270 
PEOFESSOE H. J. S. SMITH ON THE OEDEES 
for f is transformed into such a form by a substitution of which the matrix is 
«i» (3„ y x 
P* y 2 
P« y 3? 
y n y 2 , y 3 denoting any three numbers which render the determinant of that matrix equal 
to -(- 1. 
Let 
F'= P# 2 + Py 2 -j- P V -f- 2Qyz + 2Q!xz + 2Q!'xy 
be the primitive contravariant of/*', so that, in particular, 
QP "=q" 2 -pp'; (15) 
multiplying the equations 
P'P"- Q 2 —Ap, -j 
QQ' -P"Q"=A/', l (16) 
PP"- Q! 2 =Ap' J 
(which result from the contravariance of/ 7 and F) by x 2 , 2xy, y* respectively, we obtain 
-Ax (K+%''^+yy)=(Q 2 -P'PV-2(QQ , -P , 'Q , ')^+(Q' 2 -PF)y ; 
and this equation, considered as a congruence for the modulus P", becomes 
A X<P+(Q#— Q'y) 2 =0, mod P", (17) 
the coefficients of x 2 , 2#y, y 2 in the left-hand member being all divisible by P". If therefore 
<p is a binary quadratic form of determinant — flP", admitting of primitive representa- 
tion by a ternary form of order [Q, A], — A <p is a quadratic residue of P". And we 
shall now show that if <p is a primitive (and not negative) binary form of determinant 
— OP", P" being of the same sign as A and prime to A, <p admits of primitive repre- 
sentation by ternary forms of the invariants [O, A], whenever — A <p is a quadratic resi- 
due of P". 
Because —A <p is a quadratic residue of P", the congruence (17) admits of solution in 
integral numbers Q, Q'. Any solution of this congruence supplies a system of five 
numbers, P, P', Q, Q', Q", satisfying the equations (16). The greatest common divisor 
of these five numbers divides A, because p, q”, p' are relatively prime ; but P" is prime 
to A ; therefore the six numbers P, P', P", Q, Q', Q" are relatively prime. Let q and q' 
be determined by the equations 
qq"-q'p' = QQ', | ^ 
qp — //'=— OQ, J 
which are always resoluble because their determinant q" 2 —pp'= OP" is different from 
zero. Also let p" be determined by the equation 
q'Q! + qQ +p"P" = Q A (19) 
The values of q, q 1 , p" are rational ; and, if they are fractions, their denominators, when 
they are expressed in their lowest terms, are divisors of P". Substituting in (19) for 
