A theorem is established for generating by differentiation and integration the Airy stress function due to the presence of any arbitrary two-dimensional singularity near a thick layer separating two other dissimilar semi-infinite solids directly from the corresponding Airy stress function for the unbounded homogeneous solid. A systematic application of the theorem yields the dominant distant effect associated with any arbitrary influencing line singularity in a well-structured physically interpretable form. As an illustration, the problem of a concentrated line force of arbitrary orientation near the interface layer is examined from this standpoint. It is found that a force aligned normal to the interface layer produces a far-field effect of an edge dislocation with Burgers vector parallel to the interface layer, together with a semi-infinite plane of centres of dilatation, whereas a force aligned parallel to the interface layer produces a far-field effect of an edge dislocation with Burgers vector perpendicular to the interface layer together with a semi-infinite plane of concentrated couples.