Cycle integral (diagram)
The cycle integral diagram used by James Partington (1924) to explain that, in chemical thermodynamics, the following symbol
Cycle integral (symbol)or ∮ in modern symbol nomenclature, means or denotes "integration round a cycle", such that a function, say internal energy U, can be taken from position A, to point B, along one path (M) and then brought back to its starting position along a second path (N), thus completing a cycle, in each path an integration can be done, the sum of which yields the value of the cycle integral. [3]
In symbols, the mathematical operator ∮, of an integration sign with a circle in it:

 \oint \,

is called the "cycle integral", closed path integral, or circle integral. In mathematics, the symbol is used to mean that the "integration is done around a closed path". [1] In thermodynamics, means that the "integration is done around a cycle", in particular that of the Carnot cycle or heat cycle, and thus seems to take on a peculiar meaning, distinct form that of its usage in standard mathematics. [1]

History
The origin of this symbol introduction remains to be tracked down, particularly because it is often associated with the Clausius inequality, and thus with the summations of differential units of heat dQ going into and out of bodies through the boundary.

In 1901, American mathematician Edwin Wilson (Gibbs' student) is said to have used a small circle below the standard integral symbol to denote integration around a closed curve in his Vector Analysis (1901, 1909) as well as in his Advanced Calculus (1911, 1912).

In 1917, English physicist Arnold Sommerfeld is said to have used the circle integral symbol. [2]

In 1924, English chemist James Partington, in his Chemical Thermodynamics, goes into some detail into describing the the following symbol:

Cycle integral (symbol)

which he says "denotes integration round a cycle". To explain the idea of the “cycle integration”, Partington begins with the principle of conservation of energy, which according to him states that

“The change of intrinsic energy of a system undergoing any change depends solely on the initial and final states of the system and is independent of the manner in which the change from one state to the other is effected.”

The definition of intrinsic energy, according to Partington, was first described by William Thomson in 1851. [4] Intrinsic energy or rather internal energy is one of the five main characteristic functions. Characteristic functions in which the change in the function depends solely on the initial and final states of the system and is independent of the manner in which the change from one state to the other is effected are called “path independent.” Those functions that depend on the path followed are called “path dependent.”

Hence, in diagrammatic terms, shown above, the change of the system from an initial state A to a final state B can be carried out along path M, according to which the change in intrinsic energy will be ΔUM. If then the change in the system is reversed by bringing the system from state B to the original state A, but via path N, then the change in intrinsic energy U will be – ΔUN.

The two changes, according to Partington, constitute what is called “cycle”. Because U is a path dependent function and in this graphical cycle has returned to its original position on the graph, the total change of U is zero or:

      \Delta U_M - \Delta U_N = 0 \,

Partington goes on to state that the same method may be applied to any other possible direct change instead of M, but the proof depends on the existence of at least one path (N) along which the change from A to B can be reversed. If this does not exist, the principle can only be inferred. He gives the example of radioactive decay changes as one such process or change in the state of a system, in which reversible paths do not exist.

Partington later defines the circle integral for the Clausius equality or reversible version of the second law as follows: [3]

circle integral (symbol)
=
 \sum \frac{Q}{T} \,

Partington, supposedly, gives a fuller account of this in his 1920 Higher Mathematics for Chemical Students. [5]

References
1. Miller, Jeff. (2009). “Earliest Uses of Symbols of Calculus”, Jeff560.tripod.com.
2. Sommerfeld, Arnold. (1917). "Die Drudesche Dispersionstheorie vom Standpunkte des Bohrschen Modelles und die Konstitution von H2, O2 und N2", Annalen der Physick.
3. Partington, James R. (1924). Chemical Thermodynamics: An Introduction to General Thermodynamics and its Applications to Chemistry (pg. 14). D. Van Nostrand.
4. Thomson, William. (1851). “Article”, Trans. R.S. Edin. Date.
5. Partington, James. (1920). Higher Mathematics for Chemical Students (§§ 55, 115). London.

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