The most general motion of an isolated thermodynamic system K in equilibrium is determined by Landau and Lifshitz through an appeal to the equilibrium condition δS=0. It is argued that this work actually amounts to a demonstration of the internal consistency of the thermodynamic formalism in this case since the problem is soluble by an appeal to prior criteria. We use these to determine all motions consistent with equilibrium in the more general case of a system K which is not isolated. Although the temperature would of necessity be uniform if K were at rest, this is no longer obviously the case when it is moving. To emphasize this point, we first determine the possible relativistic motions of K, restricted to be translations in a fixed direction, and then determine explicitly the distribution of absolute temperature within it. It turns out to be nonuniform only when K is accelerated, and then it varies in a manner consistent with the principle of equivalence.