496] 
FOE THE NEGATIVE DETERMINANTS &C. 
53 
deducible from the classes which belong to the determinant 
omit the non-primitive classes. 
it seems better to 
The two opposite forms included in a single expression by means of the sign + 
have opposite characteristics which are for the most part unequal to each other, for 
instance 
Det. — 44; (0, — 2, 1, 1) has the characteristic (6, — 2, 8), 
(0, 2, 1, -1) „ „ (6, 2, 8), 
where (6, — 2, 8), (6, 2, 8) are unequal forms, but this is not always the case, for 
instance 
Det. — 112; (1, —1, —3, —1) has the characteristic (8, —4, 16), 
(1, D - 3, 1) „ „ (8, 4, 16), 
where (8, —4, 16) = (8, 4, 16), since each is an ambiguous form. Instead of the two 
unequal forms (1, —1, —3, —1), (1, 1, —3, 1) which correspond to the opposite (though 
equal) characteristics (8, —4, 16), (8, 4, 16), M. Arndt might have given the two forms 
(1, 2, 0, —4) and (1, —1, —3, —1) corresponding to the same characteristic (8, 4, 16); 
but then it would not have appeared at a glance that the two classes were opposite 
to each other; and I presume that it is for this reason that he has selected the two 
representative forms (1, — 1, — 3, — 1) and (1, 1, — 3, 1). It must not, however, be 
imagined that the opposite cubic forms which correspond to opposite characteristics, 
which are ambiguous (and therefore equal to each other), are always, as in the last pre 
ceding example, unequal: for example Det. — 144, there is only the form (0, — 2, — 2, 3) 
given as corresponding to the ambiguous characteristic (8, 4, 20); the opposite form 
(0, 2, — 2, — 3) corresponding to the opposite but equal characteristic (8, — 4, 20) is 
equal to (0, — 2, — 2, 3), and so does not give rise to a distinct opposite class. In 
the new tables, the sign ± is only employed in regard to opposite ambiguous 
characteristics; for instance, Det. — 4 x 28 there are given (not included in a single 
expression by means of the sign ±) the two forms (1, —1, —3, 1), (1, 1, —3, 1) corre 
sponding to the characteristic 2 (2, +1, 4). 
I remark that, in a few instances M. Arndt, in passing from the second to the 
third column, has modified the expression for a cubic form in such manner that the 
characteristic has ceased to be a reduced form; for instance, Det. — 216, he has given 
in the third column the two forms (0, +3, 2, ± 1) belonging to the characteristic 
(18, +6, 14); it would have been better, it appears to me, to preserve the expres 
sion of the second column (1, -2, -3, 0), and adopt the two representative forms 
(1, +2, —3, 0); I have accordingly made this change. 
I divide M. Arndt’s table into two tables; the first of them corresponding to the 
determinants = 0 (mod. 4), the second to the determinants = 1 (mod. 4). In the first 
table I take for the characteristic the form 
{b I 2 -ac, l(bc-ad), bd-c 2 },