Let
X
X
be an integral plane quartic curve over a field
k
k
, let
f
f
be an equation for
X
X
. We first consider representations
(
∗
)
(*)
c
f
=
p
1
p
2
−
p
0
2
cf=p_1p_2-p_0^2
(where
c
∈
k
∗
c\in k^*
and the
p
i
p_i
are quadratic forms), up to a natural notion of equivalence. Using the general theory of determinantal varieties we show that equivalence classes of such representations correspond to nontrivial globally generated torsion-free rank one sheaves on
X
X
with a self-duality which are not exceptional, and that the exceptional sheaves are in bijection with the
k
k
-rational singular points of
X
X
. For
k
=
C
k=\mathbb {C}
, the number of representations
(
∗
)
(*)
(up to equivalence) depends only on the singularities of
X
X
, and is determined explicitly in each case. In the second part we focus on the case where
k
=
R
k=\mathbb {R}
and
f
f
is nonnegative. By a famous theorem of Hilbert, such
f
f
is a sum of three squares of quadratic forms. We use the Brauer group and Galois cohomology to relate identities
(
∗
∗
)
(**)
f
=
p
0
2
+
p
1
2
+
p
2
2
f=p_0^2+p_1^2+p_2^2
to
(
∗
)
(*)
, and we determine the number of equivalence classes of representations
(
∗
∗
)
(**)
for each
f
f
. Both in the complex and in the real definite case, our results are considerably more precise since they give the number of representations with any prescribed base locus.