationes et ampliationes novae," 

 already quoted, that, if p be a prime number, the value of the symbol I— ) 



may be obtained by developing the vulgar fraction — in a continued frac- 

 tion, and considering the evenness or unevenness of a certain function of the 

 quotients and remainders which present themselves in the development, 

 Jacobi has observed (see Crelle, vol. xxx. p. 173) that a much simpler rule 

 may be obtained, by the use of his extension of Legendre's symbol to the 

 case when p is not a prime. The following is the form in which the rule 

 has been exhibited by Eisenstein (see Crelle, vol. xxvii. p. 319). Let p^Pi 

 be two uneven numbers prime to one another, and let us form by division the 

 series of equations 



