Cecil, W5DXP wrote:
"Are they assuming equal currents into each element? That`s the real
trick."

The subscript says:
"The two elements are assumed to be thin and self-resonant, with
equal-amplitude currents flowing at the feed-point."

If everything is symmetrical, the self-impedances and the mutual
impedances of the two elements should be equal, producing equal powers
into each element.

Roy Lewallen may have been the source of a caution on phasing errors
which appears on page 8-13 of the 19th edition of the ARRL Antenna Book.
Roy is named on page 8-12.

Kraus says on page 284 of the 1950 edition of "Antennas":
"It is important that the antenna power W be considered
constant.---Until the antenna power was considered constant by G.H.
Brown (Proc. I.R.E., January 1937) the advantages of closely spaced
elements were not apparent. Prior to this time the antenna current had
usually been considered constant."

Kraus took a G.H. Brown idea and ran with it producing the W8JK antenna.
He had a lot of trouble gettinng the W8JK antenna story published due to
naysayers. But it works despite its low impedance.

Best regards, Richard Harrison, KB5WZI