or in the case of atmospheric air by. The specific heat

is found by dividing the former quantity by the latter, or,

should

ν

In the first of these two equations we may give to c', its value as found by Regnault, 02375; the equation then becomes

In the second we may put for c',, according to (34), the value 0.1684, and for c, the expression given in (35); whence we have

§ 7. Numerical Calculation of the Specific Heat at constant Volume.

The formulæ developed in the last section have been applied by the author to calculate from the values which Regnault has determined by his researches for the Specific Heat at constant Pressure of a large number of gases and vapours, the corresponding values of the Specific Heat at constant Volume. In so doing he has in some sort recalculated one of the two series of numbers given by Regnault himself; who has expressed the Specific Heat at constant Pressure in two different ways, and has brought together the resulting numbers in two series, one of which is superscribed 'en poids,' and the other 'en volume.' The first series contains the values which result, if the gases in question are compared weight by weight with water, in relation to the quantity of heat required to warm them through 1o; in other words, the values of the quantities denoted above by cp. The numbers in the second series are simply obtained from those

in the first by multiplying them by the corresponding specific gravity, i. e. they are the values of the product cd.

These latter numbers were no doubt those most easily calculated from the observed values of c,; but their signification is somewhat complicated. With them the quantity of heat has for its unit the ordinary unit of heat, whilst the volume to which they refer is that which a unit-weight of atmospheric air assumes, when under the same temperature and pressure as the gas under consideration. The tediousness of the verbal description thus required makes the numbers troublesome to understand and to apply; moreover this mode of expressing the Specific Heat of gases has been used, so far as the author knows, by no previous writer. In considering gases with reference to volume, it has in all other cases been customary to compare the quantity of heat, which a given gas requires to raise its temperature through 1o, with the quantity of heat which an equal volume of atmospheric air requires under the same conditions for the same purpose, or, as briefly expressed above, by comparing the gas, volume for volume, with air. The numbers thus obtained are remarkable for their simplicity, and allow the laws which hold as to the specific heats of the gas to be treated with special clearness.

It will therefore, the author believes, be found an advantage that he has calculated, from the values given by Regnault under the heading 'en volume' for the product cd, the values of the quantity y,, defined above. All that was required for this, by (38), was to divide the values of cd by 0.2375.

He has further calculated the values of c, and y,; calculations which by equations (35) and (39) could be very simply performed, by taking from the values of the product cd the number 0.0691, and dividing the remainder by d, or by 01684, respectively.

The numerical values thus calculated are brought together in the annexed table, in which the different columns have the following signification :

Column I. gives the name of the gas.

Column II. gives the Chemical composition, and this expressed in such a way that the diminution of volume pro

duced by the combination can be immediately observed. For in each case those volumes of the simple gas are given, which must combine in order to give Two Volumes of the compound gas. Thus we assume for Carbon, as a gas, such an hypothetical volume as we must assume, in order to say that one volume of Carbon unites with one volume of Oxygen to make Carbonic Oxide, or with two volumes to make Carbonic acid. Again, when, e.g. Alcohol is denoted in the Table by CHO, this means that two volumes of the hypothetical carbon gas, six volumes of Hydrogen, and one volume of Oxygen, make up together two volumes of Alcoholic vapour. For sulphur-gas the specific gravity used to determine its volume is that found by Sainte-Claire Deville and Troost for very high temperatures, viz. 2.23. In the five last combinations in the Table, which contain Silicon, Phosphorus, Arsenic, Titanium, and Tin, these simple elements are denoted by their ordinary chemical signs, without reference to their volumes in the gaseous condition, because the gaseous volumes of these elements are partly still unknown, partly hampered with certain irregularities not yet thoroughly cleared

up.

Column III. gives the Density of the gas, using the values given by Regnault.

Column IV. gives the Specific Heat at constant Pressure as compared, weight for weight, with water, or in other words referred to a unit-weight of the gas and expressed in ordinary units of heat. These are the numbers given by Regnault under the heading 'en poids.'

Column V. gives the Specific Heat at constant Pressure compared, volume for volume, with air, calculated by dividing by 0.2375 the numbers given by Regnault under the heading 'en volume.'

Column VI. gives the Specific Heat at constant Volume compared, weight for weight, with water, calculated by equation (35).

Column VII. gives the Specific Heat at constant Volume compared, volume for volume, with air, calculated by equation

§ 8. Integration of the Differential Equations which express the first main Principle in the case of Gases.

The differential equations deduced in sections 3 and 4, which in various forms express the first main principle of the Mechanical Theory of Heat in the case of gases, are not immediately integrable, as can be seen by inspection; and must therefore be treated after the method developed in § 3 of the Introduction. In other words, the integration becomes possible as soon as we subject the variables occurring in the equation to some one condition, thus determining the path of the change of condition of the body. We shall here give only two very simple examples of the process, the results of which are important for our farther investigations.

Example 1. The gas changes its volume at Constant Pressure, and the quantity of heat required for such change is known.

In this case we select from the above equations one which contains p and v as independent variables, e.g. the last of Equations (15), which is

As the pressure p is to be constant, we put p=P1, and = 0; the equation then becomes

dp

which gives on integration (if we call v, the original value of v)

Example 2. The gas changes its volume at Constant Temperature, and the quantity of heat required for such change is known.

In this case we select an equation which contains T and v as independent variables, e.g. Equation (11), which is