ON THE THEORY OF NUMBERS. 249 



to suppose x=r throughout the whole demonstration, which is thus seen to 

 depend wholly on the properties of the same trigonometrical function \p, 

 which presents itself in the fourth demonstration ; only it will be observed 

 that here no necessity arises for the consideration of composite values of n in 

 the function \p (k, ?i) ; nor for the determination of the ambiguous sign in the 

 formula (A). In this specialized form, Gauss's sixth proof has been given by 

 Jacobi (in the 3rd edit, of Legendre's ' Thcorie des Nombres,' vol. ii. p. 391), 

 Eisenstein (Crelle, vol. xxviii. p. 41), and Cauehy (Bulletin de Ferussac, 

 Sept. 1829, and more fully Mem. dc l'lnstitut, vol. xviii. p. 451, note iv. of 

 the Memoire), quite independently of one another, but apparently without 

 its being at the time perceived by any of those eminent geometers that they 

 were closely following Gauss's method. (See Cauchy's Postscript at the end 

 of the notes to his Memoire; also a memoir by M. Lebesgue in Liouville, 

 vol. xii. p. 457; and a foot-noto by Jacobi, Crelle, vol. xxx. p. 172, with Eisen- 

 stein's reply to it, Crelle, vol. xxxv.p. 273.) 



MM. Lebesgue * and Eisenstein f have even exhibited a proof essentially 

 the same in a purely arithmetical form, from which the root of unity again 

 disappears, and is replaced by unity itself. Eisenstein considers the sum 



or un- 



C a = 2(— l J (— ) - * * ( -)> m which k v k 2 , .. ,h q denote q terms (equal 



equal) of a system of residues prime to^, the sign of summation extending 

 to every combination of the numbers k lt />'.,, . .k u , that satisfies the congruen- 

 tial condition £ 1 -J-ft a +A g +... +k q =• a, mod p. This sum is, in fact, 

 the coefficient of r a in the development of the ^th power of the function 

 k=p-\ 

 2 (~) r *> which is equivalent in value to Gauss's function ^ (l>p)- 



From the equation 2 {-) r k = ( — 1) p, it follows that 



2 ( _JrH = ( — 1) p x 2 (-W*; whence 



k=l \PJ J k=l \P/ 



C a =(-l)i(n-V(i-»(-\p~. And again, since ? (-)**] 



= 2 / - j r*t = ( - ) S (-) r*, mod q, we have the congruence 



C„ == ( - \{-£ V mod q. But these results, which, taken together, establish the 



law of reciprocity, arc obtained by Eisenstein from his arithmetical definition 

 of C u , without any reference to the trigonometrical function ^ (l,p). If we 



k=p— 1 k=p — \ /hX 



write that function in the form 2 r* 2 , instead of the form 2 ( - ) r*. 



£=0 k=\ \P/ 



we obtain from its qth power the coefficient C' a considered by M. Lc- 



* See Liouville's Journal, vol. ii. p. 253, and vol. iii. p. 113. (The proof of the law of re- 

 ciprocity will be found in sect. i. art. 5, and sect. iii. art. 2, of the memoir). See also the 

 memoir referred to in the text, Liouville, vol. xii. p. 457. 



t Crelle's Journal, vol. xxvii. p. 322. 



