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In this note, we give a new necessary 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 characterize the bounded composition operators on the anisotropic Dirichlet-type spaces 
$\mathfrak {D}_{\vec {a}}(\mathbb {D}^2)$ induced by holomorphic self maps of the bidisc 
$\mathbb {D}^2$ 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 
$C_{\Phi }:\mathfrak {D}(\mathbb {D}^2)\to A^2(\mathbb {D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the Dirichlet space of the bidisc.
We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols 
$\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.
