On Saturday, February 15, 2014 3:27:15 PM UTC-6, Sal wrote:
As built, both the pattern
and the impedance already seem to be what I want.

And it already has a counterpoise. The counterpoise is where the current flowing on the inside of the coax shield goes when it leaves the coax. The J-pole has a radiation pattern of a 1/2WL vertical, i.e. the lower 1/4WL doesn't radiate (much) and is part of the matching section and counterpoise. Quite often, the outside of the coax braid is also part of the counterpoise.
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73, Cecil, w5dxp.com