Duhamel's theorem estimates the temperature of a heat conduction body exposed to time variable heat exchange medium temperatures (T-a). Since the theorem is not applicable to variable convective, surface heat conductance (h), new general solutions are obtained for variable T-a and h. These solutions are expressed in terms of normalized temperature response functions of a body exposed to step functional medium temperatures and include continuity constants to ensure temperature continuity at each time for a change in h. Sample applications of the theorem are presented for spherical food utilizing a published analytical temperature response function.