  The `linear orbit' of a plane curve of degree $d$ is its orbit in
$\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain
an algorithm computing the degree of the closure of the linear orbit of an
arbitrary plane curve, and give explicit formulas for plane curves with
irreducible singularities. The main tool is an intersection@-theoretic study of
the projective normal cone of a scheme determined by the curve in the
projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of
the orbit closure in terms of the degrees of suitable loci related to the
limits of the curve. These limits, and the degrees of the corresponding loci,
have been established in previous work.
