the value P11, where pv, is any value obtained for the con

ข

stant in the above equation; we then get from (46):

We observe that this value of W coincides with that given in equation (42) for Q; the reason for this being that the gas, while expanding at constant temperature, absorbs only so much heat as is required for the external work.

Joule has employed the equation (48) in one of his determinations of the Mechanical Equivalent of Heat. For this purpose he forced atmospheric air into a strong receiver, up to ten or twenty times its normal density. The receiver and pump were meantime kept under water, so that all the heat which was developed in pumping could be measured in the water. The apparatus is represented in Fig. 6, in which R is the receiver, and C the pump. The vessel G, as will be easily understood, was used for the drying of the air, and the vessel with the spiral tube served to give to the air, before its entrance into the pump, an exactly known temperature. From the total quantity of heat given in the calorimeter Joule subtracted the part due to the friction of the pump, the amount of which he determined by working the pump for exactly the same length of time, and under the same mean pressure, but without allowing the entrance of air, and then observing the heat produced. The remainder, after this was subtracted, he took as being the quantity of heat developed by the compression of the air; and this he compared with the work required for the compression as given by equation (48). By this means he obtained as the mean of two series of experiments the value of 444 kilogrammetres as the Mechanical Equivalent of Heat.

This value, it must be admitted, does not agree very well with the value 424 obtained by the friction of water; the reason of which is probably to be found in the far larger sources of error attending experiments on air. Nevertheless at that time, when the fact that the work required for developing a given quantity of heat was equal under all circumstances was not yet placed on a firm basis, the agreement

of the values found by such wholly different methods was close enough to aid considerably in the establishment of the principle.

As a third case of determination of work done, we may assume that the gas changes its volume within an envelope

impermeable to heat; or, which comes to the same thing. that the change of volume takes place too rapidly to allow of the passing of any appreciable quantity of heat to or from the body during the time.

Fig. 6.

In this case the relation between pressure and volume is given by equation (45), viz.:

The curve of pressure corresponding to this equation (Fig. 7) falls more steeply than that delineated in Fig. 5. Rankine has given to this

to

d

special class of pressurecurves, which correspond to the case of expansion within an envelope impermeable to heat, the name of Adiabatic curves (from διαβαίνειν, το pass through). On the other hand Gibbs (Trans. Connecticut Academy, vol. II. p. 309) has proposed to name them Isentropic curves, because in this kind of expansion the Entropy, a quantity which will be discussed further on, remains constant. This latter form of nomenclature is the one which the author poses to adopt, since it is both usual and advantageous to designate curves of this kind according to that quantity which remains constant during the action that takes place. To effect the integration in this case, we may put, according to the above equation,

U

1

a

C

Fig. 7.

pro

p = p1v* ×

CHAPTER III.

SECOND MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT.

§ 1. Description of a special form of Cyclical Process.

In order to prove and to make intelligible the second Principle of the Mechanical Theory of Heat, we shall commence by following out in all its parts, and graphically representing in the manner already described, one special form of cyclical process. For the latter purpose we will assume that the condition of the variable body is determined by its volume v and its pressure p, and will employ, as before, a rectangular system of co-ordinates, in which the abscissæ represent volumes, and the ordinates pressures. Any point on the plane of co-ordinates will then correspond to a certain condition of the body, in which its volume and pressure have the same volumes as the abscissa and ordinate of the point. Further, every variation of the body's condition will be represented by a line, whose extreme points determine the initial and final condition of the body, and whose form shews the way in which the pressure and volume have simultaneously varied.

In Fig. 8 let the initial condition of the body, at which the cyclical process commences, be given by the point a, so that the abscissa oev, and the ordinate ea = p1 represent the initial volume and pressure respectively. By means of these two quantities the initial temperature, which we will call T,, is also fixed.

Now let the body in the first place expand, while retaining the same temperature T. If no heat were imparted to it

during expansion, it would necessarily become cooler: we will therefore assume that it is put in communication with a body K, acting as a reservoir of heat, which body has the same temperature T,, and does not appreciably vary from this during the cyclical process. From this body the variable body is supposed to draw during the expansion just sufficient heat to keep itself also at the temperature T.

The curve, which during this expansion expresses the change of pressure, is part of an isothermal curve. In order that we may give definite forms to the graphic representations of this curve, and of others yet to be described, we will, without limiting the investigation itself to any particular bodies, draw the figure as it would appear in the case of a perfect gas. Then the isothermal curve, as explained above, will be an equilateral hyperbola; and, if the expansion take place from the volume oev, to the volume of V1, we shall obtain the part ab of such an equilateral hyperbola.

1

When the volume V, has been reached, let us suppose the body K, to be withdrawn, and let the variable body be left to continue its expansion by itself, without any heat being imparted to it. The temperature must then fall, and we obtain as curve of pressure an isentropic curve, which descends more steeply than the isothermal curve. Let this expansion continue till the volume V, is reached, giving us the portion of an isentropic curve bc. The lower temperature thus attained we may call T.

2