Employing a Landauer-type picture, the chemical potential at a certain site in a multiply-connected wire system is expressed in terms of the transmission amplitudes between this site and the external electron reservoirs. The result is applied to geometries involving rings containing magnetic flux, in order to derive nonlocal quantum interference effects in the conductance. As an example, h/e oscillations are obtained in the conductance of a wire with a dangling ring and in the conductance of a resistor in series with a ring. The general form of the scattering matrix of a three-terminal fork governing the transmission of a ring is found and exemplified by an explicit calculation within the tight-binding model. This example shows that, in general, the scattering matrix is not real (as assumed in previous studies), but that the qualitative results are not sensitive to the detailed form of the matrix.