The possibility of adopting this value of μ was shown by Thomson to depend on whether or not the heat absorbed by a given mass of gas in expanding without alteration of temperature is the equivalent of the work done by the expanding gas against external pressure. The heat H absorbed by the air in expanding from volume V to another volume V' at constant temperature is the integral of Mdv taken from the former volume to the latter. But by the value of M given on p. 121, if W be the integral of pdv, that is the work done by the air in the expansion, ∂W⧸∂t = μH. The equation fulfilled by the gas at constant pressure (the defining equation for t), v = v₀(1 + Et), gives for the integral of pdv, that is W, the equation W = pv₀(1 + Et)log(V'⧸V), so that ∂W⧸∂t = EW⧸(1 + Et). Thus μH = EW⧸(1 + Et). Hence it follows that if μ = E⧸(1 + Et), the value of H will be simply W. Thus Joule's suggested value of μ is only admissible if the work done by the gas in expanding from a given volume to any other is the equivalent of the heat absorbed; or, which is the same thing, if the external work done in compressing the gas from one volume to another is the equivalent of the heat developed. This result naturally suggests the formation of a new scale of thermometry by the adoption of the defining relation T = 1⧸μ, where T denotes temperature. A scale of temperature thus defined is proposed in the paper by Joule and Thomson, "On the Thermal Effects of Fluids in Motion," Part II, which was published in the _Philosophical Transactions_ for June 1854, and is what is now universally known as Thomson's scale of absolute thermodynamic temperature. It can, of course, be made to give 100 as the numerical value of the temperature difference between 0° C. and 100° C. by properly fixing the unit of T. This scale was the natural successor, in the dynamical theory, of one which Thomson had suggested in 1848, and which was founded, according to Carnot's idea, on the condition that a unit of heat should do the same amount of work in descending through each degree. This, as he pointed out, might justly be called an absolute scale, since it would be independent of the physical properties of any substance. In the same sense the scale defined by T = 1⧸μ is truly an absolute scale. The new scale gives a simple expression for the efficiency of a perfect engine working between two physically given temperatures, and assigns the numerical values of these temperatures; for the heat H taken in from the source in the isothermal expansion which forms the first operation of the cycle (p. 120) is Mdv, and, as we have seen, the work done in the cycle is ∂p⧸∂t.dtdv, or μHdt. If we adopt the expression 1⧸T for μ, we may put dT for dt; and we obtain for the work done the expression HdT⧸T. The work done is thus the fraction dT⧸T of the heat taken in, and this is what is properly called the efficiency of the engine for the cycle.