  In this note we give a new sufficient condition for the boundedness of the
composition operator on the Dirichlet-type space on the disc, via a two
dimensional change of variables formula. With the same formula, we characterise
the bounded composition operators on the anisotropic Dirichlet-type spaces
$\mathfrak{D}_{\vec{a}}(\mathbb{D}^2)$ induced by holomorphic self maps of the
bi-disc $\mathbb{D}^2$ of the form $\Phi(z_1,z_2)=(\phi_1(z_1),\phi_2(z_2))$.
We also consider the problem of boundedness of composition operators
$C_{\Phi}:A^2(\mathbb{D}^2) \to \mathfrak{D}(\mathbb{D}^2)$ for general self
maps of the bi-disc, applying some recent results about Carleson measures on
the the Dirichlet space of the bi-disc.
