The present work reports a general method for the calculation of the polarizability of a truncated sphere on a substrate. A multipole expansion 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 obtain this set, the interaction between the island and the substrate is taken into account by the technique of image multipoles. For numerical implementation, this set is truncated at an arbitrary multipole order. The accuracy of the method is judged through the stability of the truncated sphere polarizability and the fulfillment of the boundary conditions, which are demonstrated to be satisfied in large regions of the parameter space. This method brings an improvement with respect to the Bedeaux case [M. M. Wind, J. Vlieger, and D. Bedeaux, Physica A 141, 33 (1987); M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, {\it ibid} 143, 164 (1987)] where the multipoles are located in the center of the sphere.