The authors' develop a general formulation of the problem of elastic transport between two semi-infinite systems, connected by a system of finite size, and derive expressions for the current in and the differential conductance of such a circuit in the limit of zero interactions between the carriers. These expressions are exact in the applied voltage, the coupling of the components of the circuit, and the temperature of the circuit. They then apply their results in a tight-binding approximation to three specific cases: the one-atom contact, the finite, disordered one-dimensional chain, and the generalized stacking fault.