  We provide an explicit algorithm to compute a Massey triple product relative
to a defining system for a smooth projective plane curve $X$ defined by a
homogeneous polynomial $G(\underline x)$ over a field. The main idea is to use
the description (due to Carlson and Griffiths) of the cup product for
$H^1(X,\mathbb{C})$ in terms of the multiplications inside the Jacobian ring of
$G(\underline x)$ and the Cech-deRham complex of $X$. Our algorithm gives a
criterion whether a Massey triple product vanishes or not in $H^2(X)$ under a
particular non-trivial defining system of the Massey triple product and thus
can be viewed as a generalization of the vanishing criterion of the cup product
in $H^2(X)$ of Carlson and Griffiths. Based on our algorithm, we provide
explicit numerical examples by running the computer program.
