You do not say who your broadband ISP is but I think you will find your connection is fixed to one computer. That is to say the computer that you registered the connection on. A note of the MAC address in your network card will have been made & an IP lease (I think!) issued against your network card. This means if you plug your broadband lead into another machine it will not work.
That said, it does of course depend on your ISP, you will need to check with them if you can use more than one computer on your connection, even if you are using them at different times. If you are only allowed one connection, then your best bet would be to buy a router, & this will allow you to use both your machines on the connection.