Having fixed my wifi login problems, the iBook was still having trouble communicating with my network and the internet simultaneously; it seemed that I could only get on line if I logged into the router and forced a reset.
The fix I eventually came up with was something that Be had previously told me to do with the other computers (which don't use wifi and aren't going to be used elsewhere); set the default DNS to Be's address, with the backup set as Google's server, 22.214.171.124
That worked perfectly, and I've tested that it will still work if I'm not logged in via Be, using the WiFi from the flat below (with permission), but now I'm wondering what happens if e.g. I'm not at home and Google's server goes down. It previously seemed to be OK without a default DNS server set; am I right to assume that in the absence of both it will find another by itself?