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We present results from high-resolution N-Body/SPH simulations ofrotationally supported dwarf irregular galaxies moving on bound orbits in themassive dark matter halo of the Milky Way.The dwarf models span a range in disksurface density and the masses and sizes of their dark halos are consistentwith the predictions of cold dark matter cosmogonies. We show that the strongtidal field of the Milky Way determines severe mass loss in their halos anddisks and induces bar and bending instabilities that transform low surfacebrightness dwarfs (LSBs) into dwarf spheroidals (dSphs) and high surfacebrightness dwarfs (HSBs) into dwarf ellipticals (dEs) in less than 10 Gyr. Thefinal central velocity dispersions of the remnants are in the range 8-30 km/sand their final $v/\sigma$ falls to values $< 0.5$, matching well thekinematics of early-type dwarfs. The transformation requires the orbital timeof the dwarf to be $\simlt 3-4$ Gyr, which implies a halo as massive andextended as predicted by hierarchical models of galaxy formation to explain theorigin of even the farthest dSph satellites of the Milky Way, Leo I and Leo II.Only dwarfs with central dark matter densities as high as those of Draco andUrsa Minor can survive for 10 Gyr in the proximity of the Milky Way: this isnaturally achieved within hierarchical models, where the densest objects shouldhave small orbital times due to their early formation epochs. Part of the gasis stripped and part is funneled to the center due to the bar, generating onestrong burst of star formation in HSBs and smaller, multiple bursts in LSBs.Extended low-surface brightness stellar and gaseous streams originate from LSBsand, when projected along the line of sight, can lead to overestimate themass-to-light ratio of the bound remnant by a factor $\simlt 2$,

The Metamorphosis of Tidally Stirred Dwarf Galaxies