A closer look at the trumbone tuner

Let's look at it without the coil first. First question is at what frequency is 130 mm one quarter wavelength?

We can transpose the wavelength and frequency and we find the frequency is 300,000,000 / (.13 X 4 ) or roughly 577 Mhz.

 
If we look at  Zin with a 1/4 wavelength  transmission we line see the DUAL of Zl. (2-12-19 I posted a link to Mr Turners book on transistor theory) Mr Turner explains the theory of duals in his book. The simple statement is we see the opposite at the input and output. Some text say we see the conjagate.
In other words with the load end open we see a short. With the slider on the load end we see an open feed end. With a line shorter than 1/4 wavelength we see capacitance.

So any length will resonant at some frequency and represent a reactance at other frequencies, Below 1/4 wave length we see capacitance. Adding inductance will cancel the capacitance and make the line longer electrically.

The article from wiki says adding the inductor lengthen the antenna and lowered the resonant frequency. So Amie made the loop long enough to resonant at a lower frequency by adding tuning components (mainly the loading coil and then added the shorting strap to make it adjustable.

Here are some more variations. Notice the top right antenna has and inducter sticking off the side.
While Amie found an application in the broadcast band her system would be more common in microwave. When the wavelength is measured in centimeters lead lengths becomes more critical.
The recap:
A parallel resonant circuit is high impedance. A shorted 1/4 wavelength transmission line produces a high impedance on its input. This equals a parallel resonant circuit at that wave length. At other wavelengths the impedance is lower. Using the slide makes it adjustable.