The subject treated in this paper forms a part of a more general theory of heat flow involving moving sources of heat. The present problem deals with the temperature distribution around a point source moving at a constant speed along the axis of a cylinder. It is shown that the rise of temperature produced at a given cross section of the cylinder, as a result of the passage of the heat source, tends rapidly to become constant. The factor which governs this trend is a dimensionless product of the following units: (1) The speed at which the source travels; (2) the radius of the cylinder; and (3) the converse of the heat diffusivity.