The first English presentation of the so-called Caratheodory theorem, of Greek mathematician Constantin Caratheodory's 1908 mathematical derivation of the second law or of the proof of the existence of a state function of quantities of heat, as found in chapter form given by Indian-born American astrophysicist Subrahmanyan Chandrasekhar in 1939. [4]
In mathematical thermodynamics, Caratheodory’s theorem, Caratheodory’s theory, or theorem of Caratheodory, is a sort of mathematical proof that the entropy function exists, which states that there exists an integrating factor that makes the equation of an adiabatic process exact. [1]

A more detailed statement is: [2]

“If a Pfaffian expression has the property that, in every neighborhood of a point P, there are points which cannot be connected to P along curves which satisfy the Pfaffian equation, dQ = 0, then the Pfaffian expression must admit an integrating denominator.”

History
The so-called theory of Caratheodory, as it was called was presented by Greek mathematician Constantin Caratheodory in his 1908 "Studies in the Foundation of Thermodynamics". [3]

The subject of the Caratheodory’s theorem, as it has since been called in thermodynamics, was first presented in English, in detail, in Indian-born American astrophysicist Subrahmanyan Chandrasekhar’s 1939 book An Introduction to the Study of Stellar Structure, chapter one, where he presents the laws of thermodynamics following what he calls “Caratheodory’s axiomatic standpoint” [4]

Soon thereafter, Caratheodory's mathematical version of the second law began to make its way into the various thermodynamics textbooks.

References
1. Achuthan, M. (2004). Engineering Thermodynamics (pg. 416; see: Appendix C). PHI Learning.
2. Kirkwood, John G. and Oppenheim, Irwin. (1961). Chemical Thermodynamics (pg. 36). McGraw-Hill.
3. Caratheodory, Constantin. (1908). Studies in the Foundation of Thermodynamics (Untersuchungen uber die Grundlagen der Thermodynamik). Bonn; published in: Math. Ann., 67: 355-386, 1909.
4. (a) Chandrasekhar, Subrahmanyan. (1939). An Introduction to the Study of Stellar Structure (ch. 1). University of Chicago Press.
(b) Subrahmanyan Chandrasekhar – Wikipedia.