One of the only things I can recall being referred to as a line, is the Line of Merlin Unbroken - the short rod Dumbledore has when presiding over the Wizengamot. Of course, that's not silver...

EDIT: Just expanding; the Line seems to both refer to the physical object Dumbledore holds, but also the succession of people, going back to Merlin - making Dumbledore literally a fraction of a line.