AND GENERA OF TERNARY QUADRATIC FORMS. 
289 
r, but not by r i+1 , and also render the quotients XZ - - y all quadratic residues, or all non- 
quadratic residues of r (Lemma iii. Cor. 2.). 
Lastly, let us consider the even prime 2, and let 2* be the highest power of 2 dividing O. 
Considering separately the eighteen cases of the Tables (i) and (ii), we find that of the 
231+9 systems of values of x, y, z, mod 2 i+3 , 
f X n x 2 3i+9 x systems 
(in which x, y , z are not simultaneously even, but x and z are or are not simultaneously 
ocz —■ 
even, according as the forms (/) are improperly or properly primitive) give to — ^ — 
an integral and uneven value, satisfying the supplementary character (if any) of ^ F, 
and, if the forms ( f ) are properly primitive, satisfying the congruence xz — y*= 1, mod 4, 
when O is uneven, and A uneven or unevenly even. 
For example, let 1, A=0, mod 8. Here F, or % F, has two supplementary 
characters, and of the 2 3i+9 systems of values of x, y, z, mod 2 i+3 , 
f X i X 2 3,+9 X ^7 systems, 
in which x and z are not simultaneously even, give to XZ 9 y~ an integral and uneven 
value, satisfying the supplementary characters of ^ F (Lemma iii. Cor. 3). 
Again, let £> 2, A=l, mod 2. Here F has or has not a supplementary character, 
A/+I 
according as ( — 1) 2 = — 1, or = + l. In the former case, of the 2 3{+9 systems of values 
of x , y , z , mod 2' +3 , 
f X i X 2 3i+9 X ^ systems 
• • CC Z ' 
(in which x and z are not simultaneously even) give to — an integral and uneven 
value satisfying the supplementary character of ^ F. In the latter case, of the same 
2 3i+9 systems of values, 
fxlx 2 3i+9 X^ systems, 
• • • XZ ~~~ 
in which x and z are not simultaneously even, give to — ^ — an integral and uneven value. 
Both results are comprised in the formula 
r A/+i~i y 
fXiL3 + (-l)~ J , X2 3 ‘ +9 X2 i . 
As a third example, let *=0, A — 0, mod 4, and let the forms considered be of an im- 
properly primitive order. Then OF^3, mod 4 ; and either OF=3, mod 8, or OF ^7, 
2 r 2 
