ON THE THEORY OF NUMBERS. 257 



k=P ~ 1 /k\ 

 tions relating to the function 2 ( - la?*, p denoting a real prime, and 



k = l W 



x p i 



x a root of the equation -=0. This function is a particular case of the 



well-known function (introduced by Gauss and Lagrange into the theory of 



s=p— 2 

 the division of the circle) F(0, x) = 2 fl'a;/ , where is any root of the 



s=0 



Qp-i 1 



equation — - — - — = 0, y a, primitive root of the congruence aM^sI, mod j», 



and x a root of the equation — — — =0. In the quadratic theory we assign 



to the value — 1 ; in the theory of Biquadratic Residues we put 6=i, and 

 are thus led to consider another particular form of the same function, viz. 



s—p — 2 

 t(i,x) = 2 i*xv ,p denoting a prime of the form 4w + l. 



5 = 



30. The function ¥(d,x) or F(0) is characterized by the following general 

 properties; which have been given by Jacobi, Cauchy, and Eisenstein. (See 

 Jacobi, Crelle, vol. xxx. p. 166; Cauchy, Memoire sur la Theorie des Nom- 

 bres in the Mem. de l'Acad. de l'lnstitut de France, vol. xviii. ; Eisenstein, 

 Crelle, vol. xxvii. p. 269.) 



I. F(d,x k ) = d- I * d y"F(d,x), 



II. F(0)F(0->) = 0^>, 

 nL F(0-™)F(0-») 



F(0-C"'+")) r\"j> 



where i/,(0) does not involve x, and is an integral function of with integral 

 coefficients*. The function \(j(d) satisfies the equation 



IV. ^(0)^(0-i)=p. 



Lastly, let be a. primitive root of? — =0, and in the function 



x—1 



M8)= F(8-)F(fl-) 

 F(0-'»- re ) 

 let y be written for 0; then if m and n be positive and less than^a— 1, 



Urn . l\n 

 rim denoting the continued product 1 .2 . 8 . . . m. 



Applying these equations to the particular form of the function F which 

 we have to consider here, we find 

 p—\ 



¥(i)F(-i)=i 2 p , iK0= p/_jffi> if 6to-» = i, and ?n=n=i(p-l). 



[F(0]'=i4(0 2 , 



[F(-«)3 4 =^(-0 9 ^(0^(-«)=p. 



^( y «P-D)=0, mod/>. 

 * In this equation Q~ m and 0-" are supposed not to be reciprocals. 



