But then geography comes into play. The router/modem is in the kitchen, the desktops are upstairs, and static cabling is simply not an option. So the problem is connecting an old 750MHz PC running Linux so that it can be accessed remotely (don't want an extra keyboard, mouse, video hanging around) to the upstairs desktop PC running MS Vista, without the expense of a wireless network adapter.
I tried adding a spare router and using the Vista PC as abridge, but it didn't seem to work and raised the average power consumption anyway. But an answer lies in a simple Ethernet crossover cable and Microsoft Internet Connection Sharing.
Setting up ICS is easy, although it sets up the PC as 192.168.0.1, so it may be necessary to change the IP address of the router/modem or Vista throws out a very unhelpful "can't do it 'cos I just can't!" errors.
Then the Linux network connector has to be set up. This is straightforward (delete the old one connection, and add a new one), although automatic IP address allocation using DHCP appears to be needed.
And that is it. Just need to try and set up the dreaded XDMCP ...!