The assumption on which this calculation is founded is that there is no alteration of the internal energy of the gas in consequence of expansion. If the air when raised in temperature, and at the same time increased in volume, contained less internal energy than when simply heated without alteration of volume, the energy evolved would be available to aid the performance of the work done against external forces, and less heat would be required, or, in the contrary case, more heat would be required, than would be necessary if the internal energy remained unaltered. Thus putting dW for pdv, the work done, e for the internal energy before expansion, and dH for the heat given to the gas, we have obviously the equation JdH = de + dW where de is the change of internal energy due to the alteration of volume, together with the alteration of temperature. If now the temperature be altered without expansion, no external work is done and dW for that case is zero. Let ∂e and ∂H be the energy change and the heat supplied, then in this case J∂H = ∂e + O Thus J(dH － ∂H) = de － ∂e + dW and the assumption is that de = ∂e, so that dW = J(dH － ∂H); that is, dW = J(Cp － Cv), when the rise of temperature is 1° C. and the mass of air is one gramme. This assumption requires justification, and by an experiment of Joule's, which was repeated in a more sensitive form devised by Thomson, it was shown to be a very close approximation to the truth. Joule's experiment is well known: the explanation given above may serve to make clear the nature of the research undertaken later by Thomson and Joule conjointly.