A problem in the theory of heat conduction

The first problem discussed in this paper is as follows :— “A long hollow cylinder is immersed in a medium of uniform temperature, and at a certain instant, is brought suddenly to a temperature above that of the medium, for example, by the passage through it of a stream of heated liquid. It ismaintained at that temperature, and the temperature distribution in the outside medium at any instant, and the rate of leakage of heat from the cylinder, are required.” The main difficulty arises from the fact that the law of temperature on the surface of separation is arbitrary—we shall suppose, in the first instance, that it is constant. Solutions for the external temperature distribution are apparently known only when the surface has a giveninitial distribution, which is left to adjust itself without further direct supply of heat.