In this work we derive rigorously the free energy balance for three fluid phases in a straight capillary of arbitrarily shaped cross-section. This balance is then used to derive the general equation for the capillary entry pressures of all possible two-phase and three-phase displacements. Moreover, the equation provides the criterion determining the existence of layers of the intermediate-wetting phase separating the wetting and non-wetting phases in the corners or cavities of a capillary, by also treating the spreading of such layers as a capillary displacement. For a number of combinations of interfacial tensions and contact angles, illustrating all the different relevant situations, we calculate the criteria for spreading of such a layer in the corner of a capillary with polygonal cross-section. In a capillary with a cross-section in the shape of an isosceles triangle of varying corner size, these criteria are used to determine the unique capillary entry pressures for piston-like displacement from alternative solutions of the general equation. These solutions relate to displacements in the presence or absence of layers in the various differently sized corners. (C) 2004 Elsevier Inc. All rights reserved.