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To compare the chemistries of stars in the Milky Way dwarf spheroidal (dSph) satellite galaxies with stars in the Galaxy, we have compiled a large sample of Galactic stellar abundances from the literature. When kinematic information is available, we have assigned the stars to standard Galactic components through Bayesian classification based on Gaussian velocity ellipsoids. As found in previous studies, the [alpha/Fe] ratios of most stars in the dSph galaxies are generally lower than similar metallicity Galactic stars in this extended sample. Our kinematically selected stars confirm this for the Galactic halo, thin-disk, and thick-disk components. There is marginal overlap in the low [alpha/Fe] ratios between dSph stars and Galactic halo stars on extreme retrograde orbits ( V <- 420 km s(-1)), but this is not supported by other element ratios. Other element ratios compared in this paper include r- and s-process abundances, where we find a significant offset in the [Y/Fe] ratios, which results in a large overabundance in [Ba/Y] in most dSph stars compared with Galactic stars. Thus, the chemical signatures of most of the dSph stars are distinct from the stars in each of the kinematic components of the Galaxy. This result rules out continuous merging of low-mass galaxies similar to these dSph satellites during the formation of the Galaxy. However, we do not rule out very early merging of low-mass dwarf galaxies, since up to one-half of the most metal-poor stars ([Fe/H] &LE; - 1.8) have chemistries that are in fair agreement with Galactic halo stars. We also do not rule out merging with higher mass galaxies, although we note that the LMC and the remnants of the Sgr dwarf galaxy are also chemically distinct from the majority of the Galactic halo stars. Formation of the Galaxy's thick disk by heating of an old thin disk during a merger is also not ruled out; however, the Galaxy's thick disk itself cannot be comprised of the remnants from a low-mass ( dSph) dwarf galaxy, nor of a high-mass dwarf galaxy like the LMC or Sgr, because of differences in chemistry. The new and independent environments offered by the dSph galaxies also allow us to examine fundamental assumptions related to the nucleosynthesis of the elements. The metal-poor stars ([Fe/H] &LE; - 1.8) in the dSph galaxies appear to have lower [Ca/Fe] and [Ti/Fe] than [Mg/Fe] ratios, unlike similar metallicity stars in the Galaxy. Predictions from the α-process (α-rich freeze-out) would be consistent with this result if there have been a lack of hypernovae in dSph galaxies. The α-process could also be responsible for the very low Y abundances in the metal-poor stars in dSph's; since [La/Eu] ( and possibly [Ba/Eu]) are consistent with pure r- process results, the low [Y/Eu] suggests a separate r- process site for this light (first-peak) r- process element. We also discuss SNe II rates and yields as other alternatives, however. In stars with higher metallicities ([Fe/H] &GE; -1.8), contributions from the s-process are expected; [( Y, La, and Ba)/Eu] all rise as expected, and yet [Ba/Y] is still much higher in the dSph stars than similar metallicity Galactic stars. This result is consistent with s-process contributions from lower metallicity AGB stars in dSph galaxies, and is in good agreement with the slower chemical evolution expected in the low-mass dSph galaxies relative to the Galaxy, such that the build-up of metals occurs over much longer timescales. Future investigations of nucleosynthetic constraints ( as well as galaxy formation and evolution) will require an examination of many stars within individual dwarf galaxies. Finally, the Na-Ni trend reported in 1997 by Nissen & Schuster is confirmed in Galactic halo stars, but we discuss this in terms of the general nucleosynthesis of neutron-rich elements. We do not confirm that the Na-Ni trend is related to the accretion of dSph galaxies in the Galactic halo.