In the Carnot cycle, the first operation is an isothermal expansion (AB in Fig. 12), in which the substance increases in volume by dv, and takes in from the source heat of amount Mdv. The second operation is an adiabatic expansion, BC, in which the volume is further increased and the temperature sinks by dt to the temperature of the refrigerator. The third operation is an isothermal compression, CD, until the volume and pressure are such that an adiabatic compression DA will just bring the substance back to the original state. If ∂p⧸∂t be the rate of increase of pressure with temperature when the volume is constant, the step of pressure from one isothermal to the other is ∂p⧸∂t.dt; and thus the area of the closed cycle in the diagram which measures the external work done in the succession of changes is ∂p⧸∂t.dtdv. Now, by the second law, the work done must be a certain fraction of the work-equivalent of the heat, Mdv, taken in from the source. This fraction is independent of the nature of the working substance, but varies with the temperature, and is therefore a function of the temperature. Its ratio to the difference of temperature dt between source and refrigerator was called "Carnot's function," and the determination of this function by experiment was at first perhaps the most important problem of thermodynamics. Denoting it by μ, we have the equation ∂p⧸∂t = μM ... (A) which may be taken as expressing in mathematical language the second law of thermodynamics. M is here so chosen that Mdv is the heat expressed in units of work, so that μ does not involve Joule's equivalent of heat. This equation was given by Carnot: it is here obtained by the dynamical theory which regards the work done as accounted for by disappearance, not transference merely, of heat. The work done in the cycle becomes now μMdtdv, or if H denote Mdv, it is μHdt. The fraction of the heat utilised is thus μdt. This is called the efficiency of the engine for the cycle.