The object of this note is to indicate a numerical method for finding periodic solutions of a number of important problems in conduction of heat in which the boundary conditions are periodic in the time and may be linear or non-linear. One example is that of a circular cylinder which is heated by friction along the generators through a rotating arc of its circumference, the remainder of the surface being kept at constant temperature; here the boundary conditions are linear but mixed. Another example, which will be discussed in detail below, is that of the variation of the surface temperature of the moon during a lunation; in this case the boundary condition is non-linear. In all cases the thermal properties of the solid will be assumed to be independent of temperature. Only the semi-infinite solid will be considered here, though the method applies equally well to other cases.