[17] Before his death (in 1832) Carnot had obtained a clear perception of the true state of the case, and of the complete doctrine of the conservatism of energy. [See extracts from Carnot's unpublished writings appended, with a biography, to the reprinted Memoir, by his younger brother, Hippolyte Carnot.] [18] This equation for the porous plug experiment may be established in the following manner, which forms a good example of Thomson's second definition of absolute temperature. Take pressure and volume of the gas on the supply side of the plug as p + dp and v, and on the delivery side as p and v + dv, so that dp and dv are positive. The net work done in forcing the gas through the plug = (p + dp)v － p(v + dv) = － pdv + vdp. Let a heating effect result so that temperature is changed from T to T + ∂T. Let this be annulled by abstraction of heat Cp∂T at constant pressure. (Cp = sp. heat press. const.) [It is to be understood that dv is the total expansion existing, after this abstraction of heat.] The energy e of the fluid has been increased by de = － pdv + vdp － Cp∂T. Now, since the original temperature has been restored, the same expansion dv if imposed isothermally would involve the same energy change de; but in that case heat dH (dynamical) would be absorbed, and work pdv would be done by the gas. Hence de = dH － pdv. This, with the former value of de, gives dH = vdp － Cp∂T. Thomson's work-ratio is thus pdv⧸(vdp － Cp∂T). Now suppose dp imposed without change of volume, and dT to be the resulting temperature change. The temperature and pressure ratios are dT⧸T, dp⧸p. Thus dT⧸T = dp dv⧸(vdp － Cp∂T), or (v⧸T)(dT⧸dv) = 1⧸[1 － (Cp⧸v)(∂T⧸dp)] which is Thomson's equation. The minus sign on the right arises from a heating effect having been taken here as the normal case. If the temperature T is restored by removing the heat at constant volume, a similar process gives the equation (v⧸T)(dT⧸dv) = [1 + (∂T⧸∂p)(∂T⧸dp)]⧸[1 － (Cv⧸v)(∂T⧸dp)] where dp is the change of pressure before the restoration of the temperature T, and ∂T⧸∂p is the rate of variation of T with p, volume constant. [19] "On a Universal Tendency in Nature to Dissipation of Energy," _Proc. R.S.E._, 1852, and _Phil. Mag._, Oct., 1852.