Page:Elementary Principles in Statistical Mechanics (1902).djvu/187

${\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''+{\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''+\mathrm {etc.} ={\overline {W}}.$

(480)

If $E_{0}$, $E_{1}$, and $E_{2}$ are the only ensembles, we have

$W<{\frac {\Theta _{1}-\Theta _{2}}{\Theta _{1}}}({\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}'').$

(481)

It will be observed that the relations expressed in the last three formulae between ${\overline {W}}$, ${\overline {\epsilon }}_{1}'-{\overline {\epsilon }}_{1}''$, ${\overline {\epsilon }}_{2}'-{\overline {\epsilon }}_{2}''$, etc., and $\Theta _{1}$, $\Theta _{2}$, etc. are precisely those which hold in a Carnot's cycle for the work obtained, the energy lost by the several bodies which serve as heaters or coolers, and their initial temperatures.

It will not escape the reader's notice, that while from one point of view the operations which are here described are quite beyond our powers of actual performance, on account of the impossibility of handling the immense number of systems which are involved, yet from another point of view the operations described are the most simple and accurate means of representing what actually takes place in our simplest experiments in thermodynamics. The states of the bodies which we handle are certainly not known to us exactly. What we know about a body can generally be described most accurately and most simply by saying that it is one taken at random from a great number (ensemble) of bodies which are completely described. If we bring it into connection with another body concerning which we have a similar limited knowledge, the state of the two bodies is properly described as that of a pair of bodies taken from a great number (ensemble) of pairs which are formed by combining each body of the first ensemble with each of the second.

Again, when we bring one body into thermal contact with another, for example, in a Carnot's cycle, when we bring a mass of fluid into thermal contact with some other body from which we wish it to receive heat, we may do it by moving the vessel containing the fluid. This motion is mathematically expressed by the variation of the coördinates which determine the position of the vessel. We allow ourselves for the purposes of a theoretical discussion to suppose that the walls of this vessel are incapable of absorbing heat from the fluid.