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Cross-magnetization of a ferromagnetic body is defined to occur whenever two or more independent nonparallel (crossed) magnetizing fields are applied to the body. A number of devices employing cross-magnetized components have been described in the literature. The analytical treatment of the behavior of systems containing these devices has been based generally on idealizations of the terminal properties of the cross-magnetized component, without immediate concern for the ferromagnetic processes that take place within the multipath core. The purpose of the work presented in this paper is to obtain direct predictions of the ferromagnetic behavior of such cores, from knowledge of their physical properties, their geometry, and the applied excitations. The application of crossed magnetizing fields gives rise to problems which are considerably more difficult than in ordinary unipath cores (e.g., toroids) because the total applied magnetizing field and the magnetization are not necessarily collinear, and empirical knowledge of the nonlinear relationship acquired by conventional test methods is not immediately usable. A method for the analysis of the quasi-static time-varying magnetization process is formulated, based on calculation of the potential energies of elemental closed tubes of having a constant cross-sectional area . The most likely configurations of such tubes along the various paths of the core are postulated, and the preferred states are determined by comparing the energies of elemental volumes. A method of introducing hysteresis loss into these calculations is also described. This concept of comparing the energies of elemental volumes is used to obtain models of the time-varying magnetization process in a ferrite sphere with holes along three mutually perpendicular diameters. Measured and calculated flux-MMF loops are compared for various biaxial excitations, and for a case of tri-axial excitation.