1. When a heat source travels along the surface of a semi-infinite solid with a constant velocity ν, the temperature θ in the semi-infinite solid is given by [numerical formula] where w=x-vt, β=ν/α, and the temperature distribution of the heat source can be expressed as θ(w). 2. Though the eigen-values of this partial differential equation are complex numbers, the solution can be obtained in the form of a real Fourier's integral. Thus the temperature distribution in a semi-infinite solid and heat transmission through the surface are calculated. 3. This Fourier's integral was numerically calculable by a digital computer. 4. It was much easier to solve the above partial differential equation by the accelerated Liebmann's method using a digital computer than to solve it by the Fourier's integral method, and the result obtained by one method coincided with the result obtained by the other.