730 REPORT—1863. 
minant —A*, If is the number of classes in the principal genus of impro- 
perly primitive forms of determinant —A, it follows from the theory of 
composition of quadratic forms that 2p" can always be represented primitively 
a , and that 
the exponent of the lowest power of p which is capable of such representa- 
tion is either f or a submultiple of A. Again, the equation 
sb— Za 
4p “ =2°+Ay’, if we write in it 2X+Y for w, and Y for y, becomes 
Sb— Za A +] 
7 hee a eg “e) (X, Y)’, the values of X and Y being integral. 
Assuming, then, that there exist primes of the lear form nA +1, the doubles 
of which are capable of representation by a class appertaining to the exponent 
h (an assumption which implies that —A is not an irregular determinant, 
at least in respect of its improperly primitive classes), we see that in the case 
in which A is a prime of the form 4n+3, and in which therefore there is but 
— za 
by the principal form in that genus, ¢. e. by the form (2, 1, + 
: pte os 
one genus of improperly primitive forms, 
must be equal either to the 
number of improperly primitive classes, or toa multiple of that number ; 
and as Jacobi found, upon a sufficient induction, that 2 was always equal to 
Brae) he did not scruple to enunciate the theorem as true. We know, 
however, from an account which Dirichlet has given of a communication 
made to him by Jacobi, that Jacobi never obtained a demonstration of the 
theorem; and, indeed, it would seem probable, as has been observed by Di- 
richlet, that its demonstration requires other principles (Crelle, vol. li. p, 206). 
It is hardly necessary to add that when there is more than one genus of 
forms of determinant —A, 7. ¢. In every case except when A is a prime of 
the form 4n+3, the exponent of p in the formule (A) and (B) is always a 
multiple of the least exponent for which those formule can be satisfied. 
122. Extension of the preceding Theory by Hisenstein.—In the theory of 
which an account has been given in the last article, the prime number p is 
throughout supposed of the linear form nA+1 or 4n4+1; thus in the equa- 
tions p=a°+7y*, p=x°+8y’, we have supposed p te be of the forms 7n+1 
and 8n+1 respectively. But we know that some power of every prime of 
which —A is a quadratic residue is capable of representation by the form 
wv +Ay’; and, in particular, that primes of the form 8x+3 are capable of 
representation by a+ 2y°, and primes of either of the forms 7n+2 or 7n+4 
5) 
according as (5)=41 or =—1; a result which is in accordance with the equation 
1-2=(-O) 
In the formula (B), the exponent of p, obtained by the consideration of the same pro- 
b— : : : 
duct, is A'—B'=2x GE ea) Z ae! ; A’ and B' denoting respectively the numbers of residues 
of the classes a and 4 respectively, which are inferior to 2A. 
* Crelle, vol. ix. p. 189. Jacobi counts the classes of the prime determinant —A on the 
principle of Legendre, not distinguishing opposite classes from one another. If is the 
number of improperly primitive classes so counted, we have h=2n—1, because there is but 
one improperly primitive ambiguous class. When A is of the form 8x+7, Jacobi enun- 
ciates the theorem with reference to the number of properly primitive classes, which in this 
case is equal to the number of improperly primitive classes. 
