ON THE THEORY OF NUMBERS. 321 



primitive order from which it is derived. The subdivision into genera of 

 the order of properly primitive classes depends on the principles contained in 

 the following equations, in which q is an uneven prime dividing D, m and m} 

 uneven numbers prime to q, and capable of representation by the same 

 properly primitive form of determinant D. 



('•) (?)=(f ) 



m—\ m'-\ 



(ii.) If D = 3, mod*, (-l)~^=(-l)"^ 



(iii.) If D = 2, mods, (-1) » =(-l) « . 



(iv.) If D = 6, mods, (-1) ^ +~8"=(_i) 2 + s . 



m—\ m — 1 



(v.) IfD = 4., mods, (-1) 2 =(-l)~^. 



»j— 1 m'— 1 



(vi.) IfD = 0, mods, (-1)~^=(-1)"^; and 



(-1) 8 =(_1) 8 . 



The interpretation of these symbolic formulae is very simple. Thus, tlie 

 formula (i.) expresses that — 



" The numbers prime to any prime divisor y of D which can be represented 

 by/ the same properly primitive form of det. D are either all quadratic resi- 

 dues of 5-, or else all quadratic non-residues." 



Again, the formula (iv.) expresses that "if D be of the form 8n + 6, the 

 uneven numbers that can be represented by/ are either all included in one 

 of the two forms 8m+], 8m + 3, or else in one of the two 8n — l, 8n—3." 



All the formulae are deducible by the most elementary considerations from 

 the three equations 



m = ax- + 2bxy + cy', m' =ax^'- + 2bx'y' + cy^-, 



(aar + 2bxy + cy')(ax'- + 2bx'y' + cy'") = (axx' + blxy'+x'y^+ cyy')- 

 —I>(xy'—x'yy. 



= 6. 



Thus we find immediately (^)=1, or (^^f^y And again, if D 



mod 8, the last equation shows us that axx' + b^xy'+x'y^ +cyy' is uneven • 

 and consequently vim' ^^l~Q{xy' —x'y)", modS, i. e. ?wm'ss + 3, or ss + 1,' 

 mod S, according as xy'—x'y is uneven or even ; whence m and m' are either 

 both of the forms 8m+1, 8« + 3, or else both of the forms 8w— 1, 8n—3. 



The form / is said to have ihe particular character f£.\ = -\-\, or 



^— Jlf— 1> according as the numbers (prime to §') which are represented by 



it satisfy the equation (^) = 1, ovT-j^-l- and we are to understand in 



the same way the expressions that/has the particular character (— 1 ) T~— + j ^ 

 or = — 1, &c. Every particular character of a form belongs equally to all 

 forms of the same class, and is therefore termed a particular character of the 

 class. The complex of the particular characters of a form or class constitutes 

 its complete or generic character ; and those classes which have the same com- 

 plete character are considered to belong to the same genus : so that the 

 complete character of a form is possessed not only by every form of the same 

 class, but by every form of any class belonging to the same genus. 

 1861. Y 



