For many solid dielectrics the rate of generation of heat in an alternating electric field increases approximately exponentially with the temperature. For such materials, not only the condition for a steady state and the temperature distribution in it, but also the time variation of the temperature, both when a steady state exists and when the field is above the critical field for which thermal instability occurs, is of interest and'importance. The present paper is concerned with the process of evaluating, for the transient case, the solution of the equation of heat conduction in one dimension for such a material, and gives some results.The solutions evaluated are believed to be the first calculated for a non-linear variation of generation of heat with temperature, and show a close correspondence with the results of tests on actual materials. Particularly the ultimate catastrophic rise of temperature, in fields just above the critical field which leads to breakdown, is clearly shown by the solutions. This behaviour cannot occur if the rate of generation of heat varies linearly with temperature, but these results show that it is unnecessary to postulate any other cause (such as deterioration) beyond the exponential variation of heat generation with temperature to account for it.