AND GENERA OF TERNARY QUADRATIC FORMS. 
265 
Let ^=+1 ; then if m and M are uneven numbers simultaneously represented by f 
and F, one at least of the two congruences m= — A, mod 4, M= — £2, mod 4, must be 
satisfied. Subject to this restriction, m and M may have any of the four linear forms 
8^-j-l, 3, 5, 7. 
Case (ii) The restrictions imposed on the numbers m and M by the simultaneous 
characters are exhibited in the annexed Table. 
If 
M= O', mod 4. 
m=5A, 7A, mod 8. 
m= A, 3A, mod 8. 
M=30', mod 4. 
m= A, 7 A, mod 8. 
m= 3A, 5A, mod 8. 
Except when O and A are both uneven it will be found that, in the case of any 
two properly primitive forms f and F, every representation of an uneven number by 
either of the two is simultaneous with the representation of uneven numbers by the 
f 1 - 1 A- — 1 
other. If therefore (— 1) 8 T f = ( — 1) 8 ,/ cannot represent numbers congruous to 3 A, 
mod 8, because it cannot represent them simultaneously with uneven numbers, and if 
/ 2 -l A 2 — 1 
( — 1) 8 ' V F= — ( — 1) 8 ,/ cannot represent numbers congruous to 7 A, mod 8. 
Case (iii) In this case, which is the reciprocal of the last, we have the Table, 
If 
f 2 — i n 2 -i 
(_1)— . 
m= A', mod 4. 
M=50, 70, mod 8. 
M= 0, 30, mod 8. 
m== 3A', mod 4. 
M= O, 70, mod 8. 
M=30, 50, mod 8. 
And F cannot represent numbers congruous to 30, or cannot represent numbers con- 
f 2 — i Q 2 -i n 2 -i 
gruous to 70, according as ( — 1) 8 T=( — 1) 8 , or — ( — 1) 8 . 
Case (iv) In this case both f and F represent numbers of all the four linear forms 
8^+1, 3, 5, 7. The Table, in which the modulus is everywhere 8, exhibits the restric- 
tions imposed by the simultaneous character. 
If 
p- 1 f 2 -i a 72 — i n /2 — i 
(_1)~ + -8-^ — (_1)^- + — 
fl-\ A /2 — 1 n ,2 -l 
(_!)— + -F- ^- = _(_1)— + — 
m= A' 
M=50', 70' 
M= O', 30' 
m=3A' 
M=30', 50' 
M = O', 70' 
5 A' 
M- O', 30' 
M=50 , 70' 
171=7 A' 
M- O', 70' 
M=30', 50' 
2 o 2 
