The Hertz analysis of contact stresses is extended to include the effects of friction on the interface between two elastic spheres compressed along the line connecting their centers. The problem is shown to be one of a class which requires incremental formulation. Stress functions of interest in connection with the analysis of the shear-loaded half-space in the linear theory of elasticity are developed. The distribution of shear stress needed to prevent relative slip of surficial points after they enter the contact region is found to be finite everywhere in the region. The ratio of this shear stress to the coexisting normal stress component is shown to exhibit a singularity at the edge of the contact region. This implies that when elastically dissimilar spheres are pressed together microscopic slip must occur in a narrow annulus at the boundary of the contact region.