Mayer of Heilbronn had endeavoured to determine the dynamical equivalent of heat in 1842, by calculating from the knowledge available at the time of the two specific heats of air--the specific heat at constant pressure and the specific heat at constant volume--the heat value of the work spent in compressing air from a given volume to a smaller one. The principle of this determination is easily understood, but it involves an assumption that is not always clearly perceived. Let the air be imagined confined in a cylinder closed by a frictionless piston, which is kept from moving out under the air pressure by force applied from without. Let heat be given to the air so as to raise its temperature, while the piston moves out so as to keep the pressure constant. If the pressure be p and the increase of volume be dv, the work done against the external force is pdv. Let the rise of temperature be one degree of the Centigrade scale, and the mass of air be one gramme, the heat given to the gas is the specific heat Cp of the gas at constant pressure, for there is only slight variation of specific heat with temperature. But if the piston had been fixed the heat required for the same rise of temperature would have been Cv, the specific heat at constant volume. Now Mayer assumed that the excess of the specific heat Cp above Cv was the thermal equivalent of the work pdv done in the former case. Thus he obtained the equation J(Cp － Cv) = pdv, where J denotes the dynamical equivalent of heat and Cp, Cv are taken in thermal units. But if a be the coefficient of expansion of the air under constant pressure (that is 1⧸273), and v₀ be the volume of the air at 0° C., we have dv = av₀, so that J(Cp － Cv) = apv₀. Now if p be one atmosphere, say 1.014 × 10^6 dynes per square centimetre, and the temperature be the freezing point of water, the volume of a gramme of air is 1⧸.001293 in cubic centimetres. Hence J(Cp － Cv) = (1.014 × 10^6)⧸(273 × .001293) from which, if Cp － Cv is known, the value of J can be found. In Mayer's time the difference of the specific heats of air was imperfectly known, and so J could not be found with anything like accuracy. From Regnault's experiments on the specific heat at constant pressure, and from the known ratio of the specific heats as deduced from the velocity of sound combined with Regnault's result, the value of Cp － Cv may be taken as .0686. Thus J works out to 42.2 × 10^6, in ergs per calorie, which is not far from the true value. Mayer obtained a result equivalent to 36.5 × 10^6 ergs per calorie.