AND GENERA OE TERNARY QUADRATIC FORMS. 
269 
and equation (14) becomes 
n,+ l A,+l a + A" A t +a a 2 — I A" 2 - 1 / a 
-1) * ' * * ' 2 « ’ P • X [a, 
or observing that 
fi,+A" A,+a fj, A"+I A,a+1 
and writing /and F for a and A", 
P - 1 F 2 — 1 / f\ /-p\ Oi + l 
X^— x(^)x(s;)=(-lp 
i. e. the generic character of/ satisfies the condition of possibility (11). 
Again, if/ is improperly and F properly primitive, let A" be prime to 20 A ; then the 
binary form (a, V\ a') is primitive, because A" is prime to AO, and improperly primitive, 
because /is improperly primitive. We may therefore suppose that \a is uneven and 
prime to OA", and, as before, that a and A" are positive. Multiplying together the 
equations 
and transforming the result by the law of reciprocity, we find 
(-« AJ? x(^)x(|) =(-!)' 
2 
i. e. the condition (12) is satisfied by the generic character of/. 
The case in which/ is properly and F improperly primitive, is the reciprocal of the 
preceding. 
To show that the conditions (11), (12), (13) are sufficient as well as necessary, other 
principles are required. These principles relate to the representation of binary by ternary 
quadratic forms, and will be found in the ‘ Disquisitiones Arithmeticse,’ arts. 282-284 ; 
it will, however, be convenient briefly to restate them here, in a form suited for our 
present purpose. 
Art. 10. A binary quadratic form (p, /,/) or <p is said to be represented by a ternary 
form / when/ is transformed into <p by a substitution of the type 
x=u 1 x-\-(3 1 y, 
y=u 2 x+fi 2 y, 
z=u 3 p+p 3 y. 
The representation is said to be primitive when the determinants of the matrix 
/3, 
«2> 02 
a 3? ($3 
are relatively prime. If <p is primitively represented by /, / is equivalent to a form con- 
taining <p as a part, i. e. to a form/' of the type 
/' =^ 3 +y/+p V+ Zyyz + 2/rz+ 
