It may be possible to do without dancing entirely. Instances have been known of young people passing many, many months successively, without being at any ball of any description, and no material injury accrue either to body or mind[.]Turns out that Frank really wants to dance. Those two dances he danced with Emma have made him long for a ball, and he busily tries to work out how to have a dance so that Emma, Harriet, Jane and the two Cox girls can all dance at the same time. Alas, the Weston's parlors at Randalls are too small for five couples. And, come to think of it, five couples are too few for a ball. And Mrs Weston doesn't have enough room for dancing and supper, which is her main concern as a hostess.
Despite the obvious impediments to hosting a ball at Randalls, Frank Churchill refuses to let it die.
One really ought, I think, examine why this is so? Why is he in such a lather to have a ball at all? With whom is he so very eager to dance? Putting that aside, he (once again) insists that his concern is not really his - it's his father's, he says: Mr Weston liked the idea of a ball, and will be crushed if there cannot be one.
Emma thinks to ask some questions, although not the ones I listed - she is convinced that he is in a lather to dance with her, as a matter of course.
Emma perceived that the nature of his gallantry was a little self-willed, and that he would rather oppose than lose the pleasure of dancing with her; but she took the compliment, and forgave the rest. Had she intended ever to marry him, it might have been worth while to pause and consider, and try to understand the value of his preference, and the character of his temper; but for all the purposes of their acquaintance, he was quite amiable enough. [Emphasis added.]The next day, Frank Churchill visits Emma at Hartfield to announce a new scheme - based on her strong desire to dance (again, he projects his own desires onto someone else): They are to have a ball at the Crown.
Emma savvily negotiates her father's approval of the scheme. It would be tempting to skip this portion of the chapter, but it is actually important for a couple of reasons:
1. We are shown for the first time how Mr Woodhouse responds to any new scheme - with alarm and a near-complete rejection.
2. We are shown how Emma manages to talk him around by proving (a) that there's no health risk, (b) that it's very convenient for him (and/or his horses) and (c) that the person in charge of the scheme is someone he already knows and trusts (here, Mrs Weston).
This pattern will repeat itself later in the novel when it comes to another sort of scheme entirely, and it's fun to watch Austen set up the pattern so that when we see it later, it feels vaguely familiar and is entirely in character for Mr Woodhouse and Emma.
The examination of the rooms at the Crown
In looking over the rooms at the Crown, there's some concern about where to eat - crowded in the small room adjoining the room where the dancing will take place, or down a hallway in another room, with a (probably unreasonable) fear of exposure to drafts in the hallway.
Frank immediately proposes securing another opinion, volunteering to go get the Coles - or, better yet, Miss Bates, since she lives so close by. Despite Emma pointing out that Miss Bates cannot be relied on for a useful opinion, Frank manages it so that Mr Weston directs him to bring not only Miss Bates, but also Jane Fairfax - whom he feigns not to have thought about. (Seriously, Frank, you've known her longer than you've known anybody in town except your father - and yet you didn't remember her? And yet, nobody calls bullshit on that.) And then he runs off to fetch the two ladies in question.
Our chapter ends with the news that Frank has sent to his aunt at Enscombe for permission to extend his stay a few days (so as to attend the ball), and has asked Emma to stand up with him for the first two dances, to the delight of both Westons.