It has previously been shown that the first moment of a powder EPR absorption spectrum calculated about gi, (= l/3 Tr g) is zero, subject to certain approximations (I). A rigorous proof in the high-field limit neglecting forbidden transitions that arise from nuclear state mixing was given in this paper. The theorem was then tested and verified using a rather complete computer program for simulation of powder EPR spectrum that had been developed by Pilbrow and Winfield (2). For these simulations, typical magnetic input parameters for a nitroxide radical spin label, for low spin Co2+, and for a square-planar complex of Cu2+ were used. The value g’ at which hrst moments on high and low sides are equal was compared with the known value of gim. The thrust of Ref. (I) was that even in the absence of knowledge of the spin-Hamiltonian parameters, it is possible to characterize a system by its isotropic g value. The common practice of using the crossover of a firstderivative feature at the center of the spectrum was criticized. It is obvious that the moment theorem is also valid in the liquid phase at sufficiently high fields and sufficiently fast rotational difTusion. Since the theorem holds in the rigid limit and the fast tumbling limit, the hypothesis that it holds in the intermediate slow tumbling domain would seem plausible. We have not found a general proof. However, using the slow-motion EPR simulation program developed in the laboratory of Professor J. Freed (3), we show here that empirically it holds for any isotropic rotational diffusion constant and also for anisotropic diffision. Using the same input parameters as in Ref. (I), numerous spectra were simulated as a function of both the microwave frequency and the isotropic rotational diffusion constant for Cu*+ and for spin labels, and the value of (si, g’) was calculated. The rotational correlation time was varied from lo-’ to lo-l3 s. The results for X band are shown in Fig. 1. It is noted that this computer program was developed primarily for nitroxide radical spin labels. Although it is a very complete program, it nevertheless contains approximations. The validity of these approximations is well established for spin labels, but not for copper spectra.