The Tool-Narayanaswamy-Moynihan (TNM) equation for the temperature (T) and fictive temperature (T-f) dependence of the relaxation time in glassy materials is compared with the usual nonlinear form of the Adam-Gibbs (AG) equation. It is shown that the relationship derived between the Narayanaswamy parameter x and the temperature T-2 at which the configurational entropy reduces to zero, namely x approximate to 1 - T-2/T-f, leads to unrealistic values of T-2 for many polymer glasses. This problem is resolved by expressing the configurational entropy as a function of both T and T-f, with a partitioning parameter x(s) (0 less than or equal to x(s) less than or equal to 1) controlling their respective contributions. Comparing TNM with this new nonlinear AG expression incorporating S-c(T,T-f) leads to an explicit relationship between x and x, involving T, T-2, and T-f, from which a number of predictions may be made. (1) For T approximate to T-f, i.e., for relaxations close to equilibrium, the quantity 1 - T-2/T-f is identified as the minimum possible value for x, implying that T-2 greater than or equal to T-f(1 - x), by an amount depending on the value of x(s). This resolves the apparently anomalous values of T-2. (2) For relaxations further from equilibrium, the TNM equation with constant x becomes increasingly inappropriate. (3) With increasing annealing temperature and increasing annealing time, the analysis predicts increasing values of x, as has often been reported in the literature. The origin of the dependence of S-c on T and T-f is considered from the theory of Gibbs and DiMarzio, and it is argued that typical values of x observed experimentally may be associated with the freezing-in of only a certain fraction of either flexed bonds and/or vacant lattice sites (holes) at the glass transition. Thus, it is possible to identify x(s), and indirectly x, with the relative contributions of physically meaningful parameters, such as intermolecular and intramolecular bond energies, to the freezing-in process.