Page:Messungen an Becquerelstrahlen.djvu/4

By means of this arrangement, rays of a certain velocity find quite automatically the angle, at which the compensation occurs at a given field strength, and which allows them to leave the condenser. Thus electrons of all velocities and corresponding masses will hit upon the film, and thus produce a curve which allows to determine the mass as a function of velocity. Consequently, a single exposition suffices to test the various theories of the electron. The equations of motion of the electrons assume the following form:

I	${\begin{cases}{\frac {d}{dt}}(m{\dot {x}})=0\\{\frac {d}{dt}}(m{\dot {y}})=\epsilon H{\dot {z}}\\{\frac {d}{dt}}(m{\dot {z}})=-\epsilon H{\dot {y}}\end{cases}}$

Here, $\epsilon ,\ m$ mean the specific charge and mass of the electron. The direction of increasing $x$ is the direction of the magnetic field ${\mathfrak {H}}$. While the $X$- and $Y$-axes coincide with the plane of the condenser plates, $z$ is perpendicular to this plane. Within the condenser, the direction of motion forms the angle $\alpha$ with ${\mathfrak {H}}$. Due to the spiral motion in the pure magnetic field, however, the direction deviates from the initial one, so that the impact point $P$ upon the film, lies in the direction $\theta$ which somewhat deviates from $\alpha$. If one integrates the preceding equations and if one sets $\varphi ={\tfrac {\epsilon }{m}}Ht$, then it follows:

II	${\begin{cases}{\dot {x}}=u\ \cos \alpha \\{\dot {y}}=u\ \sin \alpha \cos \varphi \\{\dot {z}}=-u\ \sin \alpha \sin \varphi .\end{cases}}$

If one furthermore sets $OD=DP=\alpha$, and

$u={\frac {F}{H\ \sin \alpha }},$

then one obtains by repeated integration:

III	${\begin{cases}x=a\ \cos \alpha +{\frac {m}{\epsilon }}{\frac {F}{H^{2}}}\varphi \cot \alpha \\y=a\ \sin \alpha +{\frac {m}{\epsilon }}{\frac {F}{H^{2}}}\sin \varphi \\z={\frac {m}{\epsilon }}{\frac {F}{H^{2}}}(1-\cos \varphi ).\end{cases}}$

If one denotes the values of $x,y,z$ and $\varphi$ (present at the impact point of the electron) by index $\Theta$, and if one furthermore sets ${\tfrac {F}{2\alpha H^{2}}}=K$, then one finds:

IV

${\begin{cases}(1)&{\frac {x_{0}}{2a}}=\cos \Theta ={\frac {1}{2}}\cos \alpha +K{\frac {m}{\epsilon }}\varphi _{0}\cot \alpha \\(2)&{\frac {y_{0}}{2a}}=\sin \Theta ={\frac {1}{2}}\sin \alpha +K{\frac {m}{\epsilon }}\varphi _{0}\sin \varphi _{0}\\(3)&{\frac {z_{0}}{2a}}=K{\frac {m}{\epsilon }}(1-\cos \varphi _{0}).\end{cases}}$

By insertion of (3) into (1) and (2) it follows:

(1a)	$\sin \Theta ={\frac {1}{2}}\sin \alpha +{\frac {z_{0}\sin \ \varphi _{0}}{2a(1-\cos \varphi _{0})}}$

(2a)	$\cos \Theta ={\frac {1}{2}}\cos \alpha +{\frac {z_{0}\varphi _{0}\cot \alpha }{2a(1-\cos \varphi _{0})}}$

$\alpha$ is given from these equations, and from that the velocity of the ray hitting at $P$.

Since $\theta$ slightly differs from $\alpha$, one proceeds just so, as if the electron (which is hitting at $P$) would have moved with the previously calculated velocity at a circular path in a vertical plane, which passes through the radium granule and $P$.

Now, the force acting in the magnetic field is:

(4)	${\frac {m\ u^{2}}{r}}=\epsilon Hu\ \sin \alpha .$

However, as it is shown by a simple calculation, it is:

(5)	${\frac {1}{r}}={\frac {a^{2}}{2z}}\left(1+{\frac {z^{2}}{\alpha ^{2}}}\right)$

Now if one considers, that according to Lorentz:

(6)	${\frac {\epsilon }{m}}={\frac {\epsilon }{m_{0}}}(1-\beta ^{2})^{\frac {1}{2}},$

then the relations (4), (5) and (6) give

(7)	${\frac {\epsilon }{m_{0}}}={\frac {2zv}{\alpha ^{2}\left(1+{\frac {z^{2}}{\alpha ^{2}}}\right)H\ \sin \alpha }}\tan \arcsin \beta$

While according to Maxwell under employment of Abraham's formula, and when we set $\tanh \delta =\beta$:

(8)	${\frac {\epsilon }{m_{0}}}={\frac {2zv}{a^{2}\left(1+{\frac {z^{2}}{\alpha ^{2}}}\right)H\ \sin \alpha }}\left\{{\frac {3}{4\beta }}{\frac {2\delta -\tan h\ 2\delta }{\tan h\ 2\delta }}\right\}$

Obviously only that theory is valid, for which ${\tfrac {\epsilon }{m_{0}}}$ is a real constant for any value of $\beta$ within the observational errors.

The experiments.

I omit at this place the experiments, which I have undertaken to test my relativity principle. The report shall suffice,