Translation:The Measurements of Kaufmann

by Max Planck

The Measurements of Kaufmann on the Deflectability of β-Rays in their Importance for the Dynamics of the Electrons

by Max Planck.

Gentlemen! Probably all physicists especially interested in the development of the newest branch of electrodynamics, the mechanics of electrons, have awaited with great interest the outcome of the subtle measurements of the electro-magnetic deflectability of β-rays of radium, performed last year by W. Kaufmann, and the expectations attached to such experiments have been met in large measure; because Kaufmann has gained from them a large amount of valuable data, and he, what must be recognized particularly grateful, has also given numerical data to the public, which is rich and reliable^[1] so that everyone is in a position to independently verify and complete the conclusions reached by Kaufmann.

I even more liked it to make use from this suggestion, as indeed the question at which Kaufmann's experiments was aimed for, is almost vital for various electrodynamic theories. It is known that there exist already a number of excellent mathematical investigations from several of these theories, and their physical meaning, of course, would be repealed at once if that theory would be defeated in the resulting competition.

§ 1. Equations of motion.

We can presuppose that the method by which Kaufmann has examined the contents of the various theories by his measurements is known. At first I was interested to see how far each of those measured deflections are removed from those, which can be calculated from the various theories on the basis of the measured "constants of apparatus" from the outset. Since I preferred not to reduce the measured deflections $({\overline {y}},{\overline {z}})$ from the start "to infinitely small deflections (y', z'), the equations of motion of the electrons had to be fully integrated. This gives for all compared theories:

${\begin{matrix}{\frac {d}{dt}}\left({\frac {\partial H}{\partial {\dot {x}}}}\right)&=&-{\frac {e}{c}}{\dot {z}}{\mathfrak {H}}\\\\{\frac {d}{dt}}\left({\frac {\partial H}{\partial {\dot {y}}}}\right)&=&e{\mathfrak {E}}\\\\{\frac {d}{dt}}\left({\frac {\partial H}{\partial {\dot {z}}}}\right)&=&{\frac {e}{c}}{\dot {x}}{\mathfrak {H}}.\end{matrix}}$

Here, H is the kinetic potential (the Lagrangian function) of a moving electron as a function of the velocity

$q={\sqrt {{\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}}}$,

${\mathfrak {(}}E)$ and ${\mathfrak {(}}H)$ is the electric and magnetic field strength, both acting in the y-direction as known functions of x, e is the electrical elementary quantum, c is the speed of light. The electrical quantities are measured in electrostatic units.

The electron moves from the radiation source:

x = x₀ = 0 y = 0 z = 0

through the diaphragm opening:

x=x₁=1,994 y=0 z=0

to the point of the photographic plate:

x=x₂=3,963 y=${\overline {y}}$ z=${\overline {z}}$.

To hit the diaphragm opening straight, one electron has to be emanated with a certain initial velocity in a certain direction from the radiation source. On that occasion, a certain curve (${\overline {y}},{\overline {z}}$) emerges on the photographic plate (x = x₂), whose points depend on a single parameter, for example on the initial velocity.

§ 2. Determination of the external field components.

The integration of the equations of motion still requires knowledge of ${\mathfrak {H}}$ and ${\mathfrak {E}}$ as functions of x. The magnetic field strength ${\mathfrak {H}}$ I assumed to be constant, and so great that the value of the "magnetic field integral" is the same as Kaufmann's. Its value is:^[2]

$\int _{x_{0}}^{x_{2}}dx\int _{x_{0}}^{x}{\mathfrak {H}}dx-{\frac {x_{2}-x_{0}}{x_{1}-x_{0}}}\int _{x_{0}}^{x_{1}}dx\int _{x_{0}}^{x}{\mathfrak {H}}dx=557,1$.

Herein we set ${\mathfrak {H}}$ constant, so it follows:

${\frac {1}{2}}(x_{2}-x_{1})(x_{2}-x_{0}){\mathfrak {H}}=557,1$

and from the values given by x₀, x₁ and x₂:

${\mathfrak {H}}=142,8$.

The electric field strength ${\mathfrak {E}}$ is zero between the diaphragm and the photographic plate, and constant between the capacitor plates at a proper distance from the edges. As Kaufmann, we connect the field strength to its value in the homogeneous part of the field and take it as unity, calling ${\mathfrak {E}}_{1}$ the field strength measured in this way. Then we have:

for $x_{1}<x<x_{2}:{\mathfrak {E}}_{1}=0$.

Between the radiation source and the diaphragm I have ${\mathfrak {E}}_{1}$ assumed to be symmetrical with respect to the center of that distance: $x={\frac {x_{1}}{2}}$, so that if one puts:

$x=\xi +{\frac {x_{1}}{2}},\ -{\frac {x_{1}}{2}}<\xi <{\frac {x_{1}}{2}},\ E_{1}(-\xi )={\mathfrak {E}}_{1}(+\xi )$.

1)

The increase of the electric field strength ${\mathfrak {E}}_{1}$ from the radiation source to its constant value 1 is assumed to be linear, as well as the decrease to the value 0 at the diaphragm. That is:

${\mathsf {for}}\ 0<\xi <\xi '\ {\mathsf {is}}\ {\mathfrak {E}}_{1}=1$ ${\mathsf {for}}\ \xi '<\xi <{\frac {x_{1}}{2}}\ {\mathsf {is}}\ {\mathfrak {E}}_{1}=\varkappa -\lambda \xi$

2)

Then, because of the continuity of ${\mathfrak {E}}_{1}$:

$\varkappa -\lambda \xi '=1$ und $\varkappa -\lambda {\frac {x_{1}}{2}}=$0.

The value of the constant ξ' I assumed to be as large, so that that the value of the "electric field integral" is the same as Kaufmann's. Its value is:^[3]

$(x_{1}-x_{1})\cdot \left\{\int _{x_{0}}^{x_{1}}{\mathfrak {E}}_{1}dx-{\frac {1}{x_{1}-x_{0}}}\int _{x_{0}}^{x_{1}}dx\int _{x_{0}}^{x}{\mathfrak {E}}_{1}dx\right\}=1,565$.

Substituting the above values, it follows:

${\frac {1}{2}}(x_{2}-x_{1})\cdot \left({\frac {x_{1}}{2}}+\xi '\right)=1,565$

and from this:

ξ = 0,593 ϰ = 2,468 λ = 2,475.

To reduce the electric field strength to the absolute electrostatic unit: ${\mathfrak {E}}$, or to the absolute electromagnetic unit: ${\mathfrak {E}}_{m}$, one has:^[4]

${\mathfrak {E}}_{m}={\mathfrak {E}}_{1}\cdot {\frac {25\cdot 10^{10}}{0,1242}}={\mathfrak {E}}\cdot 3\cdot 10^{10}$

3)

Whether the simplistic assumptions made in relation to the electric and magnetic field are really sufficient for the relevant calculations, will be shown below.

§ 3. Magnetic deflection.

If we introduce into the equations of motion (§ 1) the momentum vector (quantity of motion)

${\frac {\partial H}{\partial q}}=p$

4)

and also the electromagnetic unit of the electric field strength (${\mathfrak {E}}_{m}$) and for the electrical elementary quantum (ε), then they are:

${\frac {d}{dt}}\left(p{\frac {\dot {x}}{q}}\right)=-\epsilon {\dot {z}}{\mathfrak {H}}$

5)

${\frac {d}{dt}}\left(p{\frac {\dot {y}}{q}}\right)=\epsilon {\mathfrak {E}}_{m}$

6)

${\frac {d}{dt}}\left(p{\frac {\dot {z}}{q}}\right)=\epsilon {\dot {x}}{\mathfrak {H}}$.

7)

Because ${\mathfrak {H}}$ is constant, (5) and (7) can be integrated with respect to time t. Dividing the two resulting equations, t, p and q are entirely eliminated, and a second integration yields the equation of the trajectory of projection on the xz-plane, a circle, which goes through the points x = 0, z = 0, x = x₁, z = 0 and x = x₂, $z={\bar {z}}$ and is determined by it. The current coordinates x, z of the points of this circle can be represented as functions of one variable parameter: the angle φ which is the tangent of the circle in the direction of motion on the x-axis, and it is positive when the motion is to the side of the positive z-axis:

$x=\varrho \sin \varphi +{\frac {x_{1}}{2}},\qquad z=-\varrho \cos \varphi +{\frac {x_{1}}{2}}ctg\ \varphi _{1}$.

8)

Where

$\varrho ={\frac {x_{1}}{2\sin \varphi _{1}}}$

9)

is the radius of the circle and φ₁ the value of φ for x = x₁. In these equations it is already expressed that for x = 0 and x = x₁, z = 0. If it is considered that x = x₂, $z={\bar {z}}$, then we have the values:

$tg\ \varphi _{1}={\frac {x_{1}{\bar {z}}}{(x_{2}-x_{1})x_{2}+{\bar {z}}^{2}}}$

10)

and

$\sin \varphi _{2}={\frac {2x_{2}-x_{1}}{x_{1}}}\sin \varphi _{1}$

11)

and also $\varrho$ from (9).

By inserting (8) into (5) or (7) we have:

${\frac {p}{q}}\cdot {\dot {\varphi }}=\epsilon {\mathfrak {H}}$.

12)

Now it is:

$q^{2}=\varrho ^{2}{\dot {\varphi }}^{2}+{\dot {y}}^{2}$,

for which we can put with sufficient approximation:^[5]

$q=\varrho {\dot {\varphi }}$.

13)

Therefore:

$p=\epsilon {\mathfrak {H}}\varrho =\epsilon {\mathfrak {H}}{\frac {x_{1}}{2\sin \varphi _{1}}}$.

14)

The momentum p of each electron is independent of time and, without entering into a special theory, can be calculated from the magnetic deflection ${\bar {z}}$.

Since p is independent of time t, the same follows for the velocity q and by (12) for the angular velocity ${\dot {\varphi }}$. So angle φ is linearly dependent on time t.

§ 4. Electrical deflection.

From (6) it follows:

${\frac {p}{q}}{\frac {d^{2}y}{d\varphi ^{2}}}\cdot {\dot {\varphi }}^{2}=\epsilon {\mathfrak {E}}_{m}$

and from this according to (12) and (13)

${\frac {d^{2}y}{d\varphi ^{2}}}={\frac {\varrho }{q}}{\frac {{\mathfrak {E}}_{m}}{\mathfrak {H}}}$

15)

By this differential equation and by the condition that for φ = φ₀ (x=0) and for φ = φ₁ (x=x₁) y = 0, y is defined as a function of φ.

Between source and diaphragm, according to (1) and

$\xi =\varrho \sin \varphi$,

we have

${\mathfrak {E}}_{m}(-\varphi )={\mathfrak {E}}_{m}(+\varphi )$.

Here, the course of the curve is symmetrical, so that:

$y_{-\varphi }=y_{+\varphi }$ und $\left({\frac {dy}{d\varphi }}\right)_{-\varphi }=-\left({\frac {dy}{d\varphi }}\right)_{+\varphi }$.

The integration of the differential equation (15) yields:

$\left({\frac {dy}{d\varphi }}\right)_{\varphi _{1}}-\left({\frac {dy}{d\varphi }}\right)_{\varphi _{0}}={\frac {\varrho }{q}}\cdot {\frac {1}{\mathfrak {H}}}\int _{\varphi _{0}}^{\varphi _{1}}{\mathfrak {E}}_{m}d\varphi$

or, since φ₀=-φ₁:

$\left({\frac {dy}{d\varphi }}\right)_{\varphi _{1}}={\frac {\varrho }{q{\mathfrak {H}}}}\int _{0}^{\varphi _{1}}{\mathfrak {E}}_{m}d\varphi$.

Between the diaphragm and the plate it is ${\mathfrak {E}}_{m}=0$, so:

${\frac {dy}{d\varphi }}=const=\left({\frac {dy}{d\varphi }}\right)_{1}$,

and by integrating the last equation again:

${\bar {y}}=\left({\frac {dy}{d\varphi }}\right)_{1}\cdot (\varphi _{2}-\varphi _{1})={\frac {\varrho (\varphi _{2}-\varphi _{1})}{q{\mathfrak {H}}}}\int _{0}^{\varphi _{1}}{\mathfrak {E}}_{m}d\varphi$.

16)

The integral occurring here is given, when one considers, that according to (2):

$0<\varphi <\varphi '\qquad {\mathfrak {E}}_{1}=1$

$\varphi '<\varphi <\varphi _{1}\qquad {\mathfrak {E}}_{1}=\varkappa -\lambda \varrho \sin \varphi$,

where

$\varrho \sin \varphi '=\xi '=0,593$.

17)

Then it follows:

$\int _{0}^{\varphi _{1}}{\mathfrak {E}}_{1}d\varphi =\varphi '+\varkappa (\varphi _{1}-\varphi ')-\lambda \varrho (\cos \varphi '-\cos \varphi _{1})$

and by introduction of ${\mathfrak {E}}_{m}$ from (3):

${\bar {y}}={\frac {25\cdot 10^{10}}{0,1242}}\cdot {\frac {\varrho (\varphi _{2}-\varphi _{1})}{q{\mathfrak {H}}}}$ $\cdot \{\varphi '+\varkappa (\varphi _{1}-\varphi ')-\lambda \varrho (\cos \varphi '-\cos \varphi _{1})\}$

18)

§ 5. Various theories.

The relation between the electric deflection ${\bar {y}}$ and the magnetic deflection ${\bar {z}}$ is due to the dependence of momentum p on the velocity q, and this is given by the expression of the kinetic potential H as a function of q, which is different for various theories. I have performed the calculations only for those two theories, which are the most developed today: that of Abraham,^[6] in which the electron has the form of a rigid sphere, and that of Lorentz-Einstein,^[7] in which the "principle of relativity" possesses exact validity. For brevity I shall denote in the following the first theory as "sphere theory", the second as "relative theory". Then, according to the sphere theory, no matter whether volume charge or surface charge will be adopted, since we are concerned only with quasi-stationary motions, the kinetic potential is:

$H=-{\frac {3}{4}}\mu _{0}c^{2}\left({\frac {c^{2}-q^{2}}{2qc}}\log {\frac {c+q}{c-q}}-1\right)$

19)

(μ₁ is the mass of the electron for q = 0). Therefore:

$p={\frac {\partial H}{\partial q}}={\frac {3}{4}}{\frac {\mu _{0}c^{2}}{q}}\left({\frac {c^{2}+q^{2}}{2qc}}\log {\frac {c+q}{c-q}}-1\right)$.

20)

However, according to the relative theory:^[8]

$H=-\mu _{0}c^{2}\left({\sqrt {1-{\frac {q^{2}}{c^{2}}}}}-1\right)$.

21)

Therefore:

$p={\frac {\partial H}{\partial q}}={\frac {\mu _{0}qc}{\sqrt {c^{2}-q^{2}}}}.$

22)

Like Kaufmann, we introduce the two quantities β and u:

$\beta ={\frac {q}{c}}$ und $u={\frac {\mu _{0}c}{p}}$

23)

for the sphere theory it follows:

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

24)

and in the relative theory:

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

25)

By introduction of u instead of p, equation (14) for the magnetic deflection becomes:

$u={\frac {\mu _{0}}{\epsilon }}\cdot {\frac {2c\ \sin \varphi _{1}}{x_{1}{\mathfrak {H}}}}$.

26)

§ 6. Numerical values.

The comparison of the observed and the theoretical values was stated by me in a way, so that for each measured magnetic deflection ${\bar {z}}$ for each of the two theories, the corresponding value of the electrical deflection ${\bar {y}}$ was calculated and compared with the observed values. Accordingly, the following table in the first column contains the magnetic deflection ${\bar {z}}$ according to Kaufmann's table VI (l. c. p. 524), the second column the corresponding values of the angle φ₁ as calculated from (10), the third column the value of u in degrees following from (26), where the following value of the ratio of charge ε to mass μ₀ [extrapolated by Kaufmann (l.c. p. 551) on the basis of Simon's number 1,865.10⁷ valid for all theories] is:

${\frac {\epsilon }{\mu _{0}}}=1,878\cdot 10^{7}$

27)

The fourth and sixth column contain the values of $\beta ={\frac {q}{c}}$ calculated from u according to (24) and (25), the fifth and seventh column contain the values of ${\bar {y}}$ following from (18), where the required φ' and φ₂ are taken from (17) and (11); and finally the eighth column contain the "observed" values of ${\bar {y}}$ according to Kaufmann's table VI.

Observed ${\bar {z}}$	φ1	u	Sphere theory		Relative theory		Observed ${\bar {y}}$
			β	${\bar {y}}$	β	${\bar {y}}$
0,1354 0,1930 0,2423 0,2930 0,3423 0,3930 0,4446 0,4926 0,5522	1,977° 2,810 3,517 4,231 4,925 5,623 6,325 6,692 7,735	0,3871 (0,3870) 0,5502 (0,5502) 0,6883 (0,6881) 0,8290 (0,8286) 0,9634 (0,9630) 1,100 (1,099) 1,23 (1,234) 1,360 (1,358) 1,510 (1,506)	0,9747 0,9238 0,8689 0,8096 0,7542 0,7013 0,6526 0,6124 0,5685	0,0262 (0,0262) 0,0394 (0,0394) 0,0526 (0,0526) 0,0682 (0,0682) 0,0853 (0,0855) 0,1054 (0,1055) 0,1280 (0,1281) 0,1511 (0,1512) 0,1823 (0,1822)	0,9326 0,8762 0,8237 0,7699 0,7202 0,6728 0,6289 0,5924 0,5521	0,0273 (0,0274) 0,0415 (0,0415) 0,0555 (0,0554) 0,0717 (0,0717) 0,0893 (0,0895) 0,1099 (0,1099) 0,1328 (0,1328) 0,1562 (0,1561) 0,1878 (0,1874)	0,0247 0,0378 0,0506 0,0653 0,0825 0,1025 0,1242 0,1457 0,1746

First, in order to enable a comparison of my method of calculation with that of Kaufmann, I have put under the values of u as well as under the theoretical values of ${\bar {y}}$, those numbers in brackets resulting from the same quantities, when we (like Kaufmann) rely, not on the observed values ${\bar {z}}$, but on the values "reduced to infinitely small deflection" z' (l.c. Table VII, p. 529), and from which u is calculated by using Kaufmann's equations (14) and (17 ), determining the corresponding $v={\frac {u}{\beta }}$ according to each of the two theories, and then pass to y' by using Kaufmann's equation (18). Then ${\bar {y}}$ is given by Kaufmann's equation (12). For this calculation, Kaufmann's constants A and B are of course not the "constants of curve" but the "constants of apparatus", which were measured independently of the deflection experiments. The comparison of the bracketed numbers with the numbers stated above shows, that the results of Kaufmann's method of calculation differ from those of mine only very marginally, so each of the two methods supports the other in some way.

As regards the comparison of theoretical values of ${\bar {y}}$ with the observed ones, it can be seen that the latter are closer to the sphere theory than to the relative theory. However, in my opinion this can not be interpreted as a final confirmation of the first and a refutation of the second theory. Because for that it would be necessary that the deviations of the theoretical numbers from those observed, are small for the sphere theory against those of the relative theory. But this is not at all the case: on the contrary, the deviations of the theoretical numbers from each other are throughout smaller than the deviations of any theoretical number from the observed ones.

One might think now, perhaps, that the lack of agreement is caused by the employed value (27) for the ratio ε:μ₀, and that by a suitable amendment of this value a sufficient correspondence can be obtained for one of the two theories. This can be easily tested in the following way. Equation (18) gives, if one substitutes for ${\bar {y}}$ any observed value, the corresponding value of the velocity q = βc regardless of any special theory, and the corresponding value of u is derived separately for each theory by (24) or by (25), and then from (26) the ratio ε:μ₀ can be calculated. This procedure gives not only for none of the two theories constant values for ε:μ₀, but even for β it gives numbers that are unacceptable from the outset for each theory. The same is found, of course, in Kaufmann's method of calculation. Kaufmann^[9] gives two equations for the deflections y' and z' , which combined have the form:

$\beta ={\frac {E}{cM}}\cdot {\frac {z'}{y'}}$.

Where

${\frac {E}{cM}}=0,1884$

is a constant of apparatus, independent of the value of ε:μ₀ and independent of any specific 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, 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 $P{\begin{matrix}<\\=\\>\end{matrix}}1,1\cdot 10^{6}$ 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. Starke^[10]. 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.

Kaufmann: 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 Planck and me, and it is gratifying to me that the very different account of Planck 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 Lorentz's nor Abraham's theory agree with them. This conclusion is certain. Lorentz's theory is even worse than Abraham's. The deviations of Lorentz's theory (10-12 percent) are so great that they cannot be explained at any point by observational errors. So if no fundamental error exists in the observations, then Lorentz's theory is abolished. In Abraham's theory, the deviations amount to 3-5 percent; these are also outside the margin of observational error. But the possibility of errors which sums up in a way that such a difference comes out, would be, nevertheless, still possible.

Planck: If the correction of the theoretical numbers necessary for a complete explanation of the observations, is outside the observational error, then it is conceivable to me that if one take them into account and make corrections in addition to the errors of observation, we could come closer to Lorentz's theory than Abraham's. From the mere fact that the deviations of a theory are smaller, a preference would not follow for it.

Bucherer: I would like to return to the remark of the speaker that my theory is not sufficiently developed to be discussed here even further. I was intensely engaged with it and its consequences, and found that it is not substantially better than the earlier theory of Lorentz and the later theory of Lorentz. One objection is to be found in the fact that in the dispersion theory the electromagnetic masses create difficulties; in all other respects the theory of deformed electrons at constant volume and of the corresponding deformed system performs the same as the newer theory of Lorentz. With the extension of Planck's considerations on my electron, I could not say what would be the consequences of this reasoning for my electron. But I want to call attention to some theoretical points. The dynamics of the electron is not only testable by the deflection of Becquerel rays. Abraham has alluded to the need of attributung to the Lorentz-electron a special internal energy. That difficulty seems greater to me than the deviation from Kaufmann's measurements. Against Einstein's theory of relativity certain objections can be raised. He uses Maxwell's equations, but ignores that certain conditions are not met, namely the validity of the divergence theorem of electrical force.

After I had recognized that the existing theories, including mine, do not meet the requirements, I have asked me the question, if it is possible to be consistent with experience while keeping Maxwell's equations and the principle of equality of action and reaction. This is possible when we rely on the following principle: the ponderomotive force between two systems in relative uniform motion to each other, by consideration of the sign, is the force calculated from the Maxwell-Lorentz equations acting upon a system which is arbitrarily regarded as at rest. Of course, in this model the concept of the aether is not present, because as soon as I introduce relative motions and define the coordinate system in an arbitrary body of the dynamic system, I give up the aether theory. I have followed the consequences and I came to some conclusions in relation to Kaufmann's measurements, which I want to tell. First, the rigid electron would come into question. Due to the relative theory we come to the conclusion that other forces act when the Becquerel rays are directed, not parallel but obliquely to the capacitor plate. Here an easy possibility would be given to test the principle of the relative theory on the basis of Maxwell's equations, it is sufficient to let Becquerel rays fly obliquely to the electric or magnetic field. For vertical motion you get oddly enough the same forces as Lorentz. I have considered whether the deviation of Kaufmann's measurements is based on the fact that an angle is formed.

Runge: I would like to ask Mr. Planck the following: In the contradiction, which he finds when he calculated β from the first value, it has to be taken care whether a small change in the observation already produces a large change in the value. One would have to calculate the interval of β, for which there are still permissible errors of observation.

Planck: β is proportional to ${\frac {z'}{y'}}$. A small change of y' would do much because y' is small compared to z'. The errors, however, are already so great that one can not use the values; we would have to eliminate the exterior values in any case, we can not use them for the theory. To Bucherer, I want to ask a question. Can your equations be brought into a Lagrangian form? And if so, what value has the Lagrangian function H for your newer theory, did you investigate it?

Bucherer: No, I have not studied it and could not decide it at this moment.

Planck: That would be very important, as due to the Lagrangian form the equations of motion of the electron could be reduced to those of general mechanics.

Bucherer: I suspect that Maxwell's equations can be reduced to the Lagrangian form. I have not yet investigated this particular question, but I suppose it is possible, because I use Maxwell's equations without modification for the quasi-stationary motion, but I don't like to say anything definitive.

Abraham: If you look at the numbers, it is clear from them that the deviations of Lorentz's theory are at least twice as large as those of my own, so it may be said that the sphere theory represents the deflectability of the β-rays twice as good as the relative theory. (Great laughter.) When I consider what was the state of the question 5 years ago when I began my involvement with it, so I must be satisfied with the results; for I did not believe at first that the formula agrees with the experiments, and I was very surprised when Kaufmann told me one day that the formula agreed well with the more refined measurements. However, I see the advantage of the sphere over the relative theory not only in better agreement with the measurements, but also in the fact that it is a purely electromagnetic theory. We started from the question of whether the mass of the electron is a purely electromagnetic quantity. The sphere theory answers this question; it considers the energy of cathode rays as purely electromagnetic. The approaches to the electromagnetic energy density was also taken as a basis by Lorentz. However, the Lorentz electron has a kind of internal potential energy in addition to the electromagnetic energy, as I have shown and what is still not refuted. In the relative theory one would therefore not consider the cathode rays as purely electromagnetic processes, but as processes which cannot be explained by electrodynamics.

Gans: I would like to point out that any assumption about form-changes of the electron in motion brings, of course, more parameters into the theory, so that one can better adjust himself to the phenomena.

The Michelson-Morley and the Trouton-Noble experiment require a certain difference of the longitudinal and transverse dilatations, yet the ratio remains undetermined.

Yet, one could create more theories for which this ratio of the longitudinal to transverse dilatation always has different values; one would explain the phenomena of Becquerel best, but you could not say it is the best; it would only be a retroactive adjustment to the phenomena.

Planck: Abraham is right when he says the main advantage of the sphere theory would be that it would be a purely electrical theory. If this were feasible that would be very nice, for now it is only a postulate. The Lorentz-Einstein theory is based on a postulate, namely that no absolute translation can be demonstrated. It seems that the two postulates can not be united, and now it depends on which postulate is to be preferred. That of Lorentz is more sympathetic to me. It's probably the best thing when work continues in both areas and the experiments eventually give the decision.

Sommerfeld (Munich): The pessimistic view of Planck I wouldn't like to join for the time being. Because of the extraordinary difficulty of measuring, perhaps the deviations could have their reason in unknown sources of error. As to the question of principle formulated by Planck, I would suspect that the gentlemen under 40 years prefer the electrodynamic postulate, that those over 40 years prefer the mechanical-relativistic postulate. I prefer the electrodynamic. (Laughter.)

Kaufmann: To the postulate-question I would like to say that the epistemological value of the postulate of relative motion is, however, not very large, as it is only useful for uniform translation. As soon as we take into account rotation and irregular motions, we don't get along with it. Maybe one is trying to banish the aether (which is often perceived as uncomfortable) out of the world, but one has to introduce it in the case of rotational motions, such as in the case of the flattening of celestial bodies.

Planck: Of course, this is only about uniform translation. Irregular motion can already be demonstrated by mechanics, but uniform motion cannot. The requirement is, what cannot be verified in mechanics, cannot be verified in electrodynamics.

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1. Ann. d. Phys. (4) 19, 487, 1906
2. l. c. p. 525 and p. 544.
3. l. c. p. 526 and p. 547.
4. l. c. p. 547
5. l. c. p. 527.
6. M. Abraham, Ann. d. Phys. (4) 10, 105, 1903.
7. H. A. Lorentz, Versl. Kon. Akad. v. Wet Amsterdam 1904, S. 809. A. Einstein, Ann. d. Phys. (4) 17, 891, 1905. See also H. Poincaré, C. R. 140, 1504, 1905.
8. for example M. Planck, Verh. D. Phys. Ges. 8, 140, 1906.
9. l. c. p. 529, equations (14) and (15).
10. H. Starke, Verh. D. Phys. Ges. 5, 241, 1903.