$H$ plays the role of the inner energy in a system moving with constant velocity.

Let the entropy of the resting system be $S_{0}(U_{0},v)$, that of the moving one can be expressed by $S(H,v)$. The relations hold

$\left({\frac {\partial S_{0}}{\partial U_{0}}}\right)_{v}={\frac {1}{T_{0}}},\quad \left({\frac {\partial S_{0}}{\partial v}}\right)_{U_{0}}={\frac {p_{0}}{T_{0}}};$

but also

$\left({\frac {\partial S}{\partial H}}\right)_{v}={\frac {1}{T}},\quad \left({\frac {\partial S}{\partial v}}\right)_{H}={\frac {p}{T}};$

since it is indeed (at constant $\beta$)

$dS={\frac {1}{T}}(dH+pdv)$.

Since the system was adiabatically brought from the state of rest to that of motion, the entropy has the same value in both cases, thus:

$S_{0}(U_{0},v)=S(H,v)\,$

therefore also

${\frac {\partial S_{0}}{\partial U_{0}}}={\frac {\partial S}{\partial U_{0}}}={\frac {\partial S}{\partial H}}{\frac {\partial H}{\partial U_{0}}},$

$\left({\frac {\partial S_{0}}{\partial v}}\right)_{U_{0}}=\left({\frac {\partial S}{\partial v}}\right)_{U_{0}}={\frac {\partial S}{\partial H}}\left({\frac {\partial H}{\partial v}}\right)_{U_{0}}+\left({\frac {\partial S}{\partial v}}\right)_{H}.$

From that, also equations (6) and (8) are immediately given.

Thus far we have presupposed the existence of momentum, without making a special assumption concerning its value. Now we want to assume in agreement with the theory of Lorentz and Abraham, that momentum is equal to the space integral of the (absolute)