606 
NUMBERS. 
[795 
and these are said to be improperly primitive forms, or to belong to the improperly 
primitive order. The properly primitive forms are thus the six forms (1, 0, 35), 
(5, 0, 7), (3, ±1, 12), (4, ±1, 18); or we may say that there are represented hereby 
six properly primitive classes. Derived forms, or forms which belong to a derived 
order, present themselves in the case of a determinant D having a square factor or 
factors, and it is not necessary to consider them here. 
It is not proposed to give here the rules for the determination of the system 
of non-equivalent forms; it will be enough to state that this depends on the determ 
ination in the first instance of a system of reduced forms, that is, forms for which 
the coefficients a, b, c, taken positively satisfy certain numerical inequalities admitting 
only of a finite number of solutions. In the case of a negative determinant, the 
reduced forms are no two of them equivalent, and we thus have the required system 
of non-equivalent forms; in the case of a positive determinant, the reduced forms 
group themselves together in periods in such wise that the forms belonging to a 
period are equivalent to each other, and the required system of non-equivalent forms 
is obtained by selecting one form out of each such period. The principal difference in 
the theory of the two cases of a positive and a negative determinant consists in these 
periods; the system of non-equivalent forms once arrived at, the two theories are nearly 
identical. 
28. Characters of a form or class—division into genera. Attending only to the 
properly primitive forms: for instance, those mentioned above for the determinant 
— 35: the form (1, 0, 35) represents only numbers f which are residues of 5, and also 
residues of 7; we have, in fact, f=x iJ r 35t/ 2 , = x 1 (mod. 5), and also = # 2 (mod. 7). 
Using the Legendrian symbols and , we say that the form (1, 0, 35) has the 
characters 
Each of the other forms has in like manner a determinate 
character -f or — in regard to 
found that 
and also in regard to 
for each of them the characters are + + or else — — (that is, they are never + — 
or — +). We, in fact, have 
(1, 0, 35) + + 
(4, ± 1, 9) 
(5, 0. 7) 
(3, ± 1, 12) 
and we thus arrange the six forms into genera, viz. we have three forms belonging 
to the genus 
three 
characters 
-f + and of genera being one-half of all the combinations + +, , -I—, —K 
The like theory applies to any other negative or positive determinant; the several 
characters have reference in some cases not only to the odd prime factors of D but