Roy Jerome Meyers' device is documented in British patent 191301098. The Rex Research website also has a newspaper article from around 1912 re this device along with the patent text and figures.

Near the beginning of his patent he states that "I have by actual test been able to produce from a very small apparatus at comparatively low elevation, say about fifty or sixty feet above the earth's surface, a substantially constant current at a commercially usable voltage and amperage."

The patent describes two versions of the device. One appears to be an advanced version, and dominates most of the patent and figures, and the other was his first successful one (Figure 8 in the patent.) Both make use of a "specially constructed rectifier which also has the capacity of intensifying the current". This rectifier consists of coils and a Cooper-Hewitt mercury vapor lamp. The advanced version uses four of these recifiers for a bridge rectifier. The first successful version uses just one rectifier.

You may still need to read the patent to fully understand this. The purpose of the drawings below is to highlight things that I found in my analysis. They took some work to figure out, especially the wire to coil assignment which was a total surprise (see diagram of that name below), so this should save you some time.

Advanced version

Roy Meyers' first successful version

For completeness sake I've included Roy Meyers' first successful version. Meyers said he was "able to obtain a constant current of 8 volts" (conflicting terminology aside, 8 volts is pretty good.)

My attempt at replication

Testing done with simple air core coil from a crystal radio. See below for where to get all the parts. The results were nothing more than you'd get with a normal antenna, but this was only a quick test. There are many variables still to play with.

The zinc plates were purchased from Home Science Tools. Warning - they don't bend. If you try to bend them they snap in two. The bare iron wire was purchased from Boreal Northwest (they're in Canada. For US, use SK Science Kit & Boreal Laboratories.) The U-shaped magnets (what many people incorrectly refer to as horseshoe magnets) were the 100mm ones purchased from:

INDIGO INSTRUMENTS
169 Lexington Court, Unit I
Waterloo, ON, N2J 4R9
Canada
http://www.indigo.com/magnets/gphmgnts/u-magnets.html
Phone: (519) 746-4761 Fax: (519) 747-5636
Or ORDER toll-free at: 1 (877) 746-4764

Roy Meyers device and the Linden Experiment

Note the similarity between the Roy Meyers device and the Linden experiment. Might the Methernithans have studied, replicated and improved on the Roy Meyers device?