Asymptotic tail equilibria which are slowly varying in the Earth-Sun direction are constructed, including all three components of the magnetic field. These equilibria allow for spatial dependencies in Bn and By. Some of these equilibria are analyzed for stability with respect to collisionless electron tearing modes using a fluid model which predicts, to within a numerical factor of &pgr;1/2, the growth rates derived from kinetic theory. No ion tearing instability is found. The By field is demonstrated to have a destabilizing effect on electron tearing modes. In the asymptotic equilibria considered here, electron tearing modes can grow in the presence of By in those regions where the stabilizing effect of electron bounce is small. Implications for numerical simulations and observations are discussed. ¿ American Geophysical Union 1993