Yes, the client finds out the Partner name from the primary
See "Client-side Redirect" section in this MSDN whitepaper. My bold
You must specify the initial principal server and database in the connection string, and optionally the failover partner server.
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The great advantage of using the database mirroring support built into ADO.NET and the SQL Native Client driver is that you do not need to recode the application, or place special code in the application, to handle a database mirroring failover.
However, it's always good to test of course in your particular setup...