Caratheodory’s Theorem on the Second Law of Thermodynamics

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It is shown in this paper that if in every sufficiently small neighborhood of a point $x^0 $ there are points which cannot be connected with $x^0 $ by piecewise smooth integral curves of a Pfaffian equation, then in a neighborhood of $x^0 $ there is an integral surface of the equation passing through $x^0 $.This result is a generalization of a theorem proved by Caratheodory in his axiomatic treatment of thermodynamics. The proof is new.