FOR THE NEGATIVE DETERMINANTS &C. 
55 
[496 
496] 
sses according to 
rms (a, 3b, 3c, d) 
when the terms 
same thing when 
lave no common 
may be of the 
; or it may be 
O') considered 
of the properly 
cases, the cubic 
)p, pp on /jl . ip, 
inner case ; only 
itive, the orders 
composition ; viz. 
form which by 
simple form of 
rms c, d, &c., of 
symbol of com- 
linant B'~ — A'C', 
. 1 denotes that 
’regular negative 
s divisible by 3. 
Table, but the 
the number of 
to the number 
', %C 2 ), Det. DB\ 
rm corresponding 
t in a letter of 
i, [162], viz. that 
for a pp form (A, B, C) of negative determinant, there is either no corresponding cubic 
form, or else a single corresponding cubic form, according as (A, B, G) does not, or 
does, produce by its triplication the principal form; but the particular theorem, in the 
cases to which it applies, is the more convenient one: it shows at once that for a 
regular negative determinant the number of cubic forms corresponding to a properly 
primitive characteristic (or, what is the same thing, number of cubic classes of the 
order (pp or ip) on pp) is 1 or 3, according as the number of quadratic classes is 
not, or is, divisible by 3. 
The inspection of the tables gives rise to other remarks, but at present I abstain 
from pursuing the subject further; I will only notice that in some instances, for 
example Det. — 224, the classes which correspond to characteristics of the principal 
genus are partly of the order pp on pp and partly of the order ip on pp. 
Table I. of the binary cubic forms, the determinants of which are the negative numbers 
= 0 (mod. 4) from —4 to — 400. 
Det. Order 
4 x 
Classes 
ou 
Charact. 
Cornpn. 
1 
o, 
-1, 
o, 
1 
PP 
pp 
1, 
0, 
1 
1 
2 
o, 
- b 
0, 
2 
PP 
pp 
1, 
0, 
2 
1 
3 
o, 
- b 
0, 
3 
ip 
pp 
1, 
0, 
3 
1 
4 
o, 
-1, 
0, 
4 
PP 
pp 
b 
0, 
4 
1 
1, 
- b 
- 
1, 1 
PP 
2pp 
2 
(b 
0, 1) 
2. 1 
5 
0, 
-1, 
0, 
5 
PP 
pp 
1, 
0, 
5 
1 
6 
0, 
-1, 
0, 
6 
ip 
pp 
b 
0, 
6 
1 
7 
0, 
-1, 
0, 
7 
PP 
pp 
1, 
0, 
7 
1 
1, 
0, - 
2, 
2 1 
- PP 
2 ip 
2 
(2, 
± b 4) 
2 or 
1, 
0, - 
2, 
— 2 
1 
8 
0, 
-1, 
o, 
8 
PP 
PP 
1, 
o, 
8 
1 
0, 
- 2 
**9 
o, 
1 
PP 
2pp 
2 
(b 
0, 2) 
2.1 
9 
0, 
-1, 
0, 
9 
ip 
PP 
1, 
o, 
9 
1 
10 
0, 
- 1, 
o, 
10 
PP 
pp 
b 
0, 
10 
1 
11 
0, 
-1, 
o, 
11 Ì 
| 
b 
0, 
11 
1 
0, 
_ o 
_ 
1, 1 
\PP 
PP 
3, 
b 
4 
a 
0, 
— 2 
u 9 
1, 
1 J 
1 
3, 
- 
b 4 
æ 
12 
0, 
-1, 
0, 
12 
ip 
pp 
1, 
0, 
12 
l 
13 
0, 
-1, 
0, 
13 
PP 
PP 
b 
0, 
13 
i 
14 
0, 
- b 
0, 
14 
PP 
PP 
1, 
0, 
14 
l