The thermodynamics of several elemental superconductors is computed from isotropic Eliashberg theory formulated on the imaginary frequency axis. A symmary of the available experimental literature is presented and a comparison with theory is given. The small disagreements that are found are all in the direction expected from anisotropy effects. We calculate the effect of a small amount of model anisotropy on the critical temperature, critical field, and high-temperature specific heat from an exact solution of the anisotropic Eliashberg equations. These are the first such results below the critical temperature ; unlike previous analytical work, we include retardation, anisotropy in the mass enhancement, and the effect of the Coulomb repulsion in enhancing anisotropy, all of which are significant. We derive a new formula independent of any model anisotropy for the rate of decrease with impurity lifetime of the critical temperature. Finally we demonstrate how the commonly used formulas of Markowitz and Kadanoff and of Clem may give entirely misleading estimates of the gap anisotropy when used to interpret certain experiments. *Research supported by the Natural Sciences and Engineering Research Council of Canada.
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