AND GENERA OF TERNARY QUADRATIC FORMS. 
/ 2 , in which a 2 =p0 2 , A 2 =P0 2 . The fractional form 
i[PX’ + GY’+2>QAZ’] 
is then transformed into \p t by the substitution 
K_ 
©A’ 
b\ 
B, 
©A 
o, o, 
of which the determinant is Pj), and into \|/ 2 by a similar substitution of the same deter- 
minant. Either of the two forms %p„ \p 2 (and consequently either of the two f u f 2 ) is 
therefore transformable into the other, by a rational substitution prime to 20A. It will 
be found that if the signs of 0 n 0 2 , 0 15 0 2 are properly determined, the primes P, p will 
not appear in the denominators of these substitutions. 
lifi and/ 2 belong to an improperly primitive order, the preceding proof requires very 
little modification. It will suffice to consider the case in which and f 2 are impro- 
perly, F, and F 2 properly primitive. We take M r ^M 2 ^ — O, mod 4; <p 1 and i p 2 are 
then improperly primitive and have compatible generic characters ; let 2pQ\ be repre- 
sented by <p„ and 2 p6\ by <p 2 ; <!>, and <P 2 are properly primitive and of the determinants 
— 2A pffl, — 2Ap6\ ; these forms have compatible generic characters (their supplementary 
characters, in particular, being determined by those of F! and F 2 ) ; let, then, P0* be 
represented by and P0 2 by d> 2 , and let us suppose that \|/ 2 are forms equivalent 
to/„/ 2 , in which a x — 2p(f\, A"=P©h a 2 — 2pQ 2 2 , A 2 =P0 2 ; the fractional form 
^[i(P+Q)X a +(Q-P)XY+i(P + Q)Y , +i'QAZ»] 
is transformed into \p, by the substitution 
1 
2 7 
!&"©,+ a; 1^,0,-b, 
2 ©^ ’ 2 ©A 
! b"® l -A" 1 i'@j -f Bj 
2 0 ^, ’ 2 0 , 0 , 
0 , 0 
1 
0, 
and into \p t by a similar substitution. The determinant of each of these substitutions 
is P p, and the denominators of their coefficients do not contain the prime 2, because 
b". $ 2 , A", A", 0 n 0 2 are all uneven, and because 11,=^, mod 2, B 2 =# a , mod 2. Each 
of the forms f„f 2 is therefore transformable into the other by a rational substitution 
prime to 20 A. 
Art. 13. We have hitherto considered ternary forms of a negative determinant, defi- 
nite or indefinite ; we shall now confine our attention to definite forms. By a binary 
