Page:Zur Dynamik bewegter Systeme.djvu/15

 Before we perform the integration, we derive the relevant equations for the velocity components $$\dot{y}$$ and $$\dot{z}$$. In addition to the differential equations (6) with respect to the primed system we have to use:

the relations between the primed and unprimed components of the moving force $$\mathfrak{F}$$. To find them, we consider a special case, namely, an infinitely small diathermanous solid body charged with electricity e, in an arbitrary, evacuated electromagnetic field. Then, for the unprimed system:

where $$\mathfrak{E}$$ denotes the electric, $$\mathfrak{H}$$ the magnetic field intensity. The same equations apply according to the relativity principle, when all the variables, except e and c, were provided with primes. This leads with respect to the relations (13) and the relations:

$$\begin{matrix}\mathfrak{E}'_{x'}= & \mathfrak{E}_{x} & \mathfrak{H}'_{x'}= & \mathfrak{H}_{x}\\ \\\mathfrak{E}'_{y'}= & \frac{c}{\sqrt{c^{2}-v^{2}}}\left(\mathfrak{E}_{y}-\frac{v}{c}\mathfrak{H}_{z}\right)\quad & \mathfrak{H}'_{y'}= & \frac{c}{\sqrt{c^{2}-v^{2}}}\left(\mathfrak{H}_{y}+\frac{v}{c}\mathfrak{E}_{z}\right)\\ \\\mathfrak{E}'_{z'}= & \frac{c}{\sqrt{c^{2}-v^{2}}}\left(\mathfrak{E}_{z}+\frac{v}{c}\mathfrak{H}_{y}\right)\quad & \mathfrak{H}'_{z'}= & \frac{c}{\sqrt{c^{2}-v^{2}}}\left(\mathfrak{H}_{z}-\frac{v}{c}\mathfrak{E}_{y}\right)\end{matrix}$$

the following equations between the primed and unprimed force components:

The last two relations (22) we accept as generally valid; this give in combination with (6) and (20):