necessary to combine equation (11) with formula (6), and the following simple formula is obtained :

I shall use equation (12) in order to verify the calculation made of T' by means of equations (9).

To this end I introduce in formula (6) the values T1 =283° and p'=751.63 millims., and from it deduce

"=0.527296 cub. m., p'"-396-33.

Having ", we introduce its value into equation (12); and putting C, 0.216, I find

T1-T'=3°.56,

which is just the value deduced from formulæ (9).

The identity of the numbers evidently depends on the value of C; but the value which I have chosen is rather too great, so that it leads us to take 3°.56 as a minimum. The smallest value of C, observed by M. Regnault is 0.187; this number leads to

T1—T'=4°·11,

and we must regard this value as a maximum. Messrs. Joule and Thomson have made use of the formula (12) by expressing V] and v" as functions of p, and p' by means of the law of Mariotte. In fact putting

In this formula the pressures are expressed in units of weight on the unit of surface; if we wish to express them in atmospheres, we shall take

C-T2
p 1

(n1-n') atmosphere. (14)

Such is the formula employed by the English physicists, p. 337 of their memoir*. Let us examine if it agrees with * Philosophical Transactions of the Royal Society of London, vol. cxliv. part 2.

There is a mistake in the printing of the last formula of page 337; the is omitted.

factor

P1-P2

Po

2

formula (12). Making, in this latter as in the preceding calculation, n,=4-7 atmospheres and n'=0·989, we find

T1—T'=3°•46,

which number is a little less than that deduced from formula (12); but the difference is so trifling that the approximation of formula (14) is justified.

All the preceding formulæ follow the course of the phenomena; but they can never give very exact numbers, on account of the uncertainty which prevails as to the value of some of the constants, such as a, A, K, and Cp. Thus Messrs. Joule and Thomson have taken, from Mr. Rankine,

instead of 2.4365685 and 273, which I have adopted. Consequently formula (14) gives with these numbers

T1—T'=4°·09.

On page 336 of the memoir just quoted may be read that the cooling observed with a difference of pressure of 60-601 pounds on a square inch was 5°049 at the temperature of 12°.844.

This excess of pressure is equivalent to n1―n'=4·216 atmospheres, if we assume for one atmosphere a pressure of 14:373 pounds per square inch.

Formula (14) gives the preceding number with the constants of Mr. Rankine very well; but with those which have been used in the preceding calculations we obtain under the same circumstances 4° 65, a smaller number.

It is possible that in the experiments of Messrs. Joule and Thomson the thermometer may have been placed at too small a distance from the orifice for the vis viva of the jet to be completely converted into heat. Let us see what in fact was the mode of operating.

In the large apparatus described in volumes cxliii. and exliv. of the Transactions of the Royal Society, the gas contained in a gasometer is withdrawn by a pump and forced into a long serpentine tube surrounded by water. At the end of this serpentine tube is a wooden cylinder into which cotton is pressed, 2.72 inches in length and 1.5 inch in diameter; and beyond, a tube is adjusted which reconducts the gas to the gasometer. The pump maintains the regular circulation, and keeps the pressures P, and p' constant on each side of the porous partition. The able physicists took the most minute precautions in order to obtain regular effects. A thermometer placed very near the partition received the gas after its expansion in the form of a multitude

of small jets, having certainly very little vis viva; and when the pressures were perfectly constant, this thermometer indicated a constant temperature considerably less than the external temperature.

The general effects correspond to the theory; but I do not think that the method is susceptible of very great accuracy. The influence of the sides and the porous partition on the thermometer was very great (as the experimenters observed) when there were variations of the pressure p,. Besides, the gas was not completely dry, and the correction of the effect due to moisture not very accurate. Finally the position of the thermometer must have had an enormous influence. Very close to the partition the jets were still animated by a certain velocity, and the temperature must have increased very rapidly from the partition to a certain distance, from which it again became equal to the external temperature. It appears evident to me that the effect observed on the thermometer does not indicate the temperature which the gas would have if it returned to rest after the expansion without external calorific action. I think that the thermometer indicated too low a temperature, and that hydrogen would have been able to produce analogous effects to those of air under the same circumstances, as occurred in my own experiments. Unfortunately, particulars of the observations made on this gas are not given.

Neither is the method employed by M. Hirn* free from all objection. Aqueous vapour on leaving the boiler passes through a tube of 5 centims. diameter, where it is superheated; it passes through an orifice of 4 millims. into a cubical wooden box, which is enveloped by two other boxes. Thus the expanded vapour circulates in the spaces between the boxes before issuing into the atmosphere; and consequently the central cavity is conveniently sheltered from the cooling action of external bodies. In this cavity the thermometer is placed; it is perfectly sheltered by a partition against the direct shock of the jet of vapour, so that the molecules of vapour only impinge upon it after they have lost their velocities. But, on account of the magnitude of the orifice, the molecules of vapour have a great velocity on each side of the orifice, and there is a great fall in the temperature, due to the velocity, between the orifice and the partition. Hence the radiation of the sides is considerable, and the jet only returns to rest after having received heat from without; and therefore the final temperature observed on the thermometer is too high. This objection was made by M. Combest.

*Exposition analytique et expérimentale de la Théorie mécanique de la chaleur, second edition, p. 177 (1865).

† Exposé des principes de la Théorie mécanique de la chaleur, par M. Combs, p. 238 (1867).

I do not think that the thermal effects at present in question can be studied very exactly by means of the thermometer. The inevitable action of the sides, alone, prevents entire exactness.

M. Hirn has given in one of his last memoirs some very simple formulæ, by means of which we can solve the problems before us*. I tried to apply them; but the results to which they lead differ from those which I have given.

Following the method invented by M. Hirn, and which is applied in the memoir quoted to the vapours of water, of bichloride of carbon, of sulphide of carbon, of alcohol, and of ether, I calculated a formula for carbonic acid,

But I have not found for the constants a, b, B values which made the formula agree sufficiently with the Table on page 272. If an agreement between them were established, that formula would take the place of the formula of Mr. Rankine, and, introduced into the preceding equations, would solve our problems. For example, in the problem of § I., formula (5) would give

The calculations which I have made in the second problem indicate that it is very distinct from the first. In order that the mechanical and thermal effects might be the same in these two problems, it would be necessary that

and this equality seems to me impossible. The contradiction may very likely be only apparent between the results at which I have arrived and those of M. Hirn. It is not in the spirit of criticism that I refer to it here. Seeking to resolve those questions which have been already treated by my excellent friend, by means of a method differing from his, I meet with differences which may be explained by the inexactness of the numerical data. The importance of formula (15), which M. Hirn has used with advantage in the study of vapours, imposed on me the comparison of the two methods; and I have made it in the hope that it may contribute to the elucidation of a delicate question of thermodynamics.

*G. A. Hirn, "Mémoire sur la Thermodynamique," Ann. de Chim. et de Phys. May 1867, p. 91.

§ III. On the internal work in a gas which undergoes expansion or compression without external calorific action, and of which the elastic force at each moment balances the pressure exerted on its surface.

Problem III.-A kilogramme of gas at temperature T1 passes T, from volume v, to volume v2 while overcoming external pressure equal at each moment to its elastic force, without there being either addition or loss of external heat: knowing v1, T1, and v2, to calculate the final temperature T2 and the internal work effected.

2

This operation belongs to the kind which are termed reversibles. It cannot be realized in practice; but a gaseous mass which pushes a piston in a cylinder, or which is compressed by the piston so quickly that the external heat may be neglected, comports itself very nearly as in the problem enunciated.

The quantity of external heat which the gas takes in changing its condition may be represented by

Q= STdp,

being a function (of two of the variables p, v, T) which Mr. Rankine has called the thermodynamic function.

dp

If v and T be taken as the variables, we have*

d being the partial derivative of p deduced from relation (2).

dT

In the present problem Q=0; hence = constant, and consequently

T. Cv2T2 dp
ке
KL2+A
T1

dv=0.

dT

(16)

This equation being combined with relation (2), T2 may be calculated when T1, v1, v2 are known. The initial and final pressures P1, P2 may be calculated by means of relation (2).

A relation between p and v may afterwards be established, which will serve to calculate the external work

*P. de Saint-Robert, Principes de Thermodynamique, p. 69 (1865).

Phil. Mag. S. 4. Vol. 40. No. 267. Oct. 1870.

U