AND GENERA OE TERNARY QUADRATIC FORMS. 
257 
Art. 4. From the identical equations 
/On *1 )x/0 2 , y„ | 
=12F(y 1 a > — Z&—X&, x^—y^), 
FOn s,)xFO, y 2 , 2 2 )— 
= A/(y 1 z 2 -z,y 2 , Zfo—Xfo x x y 2 -y x x 2 ). 
O) 
0) 
we obtain the subdivision of the Orders into Genera. If u represent any uneven prime 
dividing O, § any uneven prime dividing A, these equations imply the theorems — 
I. The numbers, prime to a, which are represented by/ are either all quadratic 
residues of a>, or all non-quadratic residues of <w.” In the first case we attribute to /the 
particular generic character ^Q = + l, in the second we attribute toy the particular 
generic character ^-)== — 1- 
II. “ The numbers, prime to S, which are represented by F, are either all quadratic 
residues of ei, or all non-quadratic residues of S.” We attribute to F the particular generic 
character = in the first case, 
(?) 
= — 1 in the second. 
If O and A are both divisible by any uneven prime, / and F will both have parti- 
cular characters with respect to that prime. These theorems are due to Eisenstein. 
Besides its particular characters with respect to uneven primes dividing O, / if pro- 
perly primitive, will have in certain cases particular characters (which we shall call sup- 
plementary) with respect to 4 and 8. If the uneven numbers represented by /are all 
= 1, mod 4, we attribute to f the particular character (— l) _r = + l; if they are all 
= 3, mod 4, we attribute to /the particular character (— lp~~ = — 1. If they are all 
/*-> 
either =1, or =7, mod 8, we attribute to / the particular character (—1) 8 = + l ; 
if they are all either = 3, or = 5, mod 8, we attribute to / the particular character 
p - 1 
( — 1) 8 =— I. Lastly, if they are all either =1, or = 3, mod 8,/ has the character 
/- 1 , p - 1 
( — 1) 2 8 = +l; if they are all either =5, or == 7, mod 8, it has the character 
/-i . P-\ 
( — 1) 2 8 = — 1. Similarly, if F is properly primitive, it will, in certain cases, 
F — 1 F 2 -l 
acquire the characters ( — 1) 2 =+l, or = — 1 ; ( — 1) 8 = + 1, or = — 1 ; 
F-l F 2 — 1 
( — 1) * + 8 = + l, or =— 1. 
The following Table is useful for ascertaining the supplementary characters of any 
proposed form. 
