Substituting these values into the expression of A, and the values of A and E into the equation for dS, the latter is as follows:
.
The condition that this expression forms a complete differential of the three independent variables q, V and T (bearing in mind that ε only depends on q and T, not on V) gives as a necessary consequence the relations:
(1)
and
(2)
where the constant a is determined by the fact that ε goes over to aT4 for q = 0, which is in accordance with the Stefan-Boltzmann radiation law.
With these values we obtain for the energy E, the pressure p and the momentum G of the moving cavity radiation as functions of the independent variables q, V and T, the following expressions:
(3)
(4)
(5)
So, for example, if we impart some acceleration to the cavity radiation, while its volume V is kept constant and no heat is supplied from outside so that also the entropy S remains constant, the temperature T of the radiation is decreased by (2) in the ratio
↑According to K. Von Mosengeil, l.c. equation (24*) it is namely: