\begin{align} dU &= TdS - pdV \\ dF &= d(U-TS) \\ &= -SdT - pdV \end{align}
\begin{align} \left( \frac{\partial U}{\partial V} \right)_T &= T \left( \frac{\partial S}{\partial V} \right)_T - p \\ &= -T \frac{\partial^2 F}{\partial V \partial T} - p \\ &= -T \frac{\partial^2 F}{\partial T \partial V} - p \\ &= T \left( \frac{\partial p}{\partial T} \right)_V - p \end{align}
(2) で得た式に \begin{align} U(T,V) = V \tilde{u}(T) , \ \ p(T,V) = \frac{1}{3} \tilde{u}(T) \end{align} を代入すると、 \begin{align} \tilde{u}(T) &= T \cdot \frac{1}{3} \frac{d \tilde{u}(T)}{dT} - \frac{1}{3} \tilde{u}(T) \\ \therefore \ \ \frac{d \tilde{u}}{\tilde{u}} &= \frac{4}{T} dT \\ \therefore \ \ \log \tilde{u} &= 4 \log T + C \ \ \ \ \ \ \ \ \text{ ( $C$ は積分定数 )} \\ &= \log T^4 + C \end{align} となるので、 $\tilde{u}$ は $T$ の4乗に比例することがわかる。