In thermodynamics, the second law of thermodynamics quantifies the energetic nature of irreversibility in heat driven transformation processes, particularly in the Carnot cycle. In concise form, the second law, as formulated mathematically in 1862 by German physicist Rudolf Clausius, states that in a cyclical heat-driven process which is in any way possible the following relation will always hold:

$~ \int \frac{dQ}{T} \ge 0 ~$

where dQ is an element of the heat given up by a body to any reservoir of heat during its own changes, heat which it may absorb from a reservoir being here reckoned as negative, and T is the absolute temperature of the body at the moment of giving up this heat. [1] The quantity "dQ/T" is called entropy. The second law, in a general sense, arose out of a need to correct French physicist Sadi Carnot's 1824 view of the concept of "re-establishment of equilibrium in the caloric" as being the central principle of the operation of the steam engine.

Notations
After the 1875 Lectures on the Mechanical Theory of Heat by German mathematician Carl Neumann, indicating, as Clausius did, that the differentials δQ and δW are path dependent (inexact differentials), the symbols đ (d-crossbar) or δ (small delta), the latter used more often in the modern sense, began to be used wherever heat or work differentials were found. Hence, the second law began to be written as:

$~ \int \frac{\delta Q}{T} \ge 0 ~$

Modern formulations
There are a number of different modern formulations of the second law, such as found in the works of German engineer and physicist Kolumban Hutter; examples of which can be found in the further reading section below. [2]

References
1. (a) Clausius, Rudolf. (1862). "On the Application of the Theorem of the Equivalence of Transformations to Interior Work", (pp. 215-250).
(b) Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
(c) Clausius, Rudolf. (1879). The Mechanical Theory of Heat (second edition). London: Macmillan & Co.
2. Kolumban Hutter (Curriculum Vitae) – Alert.Epfl.ch.