340 report — 1865. 



by the second, 2 is or is not a biquadratic residue of p according as ft is even 

 or uneven in the equation p = (4a + l) 2 -f 16/3 2 *. We infer therefore that 

 a 4- ft is even, or (since a 4- b + a is even by virtue of the congruence (4a + 1)" + 

 86 2 =(4a + l) 2 , mod 16) that a+ft + b is even. This result is thus gene- 

 ralized by Jacobi. 



"For any number P the sum S( — 1)', i. e., the excess of the number of 

 solutions of the equation P=(4a + l) 2 -[-86 2 in which 6 is even above the 

 number of solutions in which b is uneven, is equal to the sum S( — l) a+ ^, i. e. 

 to the excess of the number of solutions of the equation P= (4a + l) 2 + 16/3 2 , 

 in which a +/3 is even above the number of solutions in which a+/3 is uneven.'' 



The generalized theorem is expressed analytically by the equation 



\;(- l)"a (4m+lj2+8 " 2 =S(— l)' H+ "rt (4 '" +1 ' 2+1C " 2 , . . (2) 



in which the summations extend to all values of m and n from — oo to -f go . 

 But this equation is an elliptic formula ; for, on dividing by q, and writing 

 q for q s , it becomes 



S( _ 1)» ( f x 2f/"' 2 +"' = 2( - 1 )V»*2( - 1 )"' <f " 2 + ■ 



which is included in the equations (28) of art. 125, and is therefore a corol- 

 lary from the fundamental property of the Theta functions expressed in equa- 

 tion (7) of art. 124. We infer at the same time, from the equations (28), 

 that either of the sums 2( — iy»+>'r/ il "+ 1 >°+ lc " 2 or 2(-l)'y 4 '" +l/!+8 " 2 is equal to 



m=x 

 the infinite product 9; II (1 — ,/'")(l — ^ 1C "'). 



}« = 1 

 We thus arrive at an analytical proof of Jacobi's theorem, including, as a 

 particular case, a proof of the identity of Gauss's two criteria. But the con- 

 tinuation of Jacobi's memoir was intended to contain direct arithmetical 

 demonstrations (which, however, have never been published) of the theorems 

 of which the equation 2( — 1) 4 = S( — l) a+p is an example. He says, 

 " Though these arithmetical demonstrations of results obtained analytically 

 present no essential difficulty, yet they are sometimes of a complicated cha- 

 racter, and require peculiar classifications of numbers which perhaps may be 

 of use in other researches. We have here a certain amount of freedom in the 

 choice of methods, so that the proofs can easily be varied" f. Probably one 

 of these methods was that employed by Dirichlet in his earliest arithmetical 

 memoir, to which Jacobi expressly refers. In this memoir J (written when 

 only the enunciations of Gauss's criteria for the biquadratic character of 2 had 

 been published) Dirichlet gives a demonstration of the first criterion, which 



* Theoria Residuorum Biquadraticorum, arts. 13-21. To the second criterion we have 

 already referred in this Report (art. 24, and in the additions to Part I., printed at the end 

 of Part II.) ; the first is more elementary, and is inferred from the equation p — (4a 4- 1) 2 + 

 8£ 2 ,iu which p is a prime of the form 8/c+l. Raising each side of the congruence — 84 2 = 



, Ezl En! 



(4«+l) 2 , modp, to the power ^j-, and observing that 2 2 = (~J = 1, ( — 1) 4 =1, 



we fiBd iVgJ-ptfcl). Butifi=2V A where/3 is uneven, g) = (|)=g)=l, 



because ^ ( 4«+l) 2 , mod (S ; and ( 4 -^) = (^) =(t^) - (^-(-l* 

 y-' 



Hence 2 4 =( — 1)", modp, which is Gauss's first criterion. 

 t Mathematische Werke, vol. ii. p. 73. 

 t Crelle's Journal, vol. iii. p. 35. 



