Page:Zur Dynamik bewegter Systeme.djvu/23

For those values applying to q = 0, by definition (§ 13) the expressions follow:

$E'_{0}={\frac {ac^{5}T^{4}V}{(c^{2}-q^{2})^{\frac {5}{2}}}},\quad p'_{0}={\frac {ac^{4}T^{4}}{3(c^{2}-q^{2})^{2}}},\quad S'_{0}={\frac {4ac^{4}T^{3}V}{3(c^{2}-q^{2})^{2}}}$,

and with their aid by (39), (40), (43) and (46) the values valid for any velocity q follow:

$P={\frac {ac^{4}T^{4}}{3(c^{2}-q^{2})^{2}}},\ S={\frac {4ac^{4}T^{3}V}{3(c^{2}-q^{2})^{2}}}$, $E={\frac {ac^{4}(3c^{2}+q^{2})}{3(c^{2}-q^{2})^{3}}}T^{4}V,\ G={\frac {q}{c^{2}}}(E+pV)={\frac {4ac^{4}q}{3(c^{2}-q^{2})^{3}}}T^{4}V$

in accordance with the equations of § 1

§ 16.

By momentum G, also its inertial mass is determined. This quantity, which plays in pure mechanics such a fundamental role, is degraded to a secondary expression within the general dynamics. For, once the momentum is no longer proportional to the velocity, the mass of a body is no longer constant; also we are led to completely different dependencies of mass on velocity, depending on whether we divide momentum G by velocity q, or if we differentiate velocity q, where in this case it is necessary to specify particularly the manner in which the differentiation took place: whether isothermal, adiabatic, etc. Again, a different value for the mass is found in general, if we start from the energy E and differentiate it to ${\frac {q^{2}}{2}}$. How to designate these different expressions, is of course a matter of definition.

Here, by "mass" M of a body we want to understand that quantity of a body independent of velocity, which is obtained if the momentum G is divided by velocity q and where we set the ratio q = 0, thus in our notation by (46):

$M=\left({\frac {G}{q}}\right)_{0}={\frac {R_{0}}{c^{2}}}={\frac {E_{0}+pV_{0}}{c^{2}}}$

(48)

This quantity in general depends on the temperature T and volume V of the body.

If we set in the expression ${\frac {G}{q}}$ the velocity q not to zero, then we call the ratio, as usual,^[1] the "transverse"

1. M. Abraham, Theorie der Elektricität, II, p. 186.