496] 
51 
496. 
TABLES OF THE BINARY CUBIC FORMS FOR THE NEGATIVE 
DETERMINANTS, =0 (MOD. 4), FROM -4 to -400; AND =1 
(MOD. 4), FROM -3 TO -99; AND FOR FIVE IRREGULAR 
NEGATIVE DETERMINANTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 246—261.] 
The theory of binary cubic forms for determinants, as well positive as negative, 
has been studied by M. Arndt in the memoir “ Versuch einer Theorie der homogenen 
Functionen des dritten Grades mit zwei Variabeln,” Grunert’s Archiv, t. xvii. (1851, 
pp. 1—54) ; and in the later memoir, “ Tabellarische Berechnung der reducirten binaren 
cubischen Formen und Klassification derselben fur aile negativen Determinanten (— D) 
von D = 3 bis D = 2000,” ditto, t. xxxi. (1858), pp. 335—445, he has given a very 
valuable Table of the forms for a Negative Determinant. It has appeared to me 
suitable to arrange this Table in the manner made use of for Quadratic Forms in 
my memoir “ Tables des formes quadratiques binaires pour les déterminants négatifs 
D = — 1 jusqu’à D = — 100, pour les déterminants positifs non carrés depuis D = 2 
jusqu’à D = 99, et pour les treize déterminants négatifs du premier millier, Grelle, 
t. LX. (1862), pp. 357—372, [335]; and confining myself to the limits of the last- 
mentioned tables I deduce from that of M. Arndt the three Tables which follow. 
To 
explain the arrangement, I give in the 
first instance the following extract from 
M. Arndt’s Table: 
D. 
Eeducirte Formen mit Charakteristik. 
Klassen. 
3 
(0, 1, 1, 0) (1, 0, -1, - 1) (1, 1, 0, -1) 
(2, 1, 2) (2, 1, 2) (2, 1, 2) 
| (0, 1, 1, 0), (1, o, - 1, ± 1) 
4 
(0, 1, 0, - 1) (1, 0, -1, 0) 
] (0, - 1, 0, 1) 
n o 
(2, 0, 2) (2, 0, 2) 
7—2