Adini¡¦s elements are applied to Poisson¡¦s eigenvalue problems in the unit square with periodical boundary conditions and the leading eigenvalues are obtained from the Rayleigh quotient. The penalty techniques are developed to copy with periodical boundary conditions, and superconvergence is also explored for leading eigenvalues. The optimal convergence O(h^6) are obtained for quasiuniform elements (see [2, 21]). When the uniform rectangular elements are used, the superconvergence O(h^6+p) with p = 1 or p = 2 of leading eigenvalues is proved, where h is the maximal boundary length of Adini¡¦s elements. Numerical experiments are carried to verify the analysis made. Keywords. Adini¡¦s elements, Poisson¡¦s equation, periodical boundary conditions, eigenvalue problems.