1864.] Prof. Smith on Complex Binarij Qtiadratic Forms, 279 

 or, as we shall write it. 



Thus, supposing that^ is an uneven prime dividing D, and that m and 

 7)1 are prime to p, the numbers prime to p, which are represented by/, are 

 either all quadratic residues of ^, or else all non-quadratic residues of^;; 



in the former case we attribute to /the character — 1, in the latter 



the character pj = — 1 . 



Again, to investigate the supplementary characters relating to powers 

 of the even prime l+i, let 7n — fi-\-ifi be an uneven number, /x and ^' 

 representing real numbers, and for brevity, let 



(-if =y. 



The values of the units, or characters, a, /3, y depend on the residue of 

 m for the modulus (1 as is shown in the follov/iug Table. 



Table I. 





a = 



/3 = 



7 = 



+ 1 



+ 1 



+ 1 



+ 1 



+ z 



+ 1 



+ 1 



-1 



+ 3 



+ 1 



— 1 



+ 1 



+ 3e 



+ 1 



— 1 



— 1 



±(l-2e) .. 



-1 



+ 1 



+ 1 



±(2 + 0.... 



— 1 



+ 1 



-1 



±(1+20 



-1 



-1 



+ 1 



±(2-0.... 



— 1 



-1 



-1 



An inspection of the Table shows that, of the sixteen uneven residues of 

 (1+0^ eight have the character w=l, and eight the character w = — 1, 

 w representing any one of the seven characters a, /3, y, /3y, ay, a/3, a/3y. 

 It will also be seen that any character of a product of two uneven factors 



