Page:Zur Dynamik bewegter Systeme.djvu/19

characteristically for the process under consideration at that time: the motion of an electrically charged, diathermanous solid body in an evacuated electromagnetic field.

Finally, there may be still room for general relations between the values for the energy of the body, as well as the performed external work, and the supplied heat for both reference systems.

For the energy E' we have, by (10):

$E'={\dot {x}}'{\mathfrak {G}}'_{x'}+{\dot {y}}'{\mathfrak {G}}'_{y'}+{\dot {z}}'{\mathfrak {G}}'_{z'}+T'S'-H'$,

consequently, by substituting the previously derived relations:

$E'={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left\{E-v{\mathfrak {G}}_{x}-{\frac {v({\dot {x}}-v)}{c^{2}-v{\dot {x}}}}pV\right\}$.

(33)

As to the thermal function R defined in (30), we have in the primed reference frame the simple relationship:

$R'={\frac {c}{\sqrt {c^{2}-v^{2}}}}(R-v{\mathfrak {G}}_{x})$.)

(34)

The performed translation work from outside (at an infinitesimal reversible change of state of the body) is for the primed frame of reference:

${\mathfrak {F}}'_{x'}dx'+{\mathfrak {F}}'_{y'}dy'+{\mathfrak {F}}'_{z'}dz'=$ $={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left\{{\mathfrak {F}}_{x}dx+{\mathfrak {F}}_{y}dy+{\mathfrak {F}}_{z}dz-vdt\left({\mathfrak {F}}_{x}+{\frac {{\dot {x}}-v}{c^{2}-v{\dot {x}}}}(V{\dot {p}}+T{\dot {S}}\right)\right\}$

(35)

Furthermore, the compression work:

$-p'dV'=-{\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}pdV-{\frac {vc{\sqrt {c^{2}-v^{2}}}}{(c^{2}-v{\dot {x}})^{2}}}pVd{\dot {x}}$

(36)

finally, the added heat:

$T'dS'={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}TdS$.

(37)

§ 12.

The relations derived above between the primed and unprimed quantities can be partly represented in a more simple way, if we examine those expressions that are invariant for the transformation from one reference frame to another. Such invariants are y, z, p, s, \${\mathfrak {G}}_{y},{\mathfrak {G}}_{z},{\frac {H}{\sqrt {c^{2}-q^{2}}}},G{\frac {\sqrt {c^{2}-q^{2}}}{q}}$, furthermore the differential expressions ${\sqrt {c^{2}-q^{2}}}dt$, Hdt, Vdt, Tdt, ${\mathfrak {F}}_{y}dt,\ {\mathfrak {F}}_{z}dt,Edt-{\mathfrak {G}}_{x}dx,Rdt-{\mathfrak {G}}_{x}dx$, etc. All these quantities do not change their value, if they were replaced by the corresponding primed quantities.