AND GENERA OF TERNARY QUADRATIC FORMS. 
261 
a form equivalent to f, and having one of its principal coefficients unevenly even and 
prime to any uneven number ; (ii) that if f is properly primitive, we can find a form equi- 
valent to/, and having one of its principal coefficients prime to any given number; (iii) 
that if 12A is uneven, we may suppose this principal coefficient of either of the two 
linear forms 4^-(-l, or 4^+3, at our option. 
We shall first suppose that the forms f and F, which it is proposed to transform into 
forms cp and 0 satisfying the congruences (7), are properly primitive. Let V'=V12 2 A, 
and let us assume that in the form F, A" is prime to V', and also that A" =12, mod. 4, 
if 12A is uneven. 
Let 
mod V' ; the redundant system of congruences 
ax-\-b"y-\-V =0, 
b"x-\- dy-\-b =0 
mod V'„ 
b'x+by y!2A, 
is resoluble, admitting 12 incongruous solutions*. 
X— X, 
y— f*>_ 
mod V', 
Let 
be any one of these solutions, and let us transform /by the substitution 
x=x+\z,~ 
y=y+t*z,_ 
into an equivalent form /. The coefficients b"„ a[ are the same as a, b", a 1 ; the 
coefficients a", b[ are respectively congruous for the modulus V' to yl 2A, 0, 0 ; so 
that f x satisfies the congruence 
f x = ax ’ 1 + 2 V'xy + dy 1 + y 12 Az 2 , mod V'. 
The binary form (a, b", d) is primitive ; for if d is a prime dividing a, b", a /, it divides 
— 12 A", the determinant of ( a , b", a 1 ), and 0 2 A, the discriminant of f; it therefore divides 
12 (because A" and A are relatively prime), and is a common divisor of the coefficients of 
the primitive form /,, i. e. d= 1. Again, («, b", a 1 ) is not improperly primitive; if 12 A 
is even, this is manifest, for f x is not improperly primitive; if 12A is uneven, 12A" is by 
hypothesis of the form 4&-f 1, and there are no improperly primitive binary forms of the 
determinant — 12 A". We may now suppose that, in the properly primitive binary form 
A" 
(a, b", a’), a is uneven and prime to V' ; let /3e==— , mod V' ; then the congruences 
ax+b"^ 0, 
b"x-\-a! =/312,J 
mod V', 
are resoluble and admit of one solution. Let x = X, mod V', be that solution ; if f x be 
transformed by the substitution x=x-\-\y, the resulting form <p will satisfy the congruence 
<p = ax‘ i -\-l3Q.y 2 -\-yQ.Az 2 , mod V', 
and the forms <p and <J> will satisfy the congruences (7) for the modulus V. 
* Philosophical Transactions, vol. cli. p. 323, 
