The brilliant and surprising light phenomena in Tesla's experiments¹ have aroused general interest in Germany, especially since they were demonstrated to a larger public by Mr. P. Spiess² at the Urania in Berlin.

The success of these experiments does not require as much resources as Tesla himself used, but rather that they can be carried out using a larger induction device³ or a powerful influence machine⁴.

A short theoretical discussion of this subject is in order, at least as far as the movements of electricity in the conductors are concerned. As for the transfer of the electrical oscillations to the air or to the ether, through which the actual light phenomena are caused, further experiments to clarify the situation are required.

It will therefore be our task to determine the frequencies of these two oscillations, their initial amplitudes in the two circles and the size of the attenuation.

For this purpose, we denote the current intensities in the two circles with and , and the potential differences of the two capacitors from the capacities and with: and , the resistances with and , the induction coefficients with and , and finally the coefficients of the calculation induction with .

Then the equations apply:

After introducing and in place of and we get:

To integrate these equations one sets:

so it's:

or:

This results in a fourth degree equation for calculating :

Since we have ruled out aperiodic movements of electricity from the outset, the four roots of this equation must be of the form:

Instead of solving this equation in general, where quite complicated expressions are to be expected for the roots, we want to make use of the idea that approximately correct expressions can be found for the period of oscillation of the electrical oscillations when discharging one capacitor without induction onto a second one You get a circle if you neglect the link that contains the resistance.

The formula follows:

So the resistances are here in the same way, one obtains from equation (5)

or:

Also:

If one takes into account that under this assumption the damping factors and become zero, so that the roots of the equation (5) increase the values:

and that the oscillation times and are related to them by the equations

so it's:

From this we see that the current is always in bothCircling dissolves into two simultaneous oscillations of different oscillation durations.

Let us introduce the oscillation times and T of the two individual circles into these expressions by putting:

Furthermore, let be a time of the same order of magnitude, which by the equation:

may be defined, then:

We now move on to discussing the following two special cases.

a) Let . The two circles are arranged so that they have the same period of oscillation. This is usually referred to as resonance.

Then:

or

Accordingly, in this case too, the two oscillation times are different and one oscillation period is longer, the other smaller, than the oscillation period of the two individual circles.

b) The period of oscillation of one circle is considerably longer than that of the other. So :

But it's only 8 small, so

is. Then it's a first approximation:

In this case, the oscillation duration of the slower oscillation is close to the oscillation duration of the first circle, and is therefore longer than that, while the duration of the faster oscillation is somewhat smaller than that of the second circle.

3. We now turn to the question of how the potential energy of the electricity initially accumulated in the first capacitor is distributed during the discharge between the two circuits and within them between the two individual oscillations. With other values: the amplitudes of the individual oscillatory movements should be calculated. However, the influences of resistance should also be neglected here:

Then equations (2) take the simple form:

It is sufficient to use the following form of solutions:

To determine the amplitudes you get the equations:

Further is initial:

This follows if one uses the abbreviation:

So:

What should be emphasized here is that the amplitudes of the two different oscillations in the secondary circle are the same size.

Using the previously calculated values of and one obtains:

In particular are the amplitudes of the secondary oscillations:

In the special case of resonance:

It could therefore be advantageous to ensure that the two oscillation times of the individual circuits agree by taking a small value for the induction coefficients but using a capacitor with a larger capacity, while the relevant values are the other way around for the secondary circle are to be made.

4. In order to determine the damping of the two individual oscillations, we have to go back to the general equation (5) for 3. and calculate the roots in the form of equations (6). The numbers and are then the damping factors belonging to the oscillations and . However, instead of solving the fourth degree equation (5) in general, it is probably preferable to approximate the quantities and in the following manner determine. For that purpose we still see the values and now the equations (8) and (9) - as correct. Let us also write equation (5) in the abbreviated form:

and if you think: , then satisfies the equation:

Let us further put in equation (20) above:

but under the assumption that is small compared to , we get higher powers of and the products of with and :

From this it follows, taking into account equation (21):

In the same way one obtains the approximate value for the damping factor of the second oscillation:

We substitute their real values for the individual letters and, if we introduce the abbreviated names, we get:

In the case of resonance, these expressions take the simple form:

Since is smaller than , it follows that the damping of the shorter oscillation is greater than that of the slower one. The last two expressions can be reduced to a simpler form that is more convenient for calculating numerical examples.

The induction coefficients and of the two windings of the transformer contain the squares of the number of turns, the coefficient the products of these. You can therefore set:

where is a number factor that depends on the arrangement - at least less than one.

If you introduce this into the last formula and take into account that here

was accepted, so is

According to this, the general integrals of equations (2) could now be written in the form:

Since we have now obtained approximately correct values for , , , , it would not be difficult to determine the eight constants according to the initial conditions. I will refrain from repeating the complicated formulas.

5. On the other hand, it seemed to me not without interest to calculate the numerical values of the most important quantities for an example.

After an approximate estimate, I base the constants of the apparatus on values that correspond to an arrangement with which I could repeat the most important Tesla experiments.

If you set:

further, to satisfy the resonance condition:

(i.e. equal to 900 electrostatic capacity units)

finally:

This is what you get if you keep the previous designations:

or the vibration numbers:

a) for the individual oscillation of the two circles:

b) for the two vibrations obtained by combining them:

The two damping factors are corresponding: