A rational thermodynamic theory has been developed by I. M(JLLER during the last four years [1, 2, 3]. It is not of the after effect type. Rather, it is restricted to constitutive equations which do not involve the histories of the independent variables but may involve these variables and their time derivatives up to some order, all taken at the present time. A derived quantity in this theory is the coldness function, which is universal for a certain class of materials. It is certainly worthwhile to study the physical significance of this function. We shall do this for the simplest case, which is that of pure heat conduction in an isotropic rigid material. We shall start from a generalization of MULLER'S theory which has recentlybeen obtained by BATRA [-4]. In this theory it is assumed that the constitutive assumptions express the various dependent quantities at x, t in terms of ~k~(x, t) and 9,A(x, t) where ~(x, t) is designated as the temperature field. First we examine the situation from a purely mathematical point of view, and then in the light of the results we shall discuss to what extent the interpretation of ~(x, t) as a temperature field seems reasonable.