Ok, this cannot be that difficult.
Both computers (as in sig file) connect to internet wirelessly through Belkin ADSL Router. Superb, works fine.
Dimension desktop can see itself in the 'Workgroup' window. Inspiron can see......nothing.
Both firewalls turned off.
TCP/IP set for automatic IP and DNS. On both machines.
On occasion, I can with fiddling, get machines to see each other, and themselves. So a couple of re-starts and all is fine. Until the next day...