heat and one from heat into work, e.g. let the quantity of heat of temperature T be generated out of work, and the quantity of heat of temperature T be transformed into work, and let these two stand in such relation to each other that we may put

2

Then let us suppose in the first place that the same process as last described has been performed, whereby the quantity of heat Q of temperature T has been transformed into work, and another quantity Q, has been transferred from a body K1 to another body K. Next let us suppose a second process performed in the reverse direction, in which the last-named quantity Q, is transferred back again from K, to K1, and a quantity of heat of temperature T is at the same time generated out of work. This transformation from work into heat must, independently of sign, be equivalent to the former transformation from heat into work, since they are both equivalent to one and the same transference of heat. The quantity of heat of temperature T", generated out of work, must therefore be exactly as great as the quantity Q' found in the above equation, and the given transformations have thus been made backwards.

Finally, let there be two transferences of heat, e.g. the quantity of heat Q transferred from a body K, of temperature T to a body K, of temperature T2, and the quantity Q'1, from a body K', of temperature T', to a body K', of temperature T',, and let these be so related that we may put

Then let us suppose two Cyclical Processes performed, in one of which the quantity Q, is transferred from K, to K1, and the quantity of temperature T thereby generated out of work, whilst in the second the same quantity is again transformed into work, and thereby another quantity of heat transferred from K', to K',. This second quantity must then be exactly equal to the given quantity Q', and the two given transferences of heat have thus been done backwards.

For if

When by operations of this kind all the transformations of the first division have been done backwards, there then remain the transformations, all of the like sign, of the second division, and no others whatever. Now first, if these transformations are negative then they can only be transformations from heat into work and transferences from a lower to a higher temperature; and of these the transformations of the first kind may be replaced by transformations of the second kind. quantity of heat Q of temperature T is transformed into work, then we have only to perform in reverse order the cyclical process described in § 2, in which the quantity of heat Q of temperature T is generated out of work, and at the same time another quantity Q, is transferred from a body K of temperature T, to another body K, of the higher temperature T. Thereby the given transformation from heat into work is done backwards, and replaced by the transference of heat from K, to K. By the application of this method, we shall at last have nothing left except transferences of heat from a lower to a higher temperature which are not compensated in any way. As this contradicts our fundamental principle, the supposition that the transformations of the second division are negative must be incorrect.

2

Secondly, if these transformations were positive, then since the cyclical process under consideration is reversible, the whole process might be performed in reverse order; in which case all the transformations which occur in it would take the opposite sign, and every transformation of the second division would become negative. We are thus brought back to the case already considered, which has been found to contradict the fundamental principle.

As then the transformations of the second division can neither be positive nor negative they cannot exist at all; and the first division, whose algebraical sum is zero, must embrace all the transformations which occur in the cyclical process. We may therefore write N = 0 in equation (8), and thereby we obtain as the analytical expression of the Second Main Principle of the Mechanical Theory of Heat for reversible processes the equation

87. On the Temperatures of the various quantities of Heat, and the Entropy of the Body.

In the development of Equation VII. the temperatures of the quantities of heat treated of were determined by those of the heat reservoirs from which they came, or into which they passed. But let us now consider a cyclical process, which is such that a body passes through a series of changes of condition and at last returns to its original state. This variable body, if placed in connection with the heat reservoir to receive or give off heat, must have the same temperature as the reservoir; for it is only in this case that the heat can pass as readily from the reservoir to the body as in the reverse direction, and if the process is reversible it is requisite that this should be the case. This condition cannot indeed be exactly fulfilled, since between equal temperatures there can in general be no passage of heat whatever; but we may at least assume it to be so nearly fulfilled that the small remaining differences of temperature may be neglected.

In this case it is obviously the same thing whether we consider the temperature of a quantity of heat which is being transferred as being equal to that of the reservoir or of the variable body, since these are practically the same. If however we choose the latter and suppose that in forming Equation VII. every element of heat is taken of that temperature which the variable body possesses at the moment it is taken in, then we can now ascribe to the heat reservoirs any other temperatures we please, without thereby making any alteration in the expression With this assumption

expression fda.

as to the temperatures we may consider Equation VII. as holding, without troubling ourselves as to whence the heat comes which the variable body takes in, or where that which it gives off, provided the process is on the whole a reversible one.

dQ

goes

The expression if it be understood in the sense just given, is the differential of a quantity which depends on the condition of the body, and at the same time is fully determined as soon as the condition of the body at the moment is known, without our needing to know the path by which

the body has arrived at that condition; for it is only in this case that the integral will always become equal to zero as often as the body after any given variations returns to its original condition. In another paper*, after introducing a further development of the equivalence of transformations, the author proposed to call this quantity, after the Greek word τρоπn, Transformation, the Entropy of the body. The complete explanation of this name and the proof that it correctly expresses the conditions of the quantity under consideration can indeed only be given at a later period, after the development just mentioned has been treated of; but for the sake of convenience we shall use the name henceforward.

If we denote the Entropy of the body by S we may put

§ 8.

On the Temperature Function 7.

To determine the temperature function 7 we will apply the same method as in Chapter III. § 7, p. 81, to determine the function (T,, T). For, as the function 7 is independent of the nature of the variable body used in the cyclical process, we may, in order to determine its form, choose any body we please to be subjected to the process. We will therefore again choose a perfect gas, and, as in the above-mentioned section, suppose a simple process performed, in which the gas takes in heat only at one temperature T, and gives it out only at another T. The two quantities of heat which are taken in and given out in this case, and whose absolute values we may call Q and Q, stand by equation (8) of the last chapter, p. 83, in the following relation to each other:

On the other hand, if we apply Equation VII. to this simple cyclical process, whilst at the same time we treat the

giving out of the quantity of heat Qas equivalent to the taking in of the negative quantity Q, we have the follow

If we now take T as being any temperature whatever and T as some given temperature, we may write the last equation thus:

and the temperature function is thus reduced to a constant factor.

What value we ascribe to the constant factor is indifferent, since it may be struck out of Equation VII. and thus has no influence on any calculations performed by means of the equation. We will therefore choose the simplest value, viz. unity, and write the foregoing equation

The temperature function is now nothing more than the absolute temperature itself.

Since the foregoing determination of the function & rests on the equations deduced for the case of gases, one of the foundations on which this determination rests will be the approximate assumption made in the treatment of gases, viz. that a perfect gas, if it expand at constant temperature, absorbs only just so much heat as is required for the external work thereby performed. Should anyone on this account have any hesitation in regarding this determination as perfectly satisfactory, he may in Equations VII. and VIII. regard as the symbol for the temperature function as yet