Heat engines and heat pumps at positive and negative absolute temperatures

Inequalities for efficiencies of heat engines and for the coefficients of performance of heat pumps are obtained for positive and negative absolute temperatures. There are strong analogies between heat engines at negative (positive) temperatures and heat pumps at positive (negative) temperatures, Minor improvements are shown to be desirable in the Kelvin-Planck formulation of the second law as amended for negative temperatures. The Clausius formulation is also discussed and the term perpetuum mobile of a third kind is proposed for a class of realisable physical situations.