ON THE THEORY OF NUMBERS. 615 
(vii.) “The class compounded of the opposites of two or more forms is the 
opposite of the class compounded of those forms.” It follows from this, or 
from vi., that a class compounded of ambiguous classes is itself ambiguous. 
(viii.) Let ®,, ©, ...,-1 represent all the classes of det. D, and of a 
given order ©; and let 1, f,, f,, . . . fr—1 represent the properly primitive 
classes of the same determinant; it may then be shown that w is a divisor 
_» *@ 
of n, and that, given two classes of the order Q, there always exist |, Pro 
perly primitive classes, which, compounded with one of them, produce the 
other, Assuming, for a moment, that a form ®, exists, such that the w equa- 
tions included in the formula ®,xf=®, can all be satisfied, we see that 
each of these equations is satisfied by the same number of properly primitive 
classes f; for if the equation ®, x f=, be satisfied by & primitive classes, 
1, $5 do» + » $e-1, the equation ©, x f=,, which is, by hypothesis, satisfied 
by a single class, 7,,, 1s also satisfied by the /—1 classes f, x 9,,- +» fu X Gx—1> 
but by no other class. Since, then, the classes ®, x f, of which the number 
‘is n, represent every class of the order Q k times, we have evidently n=kw. 
It is also readily seen that every equation of the type ®, x f=, admits of k 
solutions; and thus it only remains to justify the assumption on which the 
preceding proof depends. If the order Q be derived by the multiplier m from 
a properly primitive class of determinant Ay we may take for ®, the 
m 
class represented by the form (m, 0, —Am); if Q be derived from an im- 
properly primitive class, we take for ®, the class represented by the form 
(2m, m,— ma . Representing ©, in the first case by the form (ma, mb, 
me), and in the second by the form (2ma, mb, 2mc), and supposing (as we may 
do) that a in each case is prime to 2D, we see that the forms (a, mb, m*c) and 
(a, bm, 4cm”) are properly primitive ; and we find by the formule of compo- 
sition (Art. 110), 
(m, 0, —Am) x (a, bm, em*)=(ma, mb, me) 
(2, m, —™m ax 
) x (a, bm, 4em?)=(2ma, mb, 2me) ; 
i.e. the equation ®, x f=, can be satisfied for every value of p. 
113. Comparison of the numbers of Classes of different Orders—To deter- 
mine the quotient ” of the last article, Gauss investigates the properly pri- 
Ww 
mitive classes of det. D, which, compounded with the classes (m, 0, —Am) 
and (2m, m, —m , reproduce those classes themselves. He thus em- 
2 
ploys the theory of composition to compare the number of properly pri- 
mitive classes of a given determinant with the number of classes contained 
in any other order of the same determinant; or, which comes to the same 
thing, to compare the numbers of classes, of any given orders, of two de- 
terminants which are to one another as square numbers (Disq. Arith., art. 
253-256). We have already seen (Art. 103) that the infinitesimal analysis 
of Dirichlet supplies a complete solution of this problem ; whereas, in the case 
of a positive determinant, the result in its simplest form was not obtained 
by Gauss. It has, however, been recently shown by M. Lipschitz (Crelle, 
yol. lili. p. 238) that the formule of Dirichlet may be deduced, in a very ele- 
mentary manner, from the theory of transformation, We propose in this 
2u 2 
