developed during this cycle of operations, which, as no work is performed, must be wholly expended in agitating the fluid, and reproducing by friction the heat consumed by the free expansion represented by the curve BC, which heat is measured by the indefinitely-prolonged area MCBL, which area is therefore equal to the area ABCD. Subtracting from each of these equal areas the common area BUC, and adding to each of the equal remainders the indefinitely-prolonged area LUDN, we form the areas MCDN, LBADN, which are consequently equal. Q. E. D.

51. Of the total heat of evaporation.

The symbolical expression of the preceding proposition is formed in the following manner. The area LBA DN represents the total heat of evaporation, at the temperature 7,, from the temperature 72, and is composed of two parts, as follows:

of which the first is the heat necessary to raise the liquid, whose specific heat is K1, from 72 to 71, and the second is the latent heat of evaporation

at T1

Let v be the volume of unity of weight of the vapour at the pressure P. and temperature of saturation T2; draw the ordinate v E, meeting DF in E, through which point draw the indefinitely-prolonged curve of no transmission ER: then is the area MCDN divided into two parts, as follows:

MCDN MCER+REDN= (© K„dr+Ly. (83.)

Το

in which equation 7, denotes the temperature corresponding to the point C on the curve of free expansion, and K, the specific heat of the vapour, at the constant pressure P, when above the temperature of saturation; so that the first term represents the heat abstracted in lowering the temperature of the vapour from 7 to the temperature of saturation 79, at the constant pressure P2; and the second term, the latent heat of evaporation at 7 abstracted during the liquefaction.

By equating the formulæ (82) and (83), the following equation is obtained:

which is the symbolical solution of Proposition XIX., and shows a relation between the total heat of evaporation of a fluid, the free expansion of its vapour, and the specific heat of that vapour at constant pressure.

52. Approximate law for a vapour which is a perfect gas.

If the vapour of the fluid in question be a perfect gas, and of very great volume as compared with the fluid in the liquid state, the curve BC will be nearly a hyperbola, and will nearly coincide with the isothermal curve of the higher temperature 71, to which, consequently, Te will be nearly equal; and the following equation will be approximately true:

which, when the difference between the higher and lower temperatures diminishes indefinitely, is reduced to the following:

COROLLARY.-THEOREM.

When a vapour is a perfect gas, and very bulky

as compared with its liquid, the rate of increase of the total heat of evaporation with temperature is nearly equal to the specific heat of the vapour at constant pressure.

This was demonstrated by a different process, in a paper read to the Royal Society of Edinburgh in 1850. It has not yet been ascertained, however, whether any vapour at saturation approaches sufficiently near to the condition of perfect gas to render the theorem applicable.

53. Concluding Remarks.

In conclusion, it may be observed, that the theory of the expansive action of heat embodied in the propositions of this paper contains but one principle of hypothetical origin-viz., Proposition XII., according to which the actual heat present in a substance is simply proportional to its temperature measured from a certain point of absolute cold, and multiplied by a specific constant; and that although existing experimental data may not be adequate to verify this principle precisely, they are still sufficient to prove that it is near enough to the truth for all purposes connected with thermodynamic engines, and to afford a strong probability that it is an exact physical law.

XXI.-ON FORMULÆ FOR THE MAXIMUM PRESSURE
AND LATENT HEAT OF VAPOURS.*

1. It is natural to regard the pressure which a liquid or solid and its vapour maintain when in contact with each other and in equilibrio, as the result of an expansive elasticity in the vapour, balanced by an attractive force which tends to condense it on the surface of the liquid or solid, and which is very intense at that surface, but inappreciable at all sensible distances from it. According to this view, every solid or liquid substance is enveloped by an atmosphere of its own vapour, whose density close to the surface is very great, and diminishes at first very rapidly in receding from the surface; but at appreciable distances from the surface is sensibly uniform, being a function of the temperature and of the attractive force in question.

2. Many years since I investigated mathematically the consequences of this supposition, and arrived at the conclusion, that although it is impossible to deduce from it, in the existing condition of our knowledge of the laws of molecular forces, the exact nature of the relation between the temperature and the maximum pressure of a vapour, yet that if the hypothesis be true, it is probable that an approximate formula for the logarithm of that pressure for any substance will be found, by subtracting from a constant quantity, a converging series in terms of the powers of the reciprocal of the absolute temperature, the constant and the coefficients of the series being determined for each substance from experimental data. Such a formula is represented by

where P denotes the pressure, the absolute temperature, that is, the temperature as measured from the absolute zero of a perfect gas-thermometer, A the constant term, and B, C, &c., the coefficients of the converging series.

3. On applying this formula to M. Regnault's experiments on the pressure of steam, it was found that the first three terms were sufficient to

* Read before the British Association at Liverpool, September, 1854, and published in the Philosophical Magazine, December, 1854.

represent the results of these experiments with minute accuracy throughout their whole extent; that is to say, between the temperatures of

and between the pressures of of an atmosphere, and 82 atmospheres. Formulæ of three terms were also found to represent the results of Dr. Ure's experiments on the vapours of alcohol and ether, and formulæ of two terms those of his experiments on the vapours of turpentine and petroleum, as closely as could be expected from the degree of precision of the experiments. A formula of two terms was found to represent accurately the results of M. Regnault's experiments on the vapour of mercury.

4. These formula, with a comparison between their results and those of the experiments referred to, were published in the Edinburgh New Philosophical Journal for July, 1849, in a paper the substance of which is summed up at its conclusion in the following proposition (See p. 1):

If the maximum elasticity of any vapour in contact with its liquid be ascertained for three points on the scale of the air-thermometer, then the constants of an equation of the form

may be determined, which equation will give, for that vapour, with an accuracy limited only by the errors of observation, the relation between the temperature (7), measured from the absolute zero, and the maximum elasticity (P), at all temperatures between those three points, and for a considerable range beyond them.

5. In the case of water and mercury, the precision of the experimental data left nothing to be desired. I have, however, in the table of constants at the end of this paper, so far modified the coefficients for water and mercury as to adapt them to a position of the absolute zero (274° Centigrade, or 493°2 Fahrenheit below the temperature of melting ice), which is probably nearer the truth than that employed in the original paper, which was six-tenths of a Centigrade degree lower. This modification, however, produces no practically appreciable alteration in the numerical results of the formulæ.

6. It was otherwise with respect to the other fluids mentioned, for which the experimental data were deficient in precision, so that the values of the constants could only be regarded as provisional.

7. A summary published in the Comptes Rendus for the 14th of August, 1854, of the extensive and accurate experiments of M. Regnault on the p. 269.

*

See Phil. Mag., Series 4, Vol. VIII.,

elasticities of the vapours of ether, sulphuret of carbon, alcohol, chloroform, and essence of turpentine, has now supplied the means of obtaining formulæ, founded on data as precise as it is at present practicable to obtain, for the maximum pressures of these vapours.

A synopsis of these formulæ, and of the constants contained in them, is annexed to this paper. The constants, as given in the table, are suited for millimètres of mercury as the measure of pressures, and for the scale of the Centigrade thermometer; but logarithms are given, by adding which to them they can be easily adapted to other scales.

The limited time which has elapsed since the publication of M. Regnault's experiments prevents my being yet able to bring the details of the investigation of the formulæ, and of the comparison of their results with those of experiment, into a shape suited for publication; but I shall here add some remarks on their degree of accuracy and the extent of their applicability.

8. M. Regnault explains, that his experiments were made by two methods; at low temperatures, by determining the pressure of the vapour in vacuo; at high temperatures, by determining the boiling-point under the pressure of an artificial atmosphere. For each fluid the pressures determined by both those methods were compared throughout a certain series of intermediate temperatures.

For all fluids in a state of absolute purity, the results of those two methods agreed exactly (as M. Regnault had previously shown to be the case for water).

The presence, however, of a very minute quantity of a foreign substance in the liquid under experiment was sufficient to make the pressure of the vapour in vacuo considerably greater than the pressure of ebullition at a given temperature; and it would appear, also, that a slight degree of impurity affects the accuracy even of the latter method of observation, although by far the more accurate of the two when they disagree.

9. The degree of precision with which it has been found possible to ` represent the results of the experiments by means of the formulæ, corresponds in a remarkable manner with the degree of purity in which, according to M. Regnault, the liquid can be obtained.

Sulphuret of Carbon, M. Regnault states, can easily be obtained perfectly pure. For this fluid, the agreement of the pressures computed by the formula with those determined by experiment throughout the whole range of temperature from 16° Centigrade to + 136°, is almost as close as in the case of steam.

-

Ether and Alcohol are less easy to be obtained perfectly pure. The discrepancies between calculation and experiment in these cases, though still small, are greater than for sulphuret of carbon.

For ether the formula may be considered as practically correct through