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in the first instance to be with the black body. The operations are of two kinds: (1) compression (or rarefaction) of the kind called adiabatic, that is, without communication of heat. If the volume increases, the temperature must fall, even though in the absence of pressure upon the piston no work is done, since the same energy of complete radiation now occupies a larger space. Similarly a rise of temperature accompanies adiabatic contraction. In the second kind of operation (2) the expansions and contractions are isothermal—that is, without change of temperature. In this case heat must pass, into the black body when the volume expands and out of it when the volume contracts, and at a given temperature the amount of heat which must pass is proportional to the change of volume.
The cycle of operations to be considered is the same as in Carnot's theory, the only difference being that here, in the absence of pressure, there is no question of external work. Begin by isothermal expansion at the lower temperature during which heat is taken in. Then compress adiabatically until a higher temperature is reached. Next continue the compression iso-thermally until the same amount of heat is given out as was taken in during the first expansion. Lastly, restore the original volume adiabatically. Since no heat has passed upon the whole in either direction, the final state is identical with the initial state, the temperature being recovered as well as the volume. The sole result of the cycle is that heat is raised from a lower to a higher temperature. Since this is assumed to be impossible, the supposition that the operations can be performed without external work is to be rejected—in other words, we must regard the radiation as exercising a pressure upon the moving piston. Carnot's principle and the absence of a pressure are incompatible.
For a further discussion it is, of course, desirable to employ the general formulation of Carnot's principle, as in a former paper*. If p be the pressure, 6 the absolute temperature,
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where M dv represents the heat that must be communicated, while the volume alters by dv and dd = 0. In the application to radiation M cannot vanish, and therefore p cannot. In this case clearly
where U denotes the volume-density of the energy—a function of 6 only. Hence
* "On the Pressure of Vibrations," Phil. Hag. Vol. ni. p. 338, 1902; Scientific Papers, Vol. v. p. 47.
K. VI.uid moves in any of these stages, is stable. It would appear then that no explanation is to be found in this direction.