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In order to obtain sources of high intensity radiation it is necessary to use optical systems which can only provide small areas of uniform irradiation. It is important to know what effect this limited area of irradiation has on the temperature rise of the surface since this controls the thermal damage suffered. In this paper, two forms of the problem of the conduction heat flow from a circular heating source on a semi-infinite solid are discussed. In the first, analogous to the classical problem of the electrified disk, the temperature or potential V is uniform over a circular area and zero in the remainder of the surface plane. In the second and more useful problem, the flux is uniform over the circular area and zero elsewhere in the plane. More complex boundary conditions present difficulties in analysis and, even for the two above, a complete analytical result is only possible in certain conditions. Although only circular sources are discussed, there are some results for square sources given by Jaeger, but there is little difference between them for equal areas.