I just typed a long reply, then in checking the last thing to put into it, something crashed the desktop
Thanks for all those suggestions (and for the others PM'ed to me). Some easy bits of the problem are fixed - the config screwup had reset ZoneAlarm so that the LAN and all on it was outside the trusted zone. Fixed that.
The two machines can now ping each other fine.
The desktop can "see" the laptop and attach its drives, but not the other way round.
Richard - I can't remove the $ from the sharing of folders. I can delete the C$ D$ etc sharings, but get a long error message when I do so, and they come back again when I reboot.
If I go through "My Network Places", to "Entire Network", then double-click MicrosoftWindows Network, then the network name, the laptop sees the desktop there. Double-clicking the desktop brings up a long message ending "The user has not been granted the requested logon type at this computer."
Doing the same on the desktop shows the laptop and allows attaching to it.
Net view from desktop to laptop is OK, but laptop to desktop brings up "System Error 5 has occurred. Access is denied."
I can move stuff from desktop to laptop, so one bit of the problem's fixed. But I can't print from the laptop, which is a pain.
Frustrating, this computing lark!
The Bluetooth modem arrived for the laptop and I can now surf and e-mail from it - but that too isn't straightforward: I have to use O2's SMTP server rather than my own; since I don't have an O2 email address, replying to those e-mails doesn't work. Early days with that - I've invented a workaround that does the job.