83.

Before passing on to the general integration of equation [eqn:(95)] (95) let us examine it in the manner which most easily suggests itself. According to the energy principle, the change in the radiant energy $U=Vu=V{\int }_0^{\infty }𝗎d\nu ,$ occurring on adiabatic compression, must be equal to the external work done against the radiation pressure $-p\delta V=-\frac{u}{3}\delta V=-\frac{\delta V}{3}{\int }_0^{\infty }𝗎d\nu .\mathrm{\backslash Label}[eqn](96)\text{upshape (96)}$ Now from [eqn:(94)] (94) the change in the total energy is found to be $\delta U={\int }_0^{\infty }d\nu \delta (V𝗎)=\frac{\delta V}{3}{\int }_0^{\infty }\nu \frac{\partial 𝗎}{\partial \nu }d\nu ,$ or, by partial integration, $\delta U=\frac{\delta V}{3}([\nu 𝗎]{\mathrm{\backslash Strut}}_0^{\infty }-{\int }_0^{\infty }𝗎d\nu ),$ and this expression is, in fact, identical with [eqn:(96)] (96), since the product $\nu 𝗎$ vanishes for $\nu =0$ as well as for $\nu =\infty$. The latter might at first seem doubtful; but it is easily seen that, if $\nu 𝗎$ for $\nu =\infty$ had a value different from zero, the integral of $𝗎$ with respect to $\nu$ taken from $0$ to $\infty$ could not have a finite value, which, however, certainly is the case.