  We investigate the finite size effects of a three-dimensional second-order
topological insulator with fourfold rotational symmetry and time-reversal
symmetry. Starting from the effective Hamiltonian of the three-dimensional
second-order topological insulator, we derive the effective Hamiltonian of four
two-dimensional surface states with gaps derived by perturbative methods. Then,
the sign alternation of the mass term of the effective Hamiltonian on the
adjacent surface leads to the hinge state. In addition, we obtain the effective
Hamiltonian and its wave function of one-dimensional gapless hinge states with
semi-infinite boundary conditions based on the effective Hamiltonian of
two-dimensional surface states. In particular, we find that the hinge states on
the two sides of the same surface can couple to produce a finite energy gap.
