This is clearly true; for dvdp is the area of a cycle like AB'C'D, represented in Fig. 12, for which an amount of heat H is taken in, though not in this case strictly at one temperature. And clearly, since in Fig. 12 the change from B' to B is adiabatic, H is the heat which would have to be taken in for the isothermal change AB in the Carnot cycle ABCD, which has the same area as AB'C'D. Thus the efficiency of the cycle is dvdp⧸H, and this by the former definition is dT⧸T. Or we may regard the matter thus:--The amount of heat H which corresponds to an infinitesimal expansion dv may be used in equation (A) whether the expansion is isothermal or not, if we take T as the average temperature of the expansion. Hence we have dp⧸dT = H⧸(dv.T), that is, dT⧸T = dpdv⧸H. The theorem on p. 128 is obtained by what is virtually this process. COMPARISON OF ABSOLUTE SCALE WITH SCALE OF AIR THERMOMETER The comparison which Joule and Thomson carried out of the absolute thermodynamic scale with the scale of the constant pressure gas thermometer has already been referred to, and it has been shown that the two scales would exactly agree, that is, absolute temperature would be simply proportional to the volume of the gas in a gas thermometer kept at the temperature to be measured, if the internal energy of the gas were not altered by an alteration of volume without alteration of temperature, that is, if the de － ∂e of p. 107 above were zero. Joule tested whether this was the case by immersing two vessels, connected by a tube which could be opened or closed by a stopcock, in the water of a calorimeter, ascertaining the temperature with a very sensitive thermometer, and then allowing air which had already been compressed into one of the vessels to flow into the other, which was initially empty. It was found that no alteration of temperature of the water of the calorimeter that could be observed was produced. But the volume of the air had been doubled by the process, and if any sensible alteration of internal energy had taken place it would have shown itself by an elevation or a lowering of the temperature of the water, according as the energy had been diminished or increased.