It is shown that if we know the Love's strain function due to the existence of any arbitrary axisymmetric singularity in the elastic whole space, then the corresponding Love's function due to the existence of the same singularity in the interior of one of two dissimilar semi-infinite solids separated by a dissimilar thick layer can be obtained by differentiation and integration of that for the elastic whole space. This discovery rests on the principle that to every singularity in the layered space there will correspond an infinite system of images. Remarkably, the number of images is finite when the phases have equal rigidities but different compressibilities. The analysis thus brings under one general theorem the solutions due to such influencing singularities as a normal point force, a normal point force doublet, an infinitesimal prismatic dislocation loop, a centre of dilatation and a dilatation doublet. From the theorem, we readily infer that the elastic field at large distances from any arbitrary axisymmetric influencing singularity becomes nearly that for two perfectly bonded semi-infinite solids without an interface layer.