ON THE THEORY OF NUMBERS. 521 
greatest multiple of [A] not surpassing ¥D. Similarly (2B, (24+1) B,C’) 
is equivalent to (2B, B, C), and is reduced if (2/41) B be positive, and be 
the greatest uneven multiple of [B] not surpassing D. There are, there- 
fore, as many reduced ambiguous forms as there are forms in Q; and there 
are no more, because it is readily seen that every reduced ambiguous form 
is included in one or other of the two series of forms (A, “A, C’) and 
(2B, (2k+1) B, C’) which we have obtained. But every ambiguous class 
contains two reduced ambiguous forms (Art. 94); we infer, therefore, that for 
positive as well as for negative determinants the number of ambiguous classes 
is one-half the number of the forms Q, 7. ¢. one-half of the number of assign- 
able generic characters. 
Combining this result with the theorem at the commencement of this 
article, we obtain a proof of the impossibility of at least one-half of the 
assignable generic characters. As this proof is independent of the law of 
quadratic reciprocity, we may employ the result to demonstrate that law. 
[Gauss’s second demonstration, Disq. Arith., art. 262.] Let p and q be 
two primes, and first let one of them, as p, be of the form 4n+1. If (2) 
P 
=—1, we infer that (2)=-1; for if (2)=+1, we should have w’=p, 
2 
mod. g, and consequently there would exist a form (a w,— 7 ) of det. p, 
of which the character would be (£)= —l, 2. ¢. there would be 2 genera of 
P 
forms of determinant p. Similarly, if (4)= +1, we have w*=+q, mod. p; 
ee 
and (p, w, © +4) is a form of det. +g. If +¢q be of the form 4n+1, 
there will be but one genus of forms, z.e, the principal genus; whence 
(£)= +1. These two conclusions are sufficient to establish the theorem of 
reciprocity when one of the two primes is of the form 4n+1. If both 
pand q be of the form 4n+3, there are four assignable characters for the 
determinant pg. Of these (£)=1, (Z)=1 : (2)= —l, (-)= —1; are pos- 
is q qY 
sible, as is shown by the existence of the forms (1, 0, —pq), (—1, 0, pq); 
the other two are therefore impossible. Hence in the form (p, 0, —q) we 
must have either (2)=1=(=*), or (2)=—1=(=), which ex- 
q P P 
presses the theorem of reciprocity for this case. The supplementary theo- 
rems relating to 2 and —1 can be similarly proved. 
116. Equality of the Number of Genera and of Ambiguous Classes — 
In the preceding article it has only been shown that & cannot exceed h. 
But, as we have already seen (Art. 102) that the number of actually 
existing genera is one-half the whole number of assignable generic 
characters, we know that kK=h. To prove this, by the principles of the 
composition of forms, it is sufficient to show that n=n’', 7. e. that the 
problem “ to find a class which by its duplication shall produce a given class 
of the principal genus ” is always resoluble. This problem Gauss actually 
solves (Disq. Arith., art. 286, 287); he shows, first, that any proposed 
binary form, belonging to the principal genus of its own determinant, can be 
