You will follow the Gray Lady through a few different areas of the Library. When most of the time, you might have to listen to the original Ghostbusters whine about how she keeps shushing you. You will be getting near the end of this chase when you reach a room where the bookcases start moving to block their path. Just keep walking the maze, and taking the open option until you reach the other side of the room. Yes, this means that you may end up back tracking every once and a while. No, you cannot use your proton pack to speed things up.
Eventually, you'll find yourself confronted with The Gray Lady once again. In a very frustrating move, she once again leads you deeper into the library. Continue with this game of chase for several more rooms. Don't be too thrown off when she summons both book bats and golems to make your life difficult. Handle this the way you would any other proton pack fodder and keep on chasing. Make your way into the children's reading room and use your PKE meter to find a trophy, there will also be one in the special collections room. Eventually, you will be confronted by two unpleasant ghosts with shields. Capture the shields, and toss them away before you attack them. Otherwise, you won't do any damage. Once the room is clean, put your meter back on and use it to find the secret passage in the back.