Please confirm that you are not a student at a University or similar.

The resonant frequency of anything is a characteristic of that system. It has nothing to do with the frequency at which you actually operate it. That seems a strange answer because clearly you choose the frequency at which you will operate the device depending on its natural responses.

Suppose I have a motor driving a cam which pushes a spring which in turn pushes a mass. At a certain motor speed the mass will move much more than at other frequencies. This is the resonant frequency. The resonant frequency relates to the size of the mass and the size of the spring. The speed at which you drive the motor doesn't change the characteristics of the system. We have a "natural response" and a "forced response". Look these up in a text book or google them.

I am just trying to learn this as my job is actually in PLC controls and I wanted to know more about antennas because of a new wireless network we are putting in. I bought a learn at home type book and this is one of the questions at the end of chapter.

So then based on the size is there a calculationn I could you or would it be directly porportional to the size I would assume?

If you take the simplest antenna, the half wave dipole, it is roughly half a free space wavelength from tip to tip at its most efficient/effective operating frequency. Typically antennas need to be half a wavelength across to work well. Now broad band antennas like the "log period" and the "biconical" are angled. If you put a ruler across from one side to the other you can find metal to metal distances which give a half wavelength over a large range of wavelengths. For the log periodic you can see effectively lots of half wave dipoles in parallel, which is why you get such a wide bandwidth.

The key things about an antenna are directivity and output reflection coefficient (matching). If the antenna is highly reactive it will require more matching to get power n or out. If the antenna is more directive you will get a higher output signal for a given incident field strength (higher "gain").

To be just a bit more precise, the resonant frequency of an antenna is determined by its total capacitive reactance (Xfc) and its total inductive reactance (Xfl).  The frequncy at which these two characteristics are equal is the theoretical resonant frequency.  That is why antennas can be made to operate successfully at frequencies other than their naturally resonant frequency, which is primarily determined by its physical characteristics.  So to truly determine how your antenna will function, you need to know (or calculate) the Xfl and Xfc.

right. I knew that about the cap and inductance which if I read correctly to increase you would add a cap in series and to decrease you would add an inductor which is basically a coil of wire. This way beacause of their properties caps in series react different from coils in series where as one add and the other divides.

You are correct in your description of how you could "increase" and "decrease" the resonant frequency.

Here are the basics.  The formula for determining the resonant frequency is f(res) = 1 / [2 * pi (sqrt LC)].  A practical means of using this information could be as follows:

First, you construct an antenna of specific dimensions.  You measure the capacitance.  You measure the inductance.  You plug both values into the above formula and that will tell you your resonant frequency or f(res).  Let's assume that your DESIRED resonant frequency is higher than your calculated resonant frequency.  You can see from the formula that you can either decrease the inductance L or decrease the capacitance C, or both, and the result of any of these actions will be an increase in the resonant frequency.  Any action you take that makes the denominator of the formula smaller makes the resonant frequency higher in frequency, thus more closely approaching your DESIRED resonant frequency.

Second, you can see from the formula XfL = 2 * pi * f * L that decreasing the L as in our example above decreases the inductive reactance (XfL) while raising the resonant frequency.  Increasing the L by adding a series inductor would decrease the resonant frequency, as you correctly state in your last post.

Finally, you can see from the formula XfC = 1 / 2 * pi * f * C that decreasing the C as in our example above increases the capactive reactance (XfC) while raising the resonant frequency.  Decreasing the C by adding a series capacitor would increase the resonant frequency as you correctly state in your last post.

The whole point of my original post was this.  You asked if length is the determining factor in resonant frequency.  It is ONE factor, and a major factor as well.  However, anything in the construction, materials and/or feed method that affects either the XfL or XfC or both will also be a factor.  Therefore to directly answer your original question, "...how is the resonant frequency of and (sic) antenna determined.", it is determined by the value of the total XfL (which is dependent on the total L) and the value of the total XfC (which is dependent on the total C).

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