244 report — 1859. 



= — 1; (/3) \ = 1, or = 3, mod 4; (y)p = 1, or = 3, mod 4. It has to be 

 shown that, in each of these eight cases, the symbol (- jactu ally has the value 

 which the Law of Reciprocity assigns to it. The nature of the proof in the 

 four cases in which { " 1=4-1, will be rendered intelligible by a single 

 example. 



Let (- )=1 and let \=^p= 1, mod 4. By virtue of the symbolic equa- 

 tion (£\=l, we can establish the congruence * 2 ss/>,mod \, or (which is the 



same thing) the equation x~=p + \y ; in which we may suppose x even and 

 less than X, y positive, less than X and of the form 4w + 3. From this equa- 

 tion it appears that ( -^ )=1, and ( - )=1» the symbol \y\ being here used 



with the meaning Jacobi has assigned to it. But every prime divisor of y is 

 less than X; and, therefore, by Jacobi's formula of reciprocity (which is 

 valid for all uneven numbers less than X, since by hypothesis Legendre's 



law is valid for all primes less than X), (t\ = (-)=!■ But (— ) =1 = 



fh\ (y\ ; so that, finally, (- j=l in conformity with Legendre's law. We 



have here assumed that x is prime to p ; a slight modification in the proof 

 will adapt it to the contrary supposition. 



Again, the two cases in which (P )= — 1, and X = 3, mod 4, admit of simi- 

 lar treatment. For the equation {£ J= — 1 involves also the equation {—+- \ 



= 4- 1 , because X = 3, mod 4. We have therefore the congruence x 2 = — p, 

 mod X, which will serve to replace the congruence x 2 =p, mod A, which pre- 

 sents itself in the four cases first mentioned. 



But the two remaining cases, in which (^ )= — 1, X= 1, mod 4, require 



a different mode of treatment. By a singularly profound analysis, Gauss has 

 succeeded in showing that every prime of the form 4m + 1 is a non-quadratic 

 residue of some prime less than itself. Assume, therefore, the existence of a 



prime -a, less than X, and satisfying the condition / — )= — !. This condition 

 implies that (—)= — 1 ; for if ( — J were equal to + 1, we should have f — j 



= 4-1, by one of the first four cases. Hence we may infer thatf-i-j 

 = 4-1, and may establish the congruence x° = -a p,mod X, which, treated as 

 in the preceding cases, will lead us to the conclusion that (-) ( — ) =1, i.e. 



jte £)=-i. 



