A Barden-Cooper-Schrieffer type of reduced Hamiltonian, with an arbitrary pairing interaction is studied with the help of the method of temperature-dependent Green’s functions. The equations are solved by expansion in powers ofN−1, whereN is the number of particles, and elements of a proof are given that these equations possess at most three classes of solutions : the normal solution in which all correlations are absent (to order unity), a solution which requires in addition two particle correlations and corresponds to that of BCS, and a solution in which all correlations must be retained. Only the BCS case is treated in this paper. A new derivation of the complete thermodynamics is given. By considering the perturbation by an external field, one is led to the underlying matrix structure of the theory.