In order to logon as a user of the second domain, you will need to add the domain to the Realm that has been applied to the Windows Logon Application.
For example.
The Realm 'Windows Logon Realm' is bound to the 'Windows Logon' applications.
The 'Windows Logon Realm' realm with need to include all the domains you wish to be able to log onto.
To add the domain to the realm that is used by the Windows Logon Application please click on Change domains in a realm