Many thanks for your response. You've opened up a new world to me --
but I've come to a roadblock again.

First, let me clarify that I can view both the Desktop's and the
Laptop's Alpha folder side-by-side in either computer. I can't do that
with the Beta folder, though. (In real life, there is only one Alpha
folder and about eight Beta's.)

I'm the only user of both computers, so I guess I'm the Administrator,
and I log in as the only named user. (I use the Home Edition of WinXP
SP2 amd both computers connect directly to the internet via a DSL
modem/router.

In trying to follow your suggestions, the first thing I did was to
START>SEARCH>COMPUTERS on each machine. On Desktop, there are two
entries for Laptop, and on Laptop, there are two entries for Desktop.
Why are there double entries, I wonder? Need I do something about
that? If so what -- just delete one of the entries?

Next, I did a search on "permissions" in the START>HELP window and
discovered a new world. I got a long list of hits. I then clicked on
"Grant incoming connection access rights to your computer." That
brought up instructions that began with "Open Network connections,"
which I did. The instructions then say to click on "change settings of
this connection" but I don't have such an item listed under network
tasks, so I'm at a dead end.

Could it be that such an entry exists only in WinXP Pro?

Your last sentence presents a very appealing solution, but I'm a bit
confused about how I can get the computers to treat the newly created
"Network" folder like Alpha rather than Beta -- so that I can do side
by side comparisons of the files and folders in the two Network folders
on either machine.

I wold very much appreciate you continuing comments.