ON THE THEORY OF NUMBERS. l7l 



Art. 20. Dirichlet's demonstration of the formulae (A) and (A') first 

 appeared in Crelle's Journal, vol. xvii. p. 57. Some observations in this 

 paper on a supposed proof of the same formulae by M. Libri (Crelle, vol. ix. 

 p. 187) were inserted by M. Liouville in his Journal, vol. iii. p. 3, and gave 

 rise to a controversy (in the Comptes Rendus, vol. x.) between MM. Liouville 

 and Libri. The concluding paragraphs of Dirichlet's paper contain the appli- 

 cation of the formula? (A) and (A') to the law of reciprocity (Gauss's fourth 

 demonstration). 



Art. 22. From a general theorem of M. Kummer's (see arts. 43, 44 of this 



Report), it appears that the congruence r 2 = ( — 1) 2 X, mod q, is or is not 



A— 1 



resoluble, according as q 2 =+1, or = — 1, mod X, — a result which implies 

 the theorem of quadratic reciprocity. This very simple demonstration (which 

 is, however, only a transformation of Gauss's sixth) appears first to have 

 occurred to M. Liouville (see a note by M. Lebesgue in the Comptes Rendus, 

 vol. li. pp. 12, 13). 



Art. 24. A note of Dirichlet's, in Crelle, vol. lvii. p. 187, contains an ele- 

 mentary demonstration of Gauss's criterion for the biquadratic character of 

 2. From the eauation j9=a 2 + 6 2 , we have (a + 6) 2 =2a&, mod p, and hence 

 (a + i)4 ( P- 1) = 2i^- 1 5ai(P- 1 )6iCp-i) = (2/')i(/'-i)a4(p-i),or, which is the 

 same thing, 



(^)-W"-"(f) (A) 



But ( — | = |- |=L because p=b 2 , mod a; and ( ^t_ ) = ( -£— \ or, ob- 



\p) W V p / \ a + b / 



serving that 2p=(a+by + (a— b) 2 , 



since/ 2 + 1=0, mod p. Substituting these values in the equation (A), we 

 find H^p-V =fb ab , mod/), which is in fact Gauss's criterion. 



Art. 25. In the second definition of a primary number, for "b is uneven," 

 read " b is even." Although this definition has been adopted by Dirichlet in 

 his memoir in Crelle's Journal, vol. xxiv. (see p. 301), yet, in the memoir 

 " Untersuchungen iiber die complexen Zahlen " (see the Berlin Memoirs for 

 1841), sect. 1, he has preferred to follow Gauss. 



Art. 36. In the algorithm given in the text, the remainders^, p 3 ... are all 

 uneven ; and the computation of the value of the symbol ( o ) is thus rendered 



independent of the formula (iii) of art. 28. The algorithm given by Eisen- 

 stein is, however, preferable, although the rule to which it leads cannot be 

 expressed with the same conciseness, because the continued fraction equi- 

 valent to -£-£ terminates more rapidly when the remainders are the least 

 possible, and not necessarily uneven. 



Art. 37. In the definition of a primary number, for "« = + ]," read 

 " a= — 1." But, for the purposes of the theory of cubic residues, it is 

 simpler to consider the two numbers +(a+bp) as both alike primary (see 

 arts. 52 and 57). 



Art. 38. Jacobi's two theorems cannot properly be said to involve the 



D+4a», 



