WP logs on local rather than remote

I’m trying to log on to ‘192.168.x.n/mysite/wp-admin’ from a computer at 192.168.x.m. While I can access the remote site from this computer, WP won’t log me in–it keeps logging me in to ‘192.168.x.m/mysite’–that is, to the WP site on my local computer.

What’s happening here? How do I log into the site as I’ve got on the other computer? I’ve tried clearing the Firefox cache on the local computer, but that doesn’t help.