  The present work reports a general method for the calculation of t he
polarizability of a truncated sphere on a substrate. A multipole ex pansion is
used, where the multipoles are not necessarily localized in the center of the
sphere but can freely move on the revolution axis. From the weak formulation of
the boundary conditions, an infinite set of linear equations for the multipole
coefficients is derived. To obta in this set, the interaction between the
island and the substrate is t aken into account by the technique of image
multipoles. For numerical implementation, this set is truncated at an arbitrary
mu ltipole order. The accuracy of the method is jugded through the stabil ity
of the truncated sphere polarizability and the the fulfilment of t he boundary
conditions which are demonstrated to be satisfied in large regions of the
parameter space. This method brings an improvement wit h respect to the
Bedeaux's case \cite{Wind87a,Wind87b} where the multi poles are located in the
center of the sphere.
