In thermodynamics, thermodynamic function (TR:11), symbol Phi Φ, is a function, increase or diminution of which indicates whether heat is entering or leaving a system.

Overview
In 1854,
Scottish engineer-physicist William Rankine conceived of the thermodynamics function. [1]

In 1876, James Maxwell was referring to Rankine's thermodynamic function as follows:
(Ѻ)

$\Phi = \int \frac{dq}{t}$

This was later shown to be near equivalent to German physicist Rudolf Clausius’ conception of entropy (1865). [2]

The reason, supposedly, that Rankine’s thermodynamic function received so little attention is that it is tangled up with his hypothesis of molecular vortices.

Quotes
The following are related quotes:

“The function by which the absolute temperature is multiplied in calculating the conversion of energy between the mechanical and the thermic forms, is the variation of what the author has called the ‘metamorphic function’, being on term of the ‘thermodynamic function’, which corresponds to what Clausius calls entropie.”
William Rankine (1865), “On the Second Law of Thermodynamics” [4]

“The mathematical part of the theory was improved by the introduction of the scalar value which Rankine called the ‘thermodynamic function’ and Clausius the ‘entropy’ of a body, a variable quantity, momentary increase or decrease of which indicates (in a reversible physico-chemical transformation) whether heat is leaving or entering the body at that moment, irrespective of its temperature or previous condition.”
Fielding Garrison (1909), “Josiah Willard Gibbs and his Relation to Modern Science” [3]

“This proof is consistent with the basic fact that dU is a perfect differential (U, the internal energy, is a state function and therefore independent of the course of the change). He then goes on to develop his ideas on what he calls the ‘thermodynamic function’ that he designates by the letter F, and later by the Greek letter cp. The value of the thermodynamic function remains constant over an adiabatic, just as the temperature is constant for an isothermal. Rankine's thermodynamic function is close to Clausius's entropy, not identified until eleven years later. There is, however, this important difference. As Keith Hutchison pointed out, Rankine did not recognize the degradation, or dissipation of energy.”
Donald Cardwell (c.1980), The Development of Science and Technology in the Nineteenth-Century Britain (pg. iii)

“Rankine named his state-function the ‘thermodynamic function’ in 1854. But it was already present in his papers read on Feb 2, 1850. As Hutchison explains ‘all the involved calculation had been done by 1850, though Ranking still lacked a motive to single out the function for specific attention’ (Hutchison, 1981). This was 15-years before Clausius’ claim that entropy increases.”
— Timothy Lyons (2013), Recent Themes in the Philosophy of Science (pg. 87) (Ѻ)

Carnot function
Entropy (etymology)

References
1. (a) Rankine, William. (1854). “On the Geometrical Representation of the Expansive Action of Heat, and the Theory of Thermodynamic Engines” (Ѻ) , Read before the Royal Society of London on Jan 19; in: Philosophical Transactions of the Royal Society, 144:115-76 (thermodynamic function, pg. 126); in: Miscellaneous Scientific Papers of William Rankine (editor: W.J. Millar) (pgs. 339-409). C. Griffin and Co., 1881.
(b) Rankine, William. (1859). A Manual of the Steam Engine and Other Prime Movers (pg. 312). London: Charles Griffin & Co.
(c) Maxwell, James C. (1878). “
Tait’s ‘Thermodynamics’ (I)”, (pgs. 257-59). Nature, Jan. 31.
(d) Maxwell, James C. (1878). “Tait’s ‘Thermodynamics’ (II)”, (pgs. 278-81). Nature, Feb. 07.
2. (a) Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies. (Google Books). London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
(b) Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids" (pgs. 2-3), Transactions of the Connecticut Academy, I. pp. 309-342, April-May.
(c) Webb, J. Burkitt. (1887). “Rankine’s Thermodynamic Function Φ” in: Proceedings for the American Association for the Advancement of Science (pgs. 107-10).
(d) “Maxwell’s attempt to incorporate Rankine’s function into his account of the two laws of thermodynamics”, in Maxwell on Heat and Statistical Mechanics (pg. 217), Lehigh University Press, 1995.
(e) Crosbie, Smith. (1998). The Science of Energy (pg. 264). University of Chicago Press.
3. Garrison, Fielding H. (1909). “Josiah Willard Gibbs and his Relation to Modern Science, Parts I-IV” (pdf) (pg. 476), Popular Science Monthly, Part I: 74(27):470-84, May; Part II: 74:551-61, Jun; Part III: 75:41-48, Jul; Part IV: 75:191-203, Aug.
4. Rankine, William. (1865). “On the Second Law of Thermodynamics”, Read before the British Association for the Advancement of Science, at Birmingham, Sep; in: Philosophical Magazine, Oct; in: Miscellaneous Scientific Papers (pgs. 427-31; quote, pg. 430). Charles Griffin and Co, 1881.