. 
been all derived by transformation from one and the same form (see Art. 80) ; 
the last two (which, it is to be observed, present an ambiguity of sign 
ON THE THEORY OF NUMBERS. 505 
when — is even) show that the forms B and F,*, C and F*, are respectively 
n 
identical, if we omit a numerical factor. 
Lastly, let ©,, ©,, @, be any corresponding covariants of F,, F,, F,. The 
relation of covariance gives rise to the equations 
np—q 
Sear, a, 3/0.) My, \, = O,(Y45. Yor 9,6 Ee > 2) 
mp—q 
(x, Vs, : .@,) Xx C : =6,(2,, 9 , -2,) xFPY,, aa )s 
where p and gq are the orders of the covariants in the coefficients and in the 
indeterminates respectively. Combining with these equations the values of 
q q 
B and C already given, we see that 6, x F,” and 6,xF,” are identical, ex- 
cepting a numerical factor ; 2. ¢. that ®, and ®, are either identically zero, or 
else numerical multiples of powers of F, and F,. If therefore two forms 
can be combined by multiplication so as to produce a third form transtormable 
into their product, their covariants are all either identically zero or else are 
powers of the forms themselves. There is, consequently, no general theory 
of composition for any forms other than quadratic forms, because all other 
sorts of forms have covariants which cannot be supposed equal to zero, or to 
a multiple of a power of the form itself, without particularizing the nature 
of the form. And even as regards quadratic forms, we may infer that com- 
position is possible only in cases of continually increasing particularity, as 
the number of indeterminates increases. 
106. Composition of Quadratic Forms.—Preliminary Lemmas.—The follow- 
ing lemma is given by Gauss as a preliminary to the theory of the composition 
of binary quadratic forms (Disq. Arith., art. 234) :— 
(i.) “ If the two matrices ; 
fea Hey rele 
Bila By Bay ic-B, 
and 
a =| @, aves hy 
0 Ra 
be connected by the equation 
[3 [415 
es oe 
in which the sign of equality refers to corresponding determinants in the two 
matrices; and if the determinants of | 5 | admit of no common divisor beside 
A 
| [=| 
in which the sign of equality refers to corresponding constituents in the two 
matrices, is always satisfied by a matrix |k| of the type 2x2, of which 
the determinant is /, and the constituents integral numbers.”* 
The subsequent analysis of Gauss can be much abbreviated if to this 
lemma we add three others. 
* For a generalization of this theorem, see a paper by M. Bazin, in Tiiouville, vol. xix. 
p- 209; or Phil. Trans. vol. cli. p. 295. 
unity ; the equation 
? 
a 
x5 
