2 WP installations, how to allow searching both?

We have two WP installations due to size of the site. We would like to allow a search box at the top of the site and allow the visitors to choose to search section 1 or section 2. Is it possible to add a second search box to the header which will allow access to the second database?

I’m not wishing to have one search box to search both databases at the same time. The visitor would choose either section 1 or section 2. But both search boxes would be on the header site wide.

I hope this makes sense ??