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Tesla made clear references to terminal elevation throughout his writings about wireless transmission.  It was something that was important for the task.  I'd like to know exactly why.

Tesla's wireless system is a means of transmitting electrical energy through "the natural media."  It involves the production of electrical "current impulses of relatively-low electromotive force at a generating-station, utilizing such impulses to energize the primary (I) of a transformer, [and] generating by means of such primary circuit impulses in a secondary (II) surrounded by the primary."  "[From] one terminal of the secondary the current is led by a conductor to a terminal maintained at an elevation suitable for the purposes of transmission.  The other terminal of the secondary is connected to earth."  The preceding describes the basic Tesla coil transmitter.

"At the receiving-station a transformer of similar construction is employed; but in this case the coil of relatively thin wire, constitutes the primary (III) and the coil of thick wire or cable, the secondary (IV) of the transformer." [SYSTEM OF TRANSMISSION OF ELECTRICAL ENERGY, March 20, 1900]

The basic Tesla wireless transmission-reception system.

Within the context of the Tesla wireless system, the air terminal or elevated terminal is, by definition, "a conductor or terminal, to which impulses such as those here considered are supplied, but which is otherwise insulated in space and is remote from any conducting-bodies" [ibid].  It is usually fabricated in the shape of a sphere, a torus, flat disk, or rectangle (typically with rounded corners). 

The elevated terminal serves two purposes.  First it is a reservoir for electrical charge.  Tesla puts it this way, "The elevated terminal charged to a high potential induces an equal and opposite charge in the earth and there are thus Q lines giving an average current I=4Qn which circulates locally and is useless except that it adds to the momentum" or buildup of electrical energy within the oscillating system [THE TRUE WIRELESS, May 1919].  About this term "momentum," Tesla spoke of a "fundamental law of motion, which commands acceleration and increase of momentum or accumulation of energy under the action of a continuously acting force and tendency, and is the more true as every advance weakens the elements tending to produce friction and retardation" [ON ELECTRICITY, January 27, 1897]. This phenomenon of an increase in electrical activity in the resonator over time by the periodic application of primary impulses is employed by Tesla-coil hobbyists to produce big sparks.

The second purpose of the Tesla coil transmitter's elevated terminal is to serve as one of two high-voltage electrodes, the second high-voltage electrode, that is to say, the second elevated terminal, being located at the remote receiving location.  It is connected to the receiving transformer's primary winding--the helical resonator--the lower terminal of which is connected to earth.

As to why placement of the air terminal in a position, "insulated in space" and "remote from any conducting-bodies" is a requirement for proper system performance, I agree with Gavin that, "It is to do with the way the lines of electric flux spread out."  We know that a significant portion of the current flowing in the resonator "circulates locally," adding to the buildup of electrical energy within the oscillating system.  This locally circulating current exists in the form of Maxwell's "displacement current" flowing between the air terminal and the earth in the vicinity of the transmitter.  It is also believed that in addition to this local electric field, "A relatively small number [--often a very small fraction--] of [electric field] lines" originating at the transmitter's air terminal "go off to great distance."  Some of these electric field lines reach the remote receiving location and terminate on the receiver's air terminal.  Elevating the air terminals simply increases the proportion of interconnecting electric field lines by reducing the value of the two local shunt capacitances, Csec1 at the transmitter and Cpri2 at the receiver.

Redrawing of a Simplorer circuit model of the Tesla wireless system created by the Nevada Lightning Laboratory.

Credit: Leyh, G. E. and M. D. Kennan, Nevada Lightning Laboratory, "Efficient Wireless Transmission of Power Using Resonators with Coupled Electric Fields," 40th North American Power Symposium , University of Calgary, Calgary, Canada, Sept., 2008.

It may be argued that at a distance of one-sixth wavelength or so, the amount of electrical energy flowing across C_coupled should diminish to a vanishingly small quantity and the displacement current will cease to exist.  Real-world experiments show this not to be the case.  Electrical energy continues to pass between the Tesla coil transmitter and Tesla coil receiver as they are further separated, even in the demonstrated absence of radio waves.

Tesla himself pointed out that the connection between transmitter and receiver need not exist entirely as true conduction currents.  "As to the elevation of the terminals . . . In some cases when small amounts of energy are required the high elevation of the terminals may not be necessary, since, especially when the frequency of the currents is very high, a sufficient amount of energy may be collected at that terminal by electrostatic induction." [SYSTEM OF TRANSMISSION OF ELECTRICAL ENERGY.]

I'm sure that you would like a more specific answer.  The fact of the matter is, this will only come from a rigorous study of the influence of elevation upon capacity as first carried out by Nikola Tesla in 1899.

Carrying on a little farther, elevating the top-load or air terminal slightly above the Tesla coil secondary on the upper end of an electrical conductor creates what appears to some as an electrically short antenna.   In addition to reducing the chance of breakout, increasing the diameter column diameter should also reduce the production of radio waves, aka "electromagnetic radiation," by the Tesla transmitter.  Just how efficient a radiator is that short conductor?  This is easily answered by measuring the radio-wave emissions of the Tesla-wave transmitter.

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