Page:Die Kaufmannschen Messungen.djvu/6

theory. If one now take from table VII (p. 529), for example, z' = 0.1350 and y' = 0.0246, the result is:

$\beta =0,1884\cdot {\frac {0,1350}{0,0246}}=1,034$,

which is a priori not compatible with any of the theoretical formulas.

Thus, nothing seems to remain but the assumption that in the theoretical interpretation of the measured quantities there is still a major gap, which must first be filled before the measurements can be used as a final decision between the sphere theory and the relative theory. One might think of different ways, none of which I would like to discuss any closer, because it seems to me that the physical foundations for any of them are too unsure.

§ 7. Difference between the theories for rays of a certain magnetic deflectability.

However, I would like to bring in another point in more detail: that is the question in which areas of the "radiation spectrum" a decision between the conflicting theories will be possible at first. It seems that it is considered to be fairly common that the greatest differences of the theories are found using the fastest rays. This view apparently arises from the circumstances, that the momentum quantities p derived from the equations (20) and (22) for both theories, are more different from each other the closer β is coming to 1 - but this is incorrect; in some circumstances the very opposite is true. Because in the measurements we do not compare the observed values of p with the expected theoretical values of p at a certain β, but we compare, for example as with Kaufmann's measurements, the observed values of the electrical deflectability with the expected theoretical values of the electrical deflectability, at a certain magnetic deflectability, and that is something completely different.

When an electron ray is characterized by its magnetic deflectability, then this means that we denote to it a specific value of momentum p; since by (14) p is directly determined by the radius of curvature $\varrho$. To a certain value of p, to which by (23) also corresponds a certain value of u, there belong (according to the two theories) different values of β. We denote them by β and β', β may apply for the sphere theory, β' for the Relative theory, so by (24) and (25) we have:

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

It follows that always:

β' < β.

So, a ray of a certain magnetic deflectability possesses a smaller velocity in the relative theory than in the sphere theory.

Now consider the electrical deflectability in both theories. The electric beam in a certain (not too great) distance x is, as we find directly from (6), proportional to the ratio ${\frac {u}{\beta }}$. The expected electrical deflectabilities in the two theories have therefore the difference:

${\frac {u}{\beta '}}-{\frac {u}{\beta }}>0$.

A beam of a certain magnetic deflectability will be more deflected in the relative theory than in the sphere theory, and the difference is the greater, the greater the magnetic deflectability is. Of course this applies, as well as the following analog principles below, to the absolute difference, not to the percental difference. To illustrate this we can use the calculated values of ${\bar {y}}$ for both theories from the table above, their difference increases with increasing ${\bar {z}}$.

For u = 0 (magnetic deflection is zero) we have:

${\frac {u}{\beta '}}-{\frac {u}{\beta }}=0$.

For u = ∞ (magnetic deflection equal to infinity) we have:

${\frac {u}{\beta '}}-{\frac {u}{\beta }}={\frac {1}{10}}$.

Because an experimental decision between the two theories is the more likely, the more their results differ, we can assume that measurements of the electrical deflectability, which should lead to a decision between the theories, is more appropriate to conduct with cathode rays than with Becquerel rays.

§ 8. Difference between the theories for cathode rays of a certain discharge potential.

If we use homogeneous cathode rays for the deflection experiments, then,