Abstract The properties of complete equations of state are well known, but those of incomplete equations of state have received little attention. This paper uses thermodynamic identities to establish the relationships between and the properties of incomplete equations of state. The thermodynamic identities of the Gibbs' equation for temperature and pressure are transformed into linear partial differential equations that are integrated formally by the method of characteristics. The formal solutions of these equations define the relationships between incomplete equations of state and hence demonstrate the structure of classical thermodynamics. The method of integration is convenient for calculating one incomplete equation of state from the other and for specifying the thermodynamic data necessary to make the calculation. These calculations are applicable to shock wave studies; they are also important to the problem of characterizing an incomplete unreactive thermodynamic system, since any two incomplete equations of state are equivalent to a complete equation of state for a nonreactive fluid.