AND GENERA OF TERNARY QUADRATIC FORMS. 
279 
As, instead oif and F, we may consider any forms equivalent to f and F, we may sup- 
pose that/' and F satisfy, for any assigned powers of the uneven primes dividing OA, 
the congruences of Art. 5, 
F^fiyQ.Ax 2 -j- ay Ay 2 + a/3 z 2 , 
007 = 1 - 
The congruence F=0, mod S, is then satisfied by S 2 systems of values of x, y, z , mod S ; 
for z must be divisible by but x and y may have any values, mod & ; F is therefore 
prime to <5 for the remaining — 1) systems of values of x, y, 2, mod S. 
(2) If a is an uneven prime dividing O, but not A, F is prime to O, for u[u— 1) 
^ s y stems °f va l ues of %■> y i %■> mod 
For if F=0, mod at, x may have any value, mod &>, but y and z must have values satis- 
fying the congruence yAy 2 -\-fiz 2 =Q, mod a>. If ^ = — 1, the only values of y and 
z that satisfy this congruence are y= 0, 2=0, mod*/; and the congruence F=0, mod ft), 
is satisfied by a> systems of values of x , y, 2, mod u. If ^ = + the congruence 
yAy 2 -j-(3z 2 =0, mod ft), is satisfied by 2&>— 1 systems of values of y and 2; in this case 
therefore the congruence F=0, mod u, admits of u(2w— 1) solutions. And, observing 
that ^ ^ a S j — we find in both cases alike that F is prime to u for 
ft)(<y — I)^ft)— systems of values of x, y, 2, mod*). 
(3) It is evident from the congruence 
F=A.r 2 -(- A'y 2 + AV, mod 2, 
in which one at least of the numbers A, A', A" is uneven, that F acquires an uneven 
value for 4 out of the 8 systems of values, mod 2, which can be attributed to x, y, 2. 
(4) If OA is uneven, the number of solutions of the congruence OF=l, mod 4, is 
8(2— T). 
For this congruence may be written in the form (art. 6) 
ax 2 -f- (3y 2 + y2 2 = 1 , mod 4, 
a, (3, 7 representing uneven numbers which satisfy the congruence cc-\-(3-\-y-\-l=0, 
mod 4. Of the three numbers x , y, 2 one must be uneven, the other two even. The 
number of solutions in which x is uneven, y and 2 even, is 8 or 0, according asa= + 1, 
or = — 1, mod 4. The whole number of solutions is therefore 
12+4[(— lp>( — !)“+(— 1)^]- 
i. e. 24, or 8, according as the congruences a,=fi=y=l, mod 4, are, or are not satisfied; 
or again (Art. 6), according as ¥= — 1, or ¥=+1. The congruence OF=l, mod 4, 
admits, therefore of 8(2 — ¥) solutions. 
mdccclxvii. 2 Q 
