Basically correct, but for the main PC near the main router, there is no absolute requirement to use wireless. It can plug into a spare Ethernet port on the TP-Link 8817

Thank you very much indeed, you have been very helpful

If you have a need to occasionally use a wired Ethernet port in another room in your house, consider investing in a set of powerline adapters.

A powerline adapter plugs into your house power sockets, and has an Ethernet port on it as well. You need a set, as the one will need to plug into your ADSL router to provide the connection to the power line, and of course the other for your to plug your PC/Laptop into. A set generally goes for ~£25 but sometimes can be found cheaper.

so Router -> Ethernet cable -> powerliner adapter in power outlet -> other room Powerline adapter in power socket -> Ethernet cable -> PC

As has been said, those are modem/routers, and you have nearly all you need.

If you get a couple of powerline adapters as also suggested by ih8modem, (there are pricey ones that I linked to earlier that leave the power socket still available), you can have both computers online at the same time.

In reply to a post by geoffo:

does anyone know if I could have both computers on at the same time accessing the internet?

You can only use 1 modem on a phone line. To get 2 computers online at the same time you need a network (referred to as a LAN).

Which the solution(s) offered provided .

...but never mentioned.

21:11 last night I may not have said the exact words, but the gist in less tech terms is the same

My point was that precisely because LAN had not been mentioned, the OP needed to know what it is. Even I haven't made that utterly clear, but it was an attempt .