Well technically it is possible, just like i have my PS2 network adaptor plugged into my router.
However, keep some things in mind:
- Take the right cables with you: there are patch and crossed ethernet cables. I'm not sure which one the PS2 uses, bit it could be diffrent from the cable that the school provides you, so use your own cable you got together with the network adaptor.
- Most schools have firewalls running, there is a good change that th post/site/host that the PS2 network adaptor is blocked. If you want to be sure about this you could as the network administratos there, otherwise just try to connect, if it keeps on failing they probally blocked it.
- You might or may not have to fill in the IP adresses of DHCP Gateway and DNS server, it all depends on how everything is configured there. If it doesn't get discovered automatically you could ask somebody there (just say that you want to connect your laptop if they have a policy against consoles, or lookup the correct value at a PC in the school that has internet access.
Firewall would be the biggest problem, if the block the ports you need it will be impossible to play online.
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