AND GENEEA OF TEENAEY QUADEATIC FOEMS. 259 
with the same signs as O and A. Similarly, the obelisk prefixed to a character of f or 
f-i n'-i 
F indicates that that character is attributable to/orF only when ( — 1) 2 =—( — 1) 2 
/- 1 A 1 — 1 
in the first case, and ( — 1) 2 =— ( — 1)» in the second. 
The use of the Table is most easily explained by an example. Let the proposed form 
be 
f=2x 2 +lf + 7z 2 ~2yz; 
its invariants are [2, 24], and its primitive contravariant is 
F=24tf 2 +7/+7s 2 +2^. 
F-l 
Since 0=2, mod 4, A=0, mod 8, F has the supplementary characters ( — 1) 2 and 
F 2 — 1 
( — 1) 8 ; the values of these characters are found by an inspection of the coefficients, 
f-i n'-i 
and are —1 and 4-1 respectively. Again, since Q'=l, ( — 1) 2 =—( — 1) 2 ; the 
character ( — l) ” 8 ” is therefore attributable to/, and an inspection of its coefficients 
/*-r 
shows that (—1) 8 = +l. 
The demonstration of the assertions implied in the Table (so far as they relate to 
supplementary characters) is obtained without difficulty from the equations (5) and (6). 
It will suffice to consider one case as an example of the rest. Let / and F be both 
properly primitive, and let 0=2Q' = 2, mod 4; A=0, mod 8. If M^F^,, y } , z,), 
M 2 =F(;r 2 , y 2 , z 2 ) are two uneven numbers represented by F, we infer from equation (6) 
( dF dF dF\ 
x 'dx Jr y x 7hj JrZl dz) * s an uneven num ber, and consequently that M,xM 2 =l, 
mod 8 ; M, and M 2 are therefore congruous to one another, mod 8 ; i. e. all uneven 
F-i 
numbers represented by F are congruous, mod 8, or F has the characters ( — 1) 2 and 
F 2 — 1 
( — 1) 8 . To prove that/* has the supplementary character attributed to it in the 
Table, we observe first of all that F cannot represent unevenly even numbers ; for, if 
possible, let F(ar„ y x , z,) be unevenly even, and let F(# 2 , y 2 , z 2 ) be any uneven number 
represented by F ; then in the equation (6) we have a square congruous, mod 8, to an 
unevenly even number, which is impossible. Now let m x =f{x x , y x , z,), m 2 =/(# 2 , y 2 , z.J 
( ' j r Jf jf \ 
Xl dr -^y'dy ~^ Zl d^) 
uneven in equation (5) ; and considering that equation as a congruence for the mo- 
n'-i F— i 
dulus 8, we find 2 =1, or m,Xw 2 =l-l-2x( — 1) 2 + 2 , according as 
f- i n'-i 
is evenly even, or uneven. If, then, ( — 1) 2 =( — 1) 2 , m x xra 2 is =1, or= 3, mod 8 ; 
i. e. the uneven numbers represented by/* are either all of one or other of the linear 
forms 8&-J-1, 8^+3, or else all of one or other of the linear forms 8#+5, 8^+7 ; so 
/-i, / 2 -! f-i n'-i 
that / has the supplementary character ( — 1) 2 8 . But if (—1) 2 =—( — 1) 2 , 
