Milly goes to the National Gallery because she feels like it's a private place where people don't usually go. She spends most of her time there people-watching and thinking about the lives of others. She envies some of the regular people for getting to live regular lives, and wishes she could do the same.
Milly listens to a mother talking with her daughters about whether something is "handsome" and feels a close connection to them. When they leave, she glances around the gallery again and finds her gaze stopped by a couple.
And who does it turn out to be? Merton Densher and Kate Croy, hanging out alone in a place where not many people would recognize them.
When they greet each other, Kate finds out that Merton actually knows Milly, which is a big surprise to her. Not knowing what else to do, Milly invites them back to her hotel for lunch.
When she brings them back, Susan is there waiting to tell her what Sir Luke has said. But she can't with Merton and Kate there.
As the meeting drags on, though, Milly becomes re-convinced that Kate truly doesn't return Merton's affection. Slowly but surely, though, Kate and Merton's place in Milly's imagination is replaced by thoughts of Sir Luke Strett and the news he has given Susan.
And that brings us to the end of Volume 1 of this book. Give yourself a pat on the back for getting through it.