AND GENERA OE TERNARY QUADRATIC EORMS. 
275 
capable of primitive representation by a given genus of ternary forms of the improperly 
primitive order [O, A], of which the contravariant characters coincide with the 
characters of M. And in both cases no other primitive form (if M is prime to O, 
no other form, primitive or derived) of determinant — OM is capable of such repre- 
sentation. 
Art. 12. By a rational substitution we shall understand in this article a substitution 
of which the determinant is unity, and of which the coefficients are rational. If the 
common denominator of the coefficients is prime to any number m, we shall say that the 
substitution is prime to m. 
If/*, and f 2 are ternary forms, having integral coefficients, of which/! is a form of the 
invariants (O, A), and is transformed by a rational substitution, prime to 20A, into f 2 , 
f 2 is a form of the same invariants, of the same order, and of the same genus as /,. This 
may be proved nearly in the same way in which it is proved that equivalent forms have 
the same invariants and are of the same order and genus ; it is only necessary to observe 
that F! and F 2 , as well as/j and / 2 , are transformable into one another by rational sub- 
stitutions, prime to 2QA. The converse proposition, 
“ If f i and /^ are two forms of the same invariants (O, A), of the same order, and of 
the same genus, they are transformable, each into the other, by rational substitutions 
prime to 20A,” is also true, and is of importance in the present theory, because it 
establishes the completeness of the enumeration of the generic characters of ternary 
forms. To avoid the introduction, in this place, of principles relating to quaternary 
quadratic forms, we shall give an indirect demonstration of it, depending on the follow- 
ing lemma which relates to binary quadratic forms. 
“ If <p„ (p.j are two primitive binary quadratic forms of the same determinant, and of 
the same genus, the resolubility of the equation <p,(#, ?/)=M implies the resolubility of 
the equation <p 2 (a, y)=M z 2 ; and in the solution of this equation the value of z may be 
supposed prime to any given number 
Because p, and <p 2 are of the same genus, <p 2 is transformable, by a bipartite linear 
substitution, into the product X representing a properly primitive form of the 
principal genus (Disq. Arith. art. 251). But % is transformable, by a quadratic sub- 
stitution, into the square of a properly primitive form \}/ (ibid. art. 287). Therefore, by 
a mixed quadratic and linear substitution, <p 2 is transformed into the product \J/ 2 X £>,. 
Attributing, in this mixed substitution, to the indeterminates of ©, the values which 
satisfy the equation <p,=M, and to the indeterminates of any values whatever for 
which \]y acquires a value z prime to k, we obtain a solution of the equation p 2 =Mz 2 . 
Let us first suppose that the given ternary forms/’, and/ 2 belong to the properly pri- 
mitive order of the invariants (H, A); let M,, M 2 be two numbers of the same sign as 
A, prime to 20A, and primitively represented by F,, F 2 respectively ; we may suppose 
that M,=M 2 , mod 8 ; and that the representations of M, and M 2 are simultaneous with 
the representations of uneven numbers by/*, and/ 2 . Let <p„ <p. 2 be two binary quadratic 
forms, of the determinants — OM„ — QM 2 respectively, represented by/, and/j, simul- 
