ON THE THEORY OF NUMBERS. 259 



of those symbols is + 1. By combining with these results the supplementary 

 equation ( - — ) =*'~ il " -i; , in which a-\-ia' denotes any primary uneven 

 number, and also the self-evident equations, 



c (a ± bi) = (ac + bd) ± bi(c ± di) 



a (c + di) = (ac -f bd) + di (a ± bi), 



(a+bi\ /a—bi\ _. 

 c+di)\c—di)~ 



Eisenstein* investigates a relation between the symbols \ ) and 



/c+dt\ . 



( , ) , which, when a-\-bi and c-\-di are primary, coincides with that e\' 



pressed by the law of reciprocity. 



81. The proof in Eisenstein's second memoirf is identical in its essential 

 character with that in the first; but he has given it a purely arithmetical 

 form, independent of the theory of the division of the circle. Instead of the 



It= P- l /k\ x P-\ 

 S— 2 ( — ) x k . in which a; is a root of the equation =0, 



sum 



'ST& 



he considers the powers of the series 3 ( — ) , and arrives by a process 



h = \ VV* 



purely arithmetical at the equations (A.) and (B.) of the preceding article. 

 Thus the two forms in which lie has exhibited his demonstration are pre- 

 cisely analogous to the two expressions which he has given to Gauss's sixth 

 demonstration of Legendre's law (see above, Art. 21). 



32. The proofs of the Law of Biquadratic Reciprocity, which are taken 

 from the theory of elliptic functions, no less than those which we have just 

 considered, depend in great measure on a generalization of the principles intro- 

 duced by Gauss into his demonstrations of Legendre's law. Indeed, Gauss 

 himself tells us;): that his object in multiplying demonstrations of Legendre's 

 law, was that he might at last discover principles equally applicable to the 

 Biquadratic Theorem. It would be interesting to know whether the proof 

 which he ultimately obtained of this theorem depended only on the division of 

 the circle, or on elliptic transcendents. Jacobi appears to have believed the 

 latter ; for he expresses his opinion that his own demonstration of the Biqua- 

 dratic Theorem was widely different from that of Gauss§; and he further 

 conjectures that what induced Gauss to introduce complex numbers, as 

 modules, into the theory of numbers, was not the study of any purely arith- 

 metical question, but that of the elliptic functions connected with the Lem- 



C dx , 



niscate Integral I— — 1|, This opinion of Jacobi's will not appear im- 



* See the memoir entitled " Lois de Reciprocite," in Crelle, vol. xxviii. pp. 53-67. 



f " Einfaclier Beweiss und Verallgemeinerung des Fundamental-Theorems fiir die biqua- 

 dratischen Reste," in Crelle, vol. xxviii. p. 223. 



t See the memoir, " Thcoiematis Fundamentals Demonstrationes et Ampliationes Novae,' 

 p. 4, " Hoc ipsum incitamentura erat ut demonstrationibus jam cognitis circa residua qua- 

 (iratica alias aliasque addere tantopere studerem, spe fultus, ut ex lmiltis raethodis divcrsis 

 una vel altera ad illustrandum argumentum affine aliquid couferre posset." 



§ " Ueber die Kreistheilung," Crelle, vol. xxx. p. 171. 



II Crelle, vol. xix. p. 314, or in the ' Monatsbericht ' of the Berlin Academy for Mav 10, 

 1839. 



s2 



