ON THE THEORY OF NUMBERS, 243 



(-/)=(-*) » ( ii ) ) (|) = (-l) 8 >(-) 



P and Q denoting any two uneven numbers relatively prime, the signs of 

 which are subject to the same restrictions as the signs of p and q in the cor- 

 responding formula of Art. 16. The theorems expressed by these formulae 

 of Jacobi are very easily deducible from the formulae of Legendre, and will 

 be found in the Disq. Arith. (Art. 133). To prevent misconception, how- 

 ever, it is proper to observe, that while Legendre's equation ( — )=1 is a ne- 

 cessary and sufficient condition for the resolubility of the congruence x'^k, 

 mod p, Jacobi's equation f -1=1, where P is not a prime number, though 



a necessary, is not a sufficient condition for the resolubility of the correspond- 

 ing congruence r = ^, mod P. That congruence requires for its resolubility 



that the conditions | _ \= 1 , { _ )= 1 .... should separately be satisfied ; p Y 



p 2 . . . denoting the unequal prime factors of P. 



Gauss (who had in the course of his own early researches arrived inde- 

 pendently at the Law of Quadratic Reciprocity), before finally abandoning 

 the theory, succeeded in obtaining no fewer than six demonstrations of this 

 fundamental proposition. The first two are contained in the Disq. Arith. 

 (Art. 125-145, and Art. 262) ; the third and fourth in two memoirs pre- 

 sented in 1808 to the Society of Gottingen (Comm. Soc. Gdtt. vol. xvi. 

 p. 69, Jan. 15, and Comm. Recentiores, vol. i., Aug. 24), of which the 

 latter bears the title ' Summatio serierum quarundam singularium.' The 

 fifth and sixth appeared nine years later in the memoir entitled ' Theorematis 

 Fundamentals in doctrina de Residuis quadraticis demonstrationes et amplia- 

 tiones novae ' (Comm. Rec. vol. iv. p. 3, Feb. 10, 1817). The fourth of these 

 demonstrations is probably that which is promised in the Disq. Arith., Art. 151, 

 but which does not appear in that work, because (as it would seem) Gauss 

 had not yet succeeded in overcoming the difficulties connected with it. 



Independently of the fundamental importance of Legendre's Law of Reci- 

 procity, these demonstrations of Gauss possess such intrinsic interest, and 

 have contributed so much to the progress of the science, that we shall briefly 

 review them here. 



18. Gauss's First Demonstration. — The first demonstration (Disq. Arith. 

 Art. 125-145), which is presented by Gauss in a form very repulsive to any 

 but the most laborious students, has been resumed by Lejeune Dirichlet in 

 a memoir in Crelle's Journal (vol. xlvii. p. 139), and has been developed by 

 him with that luminous perspicuity by which his mathematical writings are 



distinguished 



1 



Let \ represent any uneven prime. The single observation that / - )= — 



\s) s ' 10ws * na ' * ne theorem of reciprocity is true for primes inferior 



to 7. To establish its universal truth, it is, consequently, sufficient to 

 show that, if true for all primes up to X exclusively, it is also true 

 for all primes up to \ inclusively. Let the theorem therefore be assumed 

 to be true for all primes inferior to \; let p be any one of those primes; 

 and let the eight cases [2x2x2 = 8] be considered separately, which 



arise from every possible combination of the hypotheses («) (P\= + l, or 



h2 



