Jimmie D wrote:
"The book shows a .25wl of antenna and a .25wl of stub,---."

The J antenna is often described as a half-wave of antenna with a
quarter-wave matching stub.

Function of the stub is to shunt the antenna at a point so that the
parallel combination of antenna and stub equal Zo.

If the antenna were only 1/4-wavelength, reflection from its
open-circuit end at its drive point may yet be compensated by a .25wl
stub.

At the 1/4-wavelength stub`s short-circuit, phase of the reflected
voltage is reversed. Voltage reflected from the open-circuit tip of the
antenna is not reversed in phase by its reflection. Both waves travel
the same distance to the point where they meet at the antenna drive
point. These tend to offset each other.

Terman describes impedance matching using stubs on page 109 of his 1955
opus:
"Thus, although a reflected wave is present in the length l1 because of
reflection from ZL, there is no reflected wave on the generator side of
the stub line as a result of the cancellation of the two reflected
waves."

Best regards, Richard Harrison, KB5WZI