Page:Die Kaufmannschen Messungen.djvu/7

 except the magnetic and electric deflectability, a third characteristic of rays can be measured: the discharge potential, and it appears appropriate to directly characterize the value of the discharge potential of the ray. In this case the question arises: In which way, as regards the magnetic and electric deflectability of a ray of a certain discharge potential, can the theories be distinguished? By the discharge potential P volt, the energy E of the ray is given, because:

E=εP · 108.

Now for any theory:

$$E=q\frac{\partial H}{\partial q}-H=qp-H$$,

so for the sphere theory by (19):

$$E=\frac{3}{2}\mu_{0}c^{2}\left(\frac{c}{2q}\log\frac{c+q}{c-q}-1\right)$$

and for the relative theory by (21):

$$E=\mu_{0}c^{2}\left(\frac{c}{\sqrt{c^{2}-q^{2}}}-1\right)$$.

We denote the size calculated according to the latter theory, again by using primed variables and re-introduce β and u by (23), so that the relation between β and β' is expressed by the equation:

$$\frac{3}{2}\left(\frac{1}{2\beta}\log\frac{1+\beta}{1-\beta}-1\right)=\frac{1}{\sqrt{1-\beta'^{2}}}-1$$.

Furthermore, as previously:

$$\frac{1}{u}=\frac{3}{4\beta}\left(\frac{1+\beta^{2}}{2\beta}\log\frac{1+\beta}{1-\beta}-1\right)$$

$$\frac{1}{u'}=\frac{\beta'}{\sqrt{1-\beta'^{2}}}$$.

From these equations the results follow:

1. For the velocity:

β' < β,

i.e. a ray of a certain discharge potential possesses in the relative theory a smaller velocity as in the sphere theory.

2. For the magnetic deflectability:

u' < u,

i.e. a beam of a certain discharge potential possesses in the relative theory a smaller magnetic deflectability as in the sphere theory. The difference vanishes for infinitely great and infinitely small discharge potentials, and there is a maximum for the discharge potential P = 3,2 · 105 volt (β = 0,834). As regards the practical size of this number one may say that within the currently executable measurements the difference of the theories is the greater, the greater the discharge potentials are to which we advance.

3. For the electrical deflectability:

$$\frac{u'}{\beta'}\begin{matrix} >\\ =\\ <\end{matrix}\frac{u}{\beta}$$ für $$P\begin{matrix} <\\ =\\ >\end{matrix}1,1\cdot10^{6}$$ Volt $$\left(\beta\begin{matrix} <\\ =\\ >\end{matrix}0,987\right)$$,

i.e. a beam of a certain discharge potential possesses in the relative theory a greater, equal or smaller deflectability than in the sphere theory, depending on whether the discharge potential is smaller, equal or greater than 1,1 · 106 volt. Therefore, one may say that within the currently executable measurements, the electrical deflectability of such a ray is in the relative theory always greater than in the sphere theory, and the difference is the greater, the smaller the discharge potential is.

For P = 0 (β = β' = 0) there is especially:

$$\frac{u'}{\beta'}-\frac{u}{\beta}=\frac{1}{20}$$.

Simultaneous measurements of the discharge potentials, magnetic and electrical deflectability of the cathode rays have been known to be performed by H. . Maybe they can already be used for an examination of the two theories. However, so far I found no opportunity to dwell on this question.

Discussion.

 * As the one who is immediately concerned I'd like to add a few words. I ask you to circulate the drawn curve and the five original plates, where you see two symmetrical curve branches of the same form as in the drawing. As for the results, there is complete agreement between and me, and it is gratifying to me that the very different account of  has led to identical resulting numbers. This suggests that there are no computational errors included in my calculations. As regards the conclusion, it follows from the observational facts that neither 's nor 's theory agree with them. This conclusion is certain. 's theory is even worse than 's. The deviations of 's theory (10-12 percent) are so great that they cannot be explained at any point by