AND GENERA OE TERNARY QUADRATIC FORMS. 
263 
certain congruences for the modulus 4 or 8. The existence of the equivalent forms thus 
assumed results, in each case, from the theorem of the last article. 
Case (i) Let O = A = 1, and let <p and O satisfy the congruences 
or 
Ap =ajf -\-(3y 2 -\-yz 2 , 
0$ =j3yX 2 +a/3Y 2 +a/3Z 2 , 
mod 4. 
0<b=aX 2 +/3Y 2 -|-yZ 2 , 
a/3y=l, 
Attributing in succession to the indeterminates 
y, z 
X, Y, Z 
all systems of values, mod 2, which satisfy the congruence 
^X-f-yY+^Z^O, mod 2, 
and which render m and M simultaneously uneven, we find that in every case Am is 
congruous, for the modulus 4, to one of the numbers a, /3, y, and OM to one of the 
remaining two. Thus + 1 X ~ 1 is necessarily congruous, for the modulus 2, to 
one of the three numbers 
Q3 + l)(y+l) (y + !)(«+ 1) (« + l)(/3 + l) 
4 ’ 4 ’ 4 
But these numbers are all congruous to one another for the modulus 2, because the 
congruence ajSy=l, mod 4, implies the congruence a+/3+y+l=0, mod 4. Therefore 
the unit 'F has always the same value for every pair of uneven numbers simultaneously 
represented by f and F. 
It will be seen that T r = — 1, or ^F= +1, according as the congruences a=/3=y=l, 
mod 4, are or are not satisfied. 
Case (ii) Let 0 = 2; mod 4, A=l, mod 2, and let 
A<p =a < r 2 +2/3;y 2 +2yz 2 , mod 8, 
0'<l> = 2aX 2 -f-)3Y 2 +yZ 2 , mod 4, 
a /3y =1, mod 4. 
The admissible combinations of the values of x, y, z, X, Y, Z, mod 2, give rise to eight 
cases, 
Am = a, mod 8; O M = p, or y, mod 4, 
A?n = a+2j3, mod 8 ; O'M = — /3, or -\-y, mod 4, 
Am = a-|-2y, mod 8; O'M = (3, or — y, mod 4, 
Am = a+2/3+2y, mod 8; O'M —(3, or — y, mod 4, 
A' 2 7W 2 — 1 
and, in all of them, the value of the unit ( — 1) 5 '4 r , and therefore of the unit 
/ 2 -i 
( — 1) 8 'F, is the same, because, by virtue of the congruence a-f-(3+y+l = 0, mod 4, 
MDCCCLXVII. 2 0 
