The purpose of this thesis is to investigate the ratio of the energy gap to the critical temperature in a superconductor when the energy dependent density of states is of the Van Hove singularity type. By using the theory of Bardeen, Cooper, and Schrieffer (BCS) and considering energy gap having the pairing states be isotropic (s-wave) and anisotropic (d-wave), we find equations of the ratio of the energy gap to the critical temperature, Tc, as functions of the Fermi energy, the Debye frequency, wD, the critical temperature, and the Fermi level shift. We find that the energy gap ratio always decreases when the ratio wD/Tc increases and also when the Fermi level shift increases. The existence of the mixed pairing states between the isotropic s-wave and the anisotropic d-wave states is also studied. We find an equation of the relative pairing strength interactions which demonstrates the existence of the mixed pairing states.