ON THE THEORY OF NUMBERS. 525 
ment of the classes of the principal genus, and let Q,7=C,; it may then be 
shown that Q, will belong to a genus containing no ambiguous class, and that 
the formula |K|=|C| x (1+Q,) (1+A,)...(1+A,),im which A,,.. A,, are 
ambiguous classes, represents all the classes*, In general, if the principal 
genus contain 2 ambiguous classes (a supposition which implies that the 
determinant is irregular, having « even exponents of irregularity, and that 
there are only 24—-« genera containing ambiguous classes)—let Q,’=C, ; 
0,7=C,;...0,2=C,—it will be found that all the classes are represented by 
the formula |K!=|C| x (1+@,) (1+Q,) ..(1+Qc) (1+ Asi). . (1+A,), in 
which A,4;,...A, are ambiguous classes, and Q,, Q, . . . Qe classes belonging 
to genera containing no ambiguous class . 
A similar arrangement of the improperly primitive classes (when such 
classes exist) is easily obtained. Let 3% denote the principal class of im- 
: att D—1 
properly primitive forms, 7. ¢. the class containing the form (2, jie "5*); 
we have seen (Art. 113) that the number of properly primitive classes which, 
compounded with 3, produce 3, is either one or three. When there is only 
one such class, the number of improperly primitive classes is equal to that 
of properly primitive classes; and if |K| be the general formula representing 
the properly primitive classes, the improperly primitive classes will be repre- 
sented by 3x|K|. When there are three properly primitive classes, which, 
compounded with ¥, produce 3, the principal class will be one of them, and 
if @ be another of them, ¢” will be the third; also ¢ and q* will belong to 
the principal genus, and will appertain to the exponent 38. When the deter- 
minant is regular, instead of the complete period of classes of the principal 
genus, 1, C, C?,.. C"-1, we take the same series as far as the class 0" 
exclusively ; when the determinant is irregular, we can always choose the 
bases C,, C,, . . in such a manner that the period of one of them shall con- 
tain @ and ¢’, and this period we similarly reduce to its third part by stop- 
ping just before we come to ¢ or 9’. Employing these truncated periods, 
instead of the complete ones, in the general expression for the properly pri- 
mitive classes, we obtain an expression, which we shall call |K’|, representing 
a third part of the properly primitive classes, and such that = x |K’| represents 
all the improperly primitive classes. 
119. Tabulation of Quadratic Forms,—In Crelle’s Journal, vol. Lx. p. 357, 
Mr. Cayley has tabulated the classes of properly and improperly primitive 
forms for every positive and negative determinant (except positive squares) 
up to 100. The classes are represented by the simplest forms contained in 
them+; the generic character of each class, and, for positive determinants, 
the period of reduced forms (Art. 93) contained in it, are also given. The 
* Gauss employs a class Q, producing C, by its duplication, both when one and when 
two ambiguous classes are contained in the principal genus. The number of classes re- 
quisite for the construction of the complete system of classes is therefore in either case, 
since C, may be replaced by Q?,. 
+ The principles employed by Gauss for the arrangement of the classes of a regular 
determinant are extended in the text to irregular determinants. If the determinant have 
x! uneven exponents of irregularity, the number of classes requisite for the construction 
of the complete system of classes is x+-x’. 
+ The simplest form contained in a ciass is that form which has the least first coefli- 
cient of all forms contained in the class, and the least second coefficient of all forms con- 
tained in the class and having the least first coefficient. Ifa choice presents itself between 
two numbers differing only in sign, the positive number is preferred. In the case of an 
ambiguous class of a positive determinant, the simplest ambiguous form contained in the 
class is taken as its representative. 
