The paper considers a body, or space vehicle, of arbitrary shape, with conductivity K and specific heat c. It is first assumed that K and c are invariant with time and the body, with a given initial temperature, is assumed to be heated by a distribution of heat sources that varies with space co-ordinates and time. It is shown that a rigorous analytical solution to the problem is possible, provided that the distribution function satisfies prescribed regularity conditions.

The same problem is then considered, but with a heat-source distribution which is also a function of temperature. In this case the problem can be reduced to a Volterra-type integral equation amenable to step-by-step solution. Convective heating and radiation are specific examples of this second case.

The third case considered is that of a body whose geometry and thermal coefficients K and c are all time-dependent, and this problem is reduced to the fixed-geometry case by the addition of a hypothetical heat-source distribution which is a function of the velocity of ablation, of temperature and of its time derivative; an integral equation of Volterra type again results.

Finally, an example is given of the application of the analysis to the ablation of a conductive spherical shell.