We consider the following two mixed boundary-value problems: (1) The steady-state plane-strain thermoelastic problem of an elastic layer with one face stressfree and the other face resting on a rigid frictionless foundation; the free surface of the layer is subjected to arbitrary temperature on the part a < x < b, whereas the rest of the surface is insulated and the surface in contact with the foundation is insulated. (2) The two-dimensional electrostatic problem of the electrostatic potential due to two coplanar strips that are charged to equal and opposite potentials and that are parallel to and equidistant from a grounded strip. By the use of Fourier transforms, both problems are reduced to the solution of triple trigonometric integral equations. The closed-form solution of these triple-integral equations is obtained by using the finite Hilbert-transform technique. Closed-form expressions are obtained for the physical quantities in both problems.