126 REPORT— 1860. 



-,y° +S +f + +^- l) \ 



Vo =<£' +a< +ei' + +a? y 



7?, =a / +a / +a / + +a v , 



e-I 2e_l Se- 1 /e-1 



(y still denoting a primitive root of X), are the roots of an irreducible equa- 

 tion of order e having integral coefficients, which we shall symbolize by 



F( I/ )=f+Ay- l + A o y- 2 +...A e _ l y + A=0 



(see Disq. Arith. art. 346). This equation is of the kind called Abelian ; 

 that is to say, each of the e periods is a rational function of any other, in 

 such a manner that we may establish the equations »> t = ^(»/ )> »; 2 =0(»; 1 ), 

 V3—<p(y 2 )> •••• r h = ( p( r le-\) ; where it is to be observed that the coefficients 

 of the function are not in general integral. The determination of the 

 coefficients of the equation F(3/)=0 may be effected, for any given prime X, 

 and any given divisor e of X — I, by methods which, however tedious, present 

 no theoretical difficulty. Every rational and integral function of the periods 

 can be reduced to the form a t] -\-a l r] l +a i ri 2 + •• -\-tz e _ 1 t) e _ v If we com- 

 bine the equation 1 +r) + y 1 + T) 2 + +rj e _ 1 =0 with the e—2 equations, 



by which t?^ ,^, .... tj^ -1 are expressed in that linear form, we may elimi- 

 nate t) 2 , 7? 3 , . . . J?e_i, and shall thus obtain an equation of order e, satisfied by 

 r? , i. e. the equation of the periods, or F(y)=0. This is the method proposed 

 by Gauss (Disq. Arith. art. 346) ; M. Kumruer, instead, forms the system of 

 equations 



Vl =« /+(0,0)r, + (0,l)r, 1 + (O f 2)^+...+(0,e-l)r, e _ 1) 



Vh =» l /+(l,0> +(l,l)i, 1 +(l,2)n ll +...+(l,e-l)ji e _„ 

 Vol* =« 2 /+(2,0>; + (2,l) J?1 +(2,2> 2 + ...+(2,e-l)r, e -» 



V %-i = n e-if+(e—l,0)r) + (e—l, l)j»,+(e--l,2)>7 8 + ... +(e- 1, e— 1 )/?<,_,, 



and eliminates r?,, ti 2 , . . . Tj e _i from them. The symbol (k, h) represents the 

 number of solutions of the congruence y e ff+*=l + y e *+*, mod X, x and y 

 denoting any two terms of a complete system of residues for the modulus/; 

 rib is zero for all values of k, excepting that » =1> if/ be even, and ?>$ e =l, 

 iff be uneven *. The systems of equations con esponding to the particular 

 cases e=3, e=4, have been given by Gauss, who has succeeded in expressing 

 the values of the coefficients (k, li) in each of those cases by means of num- 

 bers depending on the representation of X by certain simple quadratic forms; 

 and has employed these expressions to demonstrate the criterion already men- 

 tioned in this Report for the biquadratic character of the number 2 f. A 

 third method has been given by M. Libri J : he establishes the formula 



\N*=\*+ t, (l +ev c Y + r h {l+e, h ) l + . . . ^_,(1 +«*-,)*, 



in which N& represents the number of solutions of the congruence 



* Liouville's Journal, vol. xvi. p. 404. 



t Disq. Arith. art. 358, and Theor. Res. Biq. arts. 14-22. 



X See the memoir " Sur la Theorie des Nombres," in his ' Memoires de Mathematique et 

 de Physique,' pp. 121, 122. The notation of the memoir has been altered in the text. See 

 also M. Lebesgue, in Liouville's Journal, vol. ii. p. 287, and vol. iii. p. 113. 



