  It is shown that the partition function of the 2d Ising model on the dual
finite lattice with periodical boundary conditions is expressed through some
specific combination of the partition functions of the model on the torus with
corresponding boundary conditions. The generalization of the duality relations
for the nonhomogeneous case is given. These relations are proved for the
weakly-nonhomogeneous distribution of the coupling constants for the finite
lattice of arbitrary sizes. Using the duality relations for the nonhomogeneous
Ising model, we obtain the duality relations for the two-point correlation
function on the torus, the 2d Ising model with magnetic fields applied to the
boundaries and the 2d Ising model with free, fixed and mixed boundary
conditions.
