Well, my formula is far from scientific, but if you're
 going to build one, I'll guarantee that it will work for
 VHF and above.  Here's my simplified, seat-of-the-pants
 J-Pole design strategy that I've used with excellent results:
 
 First, a J-Pole is defined as a 1/2 length radiating
 element, fed by a 1/4 wavelenght parallel transmission
 line.  This yields the following picture (viewed from the
 side):
 
          ------------------------------------------------
 base ->  |
          ----------------
 
 It is related to the end-fed Zepp trailing wire antenna,
 once used on Blimps.  The main idea calls for the longer
 element to be (1/2 + 1/4) 3/4 wavelength long and the
 shorter one 1/4 wavelength at the frequency of interest.
 
 The Lower part of the J forms a parallel transmission line
 which is used as a matching section for the 1/2 wavelength
 free end of the long element.  In my design, the elements
 are 1/4 inch in diameter and are spaced about 2.5 inches
 apart yielding a transmission line impedance of around 400
 ohms give or take..
 
 A coax line is attached to the two sides of the parallel
 transmission line at a point above the base where a good
 match is obtained.  The method and positioning of this
 attachment is one of the greatest area's of discussion and
 dissent amongst J-Pole builders.  
 
 Some builders connect a coax like this:
 
 one side of coax here v
                   ----+----------------------------------------
                   |
                   ----+-----------
 one side of coax here ^
 
 With this method, the builder will usually make a hard
 electrical connection at each point the coax attaches to
 the pole.  Many designs using copper pipe for the elements
 find it handy just to solder directly to the elements.
 Most other designs will attach to a couple of screws.  The
 design in the Antenna Book (ARRL), uses a sliding bar which
 can be moved up and down the J and secured with set screws
 at the matching point.
 
 There is often argument as to which part of the coax (the
 center conductor or the shield) should be attached to which
 arm of the J.  In the ARRL Antenna Book, their schematic
 shows the center conductor on the short arm while the photo
 of the finished product shows the center conductor attached
 to the long arm.  I always attach the center conductor to
 the long arm.  I'm fairly sure, however, that it does not
 matter because what the RF sees is really something like
 this:
 
     ============---------------------------------
 
 Which is parallel line transitioning to a single radiator.
 Whether this is true or not is irrelavent, however, since
 it's safe to say with certianty that if you conect your
 center conductor to the long arm it WILL work.
 
 
 My J-Poles use a different feed arrangement than the one
 just shown.  I construct my J-Poles with a SO-239 mounted
 in the base of the J, which becomes the attachment point
 for the coax shield.  I then take a 6 or 8 inch piece of
 number 12 solid insulated wire and wrap in about 4 times
 around the long element, about 4 inches up from the base.
 There is no DC connection between the feed wire and the
 element.  The end of this wire is soldered into the center
 pin of the SO-239 which is sticking up from the bottom of
 the base plate.
 
            ------0000-------------------------------
            |     |
 SO-239 -> [[[-----
            |
            -----------------
 
 This same feed arrangement is quite popular around the San
 Jose, CA area where it has been sold in large numbers by
 the American Legion at the Foothills Amateur Flea Market
 for a number of years.  It is a capacitive coupling
 arrangement, despite the look that the wire is wound into a
 coil around the long radiator.  It's a bit touchy to
 adjust, and the SWR is affected by bending the feed wire
 left or right between the radiators as necessary, or by
 adjusting the tightness of the coiled wire (the capacitor)
 around the radiator.
 
 At least one other design that I've seen uses a variable
 capacitor with this same arrangement, mounted in the middle
 of the feed wire which is electrically attached to both the
 SO-239 and the radiating element.
 
 Matching the antenna to the line is not terribly
 difficult.  In my design, I made the base out of a flat
 piece of aluminum bar stock through which set screws were
 used to hold the elements in place.  This also allowed for
 some experimental lengthening and shorting of the short
 element when tuning.
 
 In a nutshell, if you make the long element 3/4 wavelengths
 long, and make the short element 1/4 wavelength long with
 some provision for adjusting it's length, then in just a
 few minutes you'll find it very easy to locate the matching
 point for the feed wire using an SWR meter.  During my
 short production run I was able to achieve a good match
 which was easily repeatable, by carefully measuring the
 feed wire and standardizing my forming method.  This
 process required no further adjustment after construction, 
 and did not require the use of an SWR meter.
 
 My final design resulted in a somewhat shorter than
 expected short element (17 inches instead of 19) but a
 perfect 1:1 match nonetheless.  I've built about 25 of
 these antennas in both 2M and 440 versions and have heard
 nothing but satisfaction from my customers.
 
 There are two additional things which come to my attention
 on the subject.  First, it ought to be possible to achieve
 a 5/8 J-Pole which would have (3dB) gain.  Secondly, there
 seems to be no commercial manufacturers of J-Poles although
 the Ringo Ranger seems to be is closely related to it.
 Ham's agree that J's work well, and will almost never pay
 more than $20 for one (which seems to be a magic price).
 
 So there you have it, a non-technical description of J-Pole
 design, derived from reading, observation and practice.  I
 hope you find it helpful.
 
 73,Fred Lloyd  AA7BQ          Fred.Lloyd@West.sun.com