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Royal Court Chess sounds like a good name for this variant.

But I'm not sure if having the Prince Piece promote into a piece having
all the benefits of a Knight and King will work if the enemy has a duty
to
capture both the original King and the Knight-King to win the game.  But
wouldn't that kind of a rule result in more draws?  I think it is easier
to play with the Prince turning into a King, and then suspending the
rules
about moving into check.

Perhaps a rule set can be dreamed up for Zillions of Games with the two
extra pieces:

Prince   = Knight+Pawn (subject to all en passant rules)
Princess = Bishop+Pawn (subject to all en passant rules)

where, if the Prince (or Princess) has a clear path, can move straight up
to the fourth rank like an ordinary pawn could, subject to any captures
en
passant by other pawns in the adjacent files.  (I can imagine some pretty
interesting chess puzzles with these kinds of pieces.)

But I am worried that promoting the Prince into a King-Knight would make
it TOO powerful, especially if it is at the same time an extra King, and
the rules against moving into check are suspended whenever there is an
extra King on the board.

Maybe you could put together a Zillions of Games rule set for something
along those lines?  I've looked at the rules for Zillions of Games, and
it looked a little too complex for me.  (I was never entirely sure if an
ordinary ASCII editor could be used for editing the variables, or whether
the names of the variables were case sensitive, or exactly how a template
was supposed to be put together.)

An interesting thing about a set up using these pieces on a 10x8 board is
that no pawns at the start of the game are left undefended.  That
resolves
a problem relating to that of Capablanca's 8x10 setup.  And having those
two new pieces with combined powers of the minor pieces, instead of the
major pieces, yet subject nevertheless to the traditional en passant
rules, insures a healthy respect for strategic play, and reduces the
emphasis on hard tactics.  Mainframe computers will also have a harder
time exhaustively analyzing a board with that kind of a set of pieces to
deal with.