Electron transmission properties of a configuration consisting of two infinite monatomic chains, which are coupled to each other by either a direct bond or through a finite atomic bridge, are studied via the tight-binding model. A method, based on the multi-channel Lippmann - Schwinger equation, is used to obtain analytical expressions for the scattering matrix, from which transmission probabilities are calculated for several special chain configurations. In particular, we study cases where the bridge is subjected to a constant potential and an electric field, which is generated by a potential difference between the two infinite chains. The results of the theory are very useful for qualitative evaluations of the transmission properties of simple atomic circuits and molecular switches.