A methodological study of the Born-Green-Yvon (BGY) approach by Fischer and Methfessel was performed for the liquid-vapour interface of pure and mixed Lennard-Jones (LJ) fluids. The bulk phase densities and concentrations obtained from the BGY calculations of the interface have to be the same as those resulting from a consistent equation of state via the equality of the chemical potentials. Deviations from this criterion indicate either deficiencies in the approximation scheme or in the numerical treatment. Regarding pure fluids, we have succeeded by an improved numerical integration for BGY in obtaining the same orthobaric densities from both methods. Moreover, at sufficiently low temperatures the BGY solutions show oscillation in the condensed phase which are interpreted as freezing. In the case of mixtures, two different approximation schemes for the calculation of the coarse-grained densities, BGY0 and BGY1, yield remarkably different vapour compositions. BGY1, however, is consistent with the equation of state results. Finally, a comparison of the phase equilibrium data and interfacial properties from this theoretical treatment with simulation results shows some differences, but is still satisfying for both pure fluids and binary mixtures.