520 REPORT—1862. 
produced by the duplication of X, K is also produced by the duplication of 
XxA,, XxA,,.. Xx Aj, but by the duplication of no other class. If, 
therefore, there be »’ classes in the principal genus which can be produced 
by duplication, the whole number of properly primitive classes is hxm’, 
i.e. hn'=kn. But n'Sn, therefore k<h. 
It may be inferred from Art. 112, vii., that all genera which contain any 
ambiguous classes contain an equal number of them. We shall immediately 
see that the number of ambiguous classes is equal to the number of genera, 
and is consequently a power of 2. The number of ambiguous classes in any 
genus is, therefore, either zero or a power of 2; and if any genus contain 
i 
: ; 5 onal 
2« ambiguous classes, such classes will exist only in pe Sonera. 
Gauss determines the number hk of properly primitive ambiguous classes 
by very elementary reasoning. He first finds the number of properly primitive 
ambiguous forms of one or other of the two types (A, 0, C) and (2B, B, C), 
and then assigns the number of non-equivalent classes in which these forms 
are contained. Let D be divisible by » different primes; and let us except 
the case D=—1. Resolving D in every possible manner into two positive or 
negative factors, having no common divisor but unity, we find 2#+! properly 
primitive forms of the type (A, 0, C); but we shall diminish this number by 
one-half by rejecting one of the two equivalent forms (A, 0, C) and (C, 0, A), 
viz. that in which [A|]>[C]. There are no properly primitive forms of the 
type (2B, B, C) unless D=3, mod. 4, or D=0, mod. 8; for one or other of 
these congruences is implied by the equation D=B (B—2C), because C is 
uneyen. Resolving D into any two factors relatively prime, if D=3, mod. 4, 
and haying 2 for their greatest common divisor, if D=0. mod. 8, we take one 
of them for B, the other for B—2C; and we obtain, in either case, 24+! pro- 
perly primitive forms of the type (2B, B, C). If BB'=—D,, it is easily seen 
that the forms (2B, B, C) and (2B’, B’, C')* are equivalent. We may thus 
diminish by one-half the number of forms of the type (2B, B, C), rejecting 
those in which [B]>[D]. We conclude, therefore, that if we now denote 
by p the number of wneven primes dividing D, we have in all 2++? ambiguous 
forms when D=0, mod. 8, 2 when D=1, or =5, mod. 8, and 24+! in every 
other case. These ambiguous forms we shall call Q, and we observe that 
their number is equal to the whole number of assignable generic characters 
Art. 98). 
To fn the number of non-equivalent classes in which these forms are 
contained, we consider separately the case of a positive and of a negative 
determinant. or a negative determinant, we diminish by one-half the 
number of the forms by rejecting the negative forms. The remaining forms, 
if of the type (A, 0, C), are evidently reduced, because A<C; if of the type 
(2B, B, C), they are also reduced, unless 2B>C, an inequality which implies 
that (C, C—B, C), to which (2B, B, C) is equivalent, is reduced (Art. 92). The 
number of [positive] ambiguous classes is, therefore, one-half the number of 
the ambiguous forms Q. 
For a positive determinant, we deduce from the forms © an equal number 
of reduced ambiguous forms. Thus (A, 0, C) is equivalent to (A, A, C’); 
and because [A]</D, this form is reduced, if /A be positive and be the 
* When the first we coefficients of a form are given, the third is given also; thus C’ 
2B’ 
sequel. The symbols [A] &c. are used, as in Art. 92, to denote the absolute values of the 
quantities enclosed within the brackets, 
is here used for @ - Similar abbreviations will be employed occasionally in the 
bd 
