256 Report — i859. 



satisfy the same congruence for either of the complex factors of Np, and are 

 therefore congruous to +z and — i, for one of those factors, and to — t and 

 +i for the other. Admitting this definition of the symbol f-\ , Gauss's 

 law of biquadratic reciprocity is expressed by the equation 



a and ft denoting two primary uneven primes, and A and B being their norms. 

 The complementary theorems relating to the unit i and the semi-even prime 

 1 + i are 



( ii.) („A\ d-»'M, (in.) (lp-\ ^((« + *~0W), 



v ' \a+ia'/i \a + ia'/i 



in which a + ia! denotes a primary uneven prime. These formulae, like those 

 of the last article, are susceptible of the same generalization which Jacobi 

 has applied to Legendre's symbol ; and we may suppose in the first that a 

 and ft are any two primary uneven numbers, prime to one another ; and in 

 the second and third that a + ia' is any primary uneven number. 



If, in the formula (i.) which expresses the law of reciprocity, a=a + ia', 

 ft=b + ib', it may be easily seen that the unit ( — 1)*(A- i)i(B-i) i s equal to 

 (_l)4(a-i)4(*-D. This gives us a second expression of the theorem. (See 

 Eisenstein, Math. Abhandl. p. 137, or Crelle, vol. xxx. p. 193.) 



Further, if we observe that every primary number is either =1, mod 4, or 

 else =3 + 2i, mod 4; and that i(A— 1)|(B— 1) and \(a— \)\{b— 1) are 

 even numbers, except both a and ft satisfy the latter congruence, we may 

 enunciate the law of biquadratic reciprocity by saying — 



" The biquadratic characters of two primary uneven prime numbers with 

 respect to one another are identical, if either of the primes be = 1, mod 4 ; but 

 if neither of them satisfy that congruence, the two biquadratic characters 

 are opposite." 



This theorem is only enunciated by Gauss, who never published his demon- 

 stration of it. " Non obstante," he observes, " summa huius theorematis sim- 

 plicitate ipsius demonstratio inter mysteria arithmetical sublimioris maxime 

 recondita referenda est, ita ut, saltern ut nunc res est, per subtilissimas tantum 

 modo investigationes enodari possit, quas limites prsesentis commentationis 

 longe transgrederentur." — Theor. Res. Biq. Art. 67. 



Soon after the publication of the theorem, its demonstration was obtained 

 by Jacobi, and communicated by him to his pupils in his lectures at Kb'nigs- 

 berg in the winter of 1836-37 (see his note to the Berlin Academy, already 

 cited in Art. 17). These lectures have unfortunately never been published; 

 but Jacobi's demonstration, from his criticism (see ibid.) on the first of those 

 given ten years later by Eisenstein, appears to have been very similar to it. 



It is to Eisenstein that we are indebted for the only published proofs of 

 the theorem in question. That great geometer (so early lost to arithmetical, 

 science — a victim, it is said, to his devotion to his favourite pursuit) has left 

 us as many as five demonstrations of it; the two earlier based on the theory 

 of the division of the circle; the three last, on that of the lemniscate. We 

 proceed to explain the principles on which each of these two classes of proofs 

 depends : — 



29. Biquadratic Residues — Researches of Eisenstein. — It is possible, as we 

 have seen, to obtain a proof of Legendre's law of Reciprocity by considera- 



