way, from atmospheric air. In these circumstances the author considers that additional weight is given to the result taken above from the figure for atmospheric air; since it will hardly be questioned that hydrogen is at least as near as atmospheric air to the condition of a perfect gas, and therefore in drawing conclusions relative to that condition the behaviour of the one is as much to be noted as that of the other. It appears therefore to be the best course (so long as fresh observations have not furnished a more satisfactory starting point for wider conclusions) to adhere to the figure which, under the pressure of one atmosphere, has been found to agree almost exactly for atmospheric air and for hydrogen; and thus to write: a = 0·003665 = 273 1 If we denote the reciprocal by a we may also write the R is here a constant which depends on the nature of the gas and is inversely proportional to its specific gravity*. T represents the temperature, provided this is measured not from the freezing point, but from a zero point lying a degrees lower. The temperature thus measured from -a we shall term the Absolute Temperature, a name which will be more * For R is proportional to the volume of a unit of weight of the gas at standard pressure and temperature; and is therefore inversely proportional to the weight of a unit of volume, i.e. to the specific gravity. (Translator.) fully explained further on. Taking the value of a given in equation (2) we obtain § 2. Approximate Principle as to Heat absorbed by Gases. In an experiment of Gay Lussac's, a vessel filled with air was put in communication with an exhausted receiver of equal size, so that half the air from the one passed over into the other. On measuring the temperature of each half, and comparing it with the original temperature, he found that the air which had passed over had become heated, and the air which remained behind had become cooled, to exactly the same degree; so that the mean temperature was the same after the expansion as before. He thus proved that in this kind of expansion, in which no external work was done, no loss of heat took place. Joule, and after him Regnault, carried out similar experiments with greater care, and both were led to the same result. The principle here involved may also be deduced, without reference to special experiments, from certain properties of gases otherwise ascertained, and its accuracy may thus be checked. Gases shew so marked a regularity in their behaviour (especially in the relation between volume, pressure, and temperature, expressed by the law of Mariotte and Gay Lussac), that we are thereby led to the supposition that the mutual action between the molecules, which goes on in the interior of solid and liquid bodies, is absent in the case of gases; so that heat, which in the former cases has to overcome the internal resistances, as well as the external pressure, in order to produce expansion, in the case of gases has to do with external pressure alone. If this be so, then, if a gas expands at constant temperature, only so much heat can thereby be absorbed as is required for doing the external work. Again, we cannot suppose that the total amount of heat actually existing in the body is greater after it has expanded at constant temperature than before. On these assumptions we obtain the following principle: a permanent gas, if it expands at a constant temperature, absorbs only so much heat as is required for the external work which it performs in so doing. We cannot of course give to this principle any greater validity than that of the principles from which it springs, but must rather suppose that for any given gas it is true to the same extent only in which the law of Mariotte and Gay Lussac is true. It is only for perfect gases that its absolute accuracy may be assumed. It is on this understanding that the author brought this principle into application, combined it as an approximate assumption with the two main principles of the Mechanical Theory of Heat, and used it for establishing more extended conclusions. More recently Mr, now Sir William Thomson, who at first did not agree with one of the conclusions so deduced, undertook in conjunction with Joule to test experimentally the accuracy of the principle*; and for this purpose instituted with great care a series of skilfully conceived experiments, which, on account of their importance, will be more fully and exactly discussed further on. These have completely confirmed the truth not only of the general principle, but also of the remark added by the author as to its degree of exactness. In the permanent gases on which they experimented, viz. atmospheric air and hydrogen, the principle was found so nearly exact that the deviations might for the purpose of most calculations be neglected; while in the non-permanent gas selected for experiment (Carbonic Acid) somewhat greater deviations were observed, exactly as might have been expected from the behaviour of that gas in other respects. After this we may with the less scruple apply the principle, as being exact for actually existing gases in the same degree as the law of Mariotte and Gay Lussac, and absolutely exact in the case of perfect gases. § 3. On the Form which the Equation expressing the first main Principle assumes, in the case of perfect gases. We now return to equation (IV), viz. : dQ = dU + pdv, in order to apply it to the case of a perfect gas, of which we assume as before one unit of weight to be given. * Phil. Trans. 1853, 1854, 1862. The condition of the gas is completely determined, when its temperature and volume are known; or it may be determined by its temperature and pressure, or by its volume and pressure. We will at present choose the first-named quantities, temperature and volume, to determine the condition, and accordingly treat T and v as the independent variables, on which all other quantities relating to the condition of the gas depend. If then we regard the energy U of the gas as being also a function of these two variables, we may write This equation, which in the above form holds not only for a gas, but for any body whose condition is determined by its temperature and volume, may be considerably simplified for gaseous bodies, on account of their peculiar properties. The quantity of heat, which a gas must absorb in expanding at constant temperature through a volume dv, is dQ dv generally denoted by dv. As by the approximate assumption of the last Section this heat is equal to the work done in the expansion, which is expressed by pdv, we have the equation: dQ dv dQ dv dv = pdv, or =p. 1 Hence we conclude that in a perfect gas the energy U is independent of the volume, and can only be a function of the temperature. dU If in equation (8) we put =0, and substitute for dv dU dT the symbol C,, it becomes dQ= C ̧dT+ pdv. (10). From the form of this equation we see that C, denotes the Specific Heat of the Gas at constant volume, since CdT expresses the quantity of heat which must be imparted to the gas in order to heat it from T to T+dT, when dv is equal to zero. As this Specific Heat = i.e. is the differ d U dT" ential coefficient with respect to temperature of a function of the temperature only, it can itself also be only a function of temperature. = În equation (10) all the three quantities T, v, and p are found; but since by equation (6) pv Rt, it is easy to eliminate one of them; and by eliminating each in succession we obtain three different forms of the equation. Eliminating p we obtain, T If we substitute this value of dv in equation (10), and then combine the two terms of the equation which contain dT, we obtain ᎡᎢ p ....... Lastly, to eliminate T, we obtain from equation (6), by differentiation, |