All the second-order differential relationships between three thermodynamic quantities that are variables of thermodynamic Maxwell equations are derived. These relationships can be used for any pure substance or mixture of fixed chemical composition and for any equation of state. The obtained equalities, which contain entropy as a variable, can be considered as the caloric equations of state. The validity of certain equalities is confirmed by the use of various mathematical methods to reach identical results at the critical point of a pure substance. Possible applications of the obtained second-order differential equations are considered.