This seems to work:
/descendant::input[@id="search_query"][2]
I go this from "XSLT 2.0 and XPath 2.0 Programmer's Reference, 4th Edition" by Michael Kay.
There is also a note in the "Abbreviated Syntax" section of the XML Path Language specification http://www.w3.org/TR/xpath/#path-abbrev that provided a clue.