The problem of two-dimensional transient heat conduction from a circular cylinder of finite length and appreciable heat capacity has been solved using a Laplace transformation with respect to time and a finite Fourier sine transformation with respect to the axial variable. A case of constant surface heat flux with the ends of the cylinder maintained at zero temperature is considered. The solution, valid for all values of time, is compared with that of Jaeger for the infinitely long cylinder. The results are of use in the evaluation of heat losses for the transient hot-wire method of determining the thermal conductivity of fluids.