"... and the two-sixth with Lizzy, and the Boulanger..." (pg. 15)
The Boulanger was a simple dance done with couples in the 18th and 19th century. It was an especially popular dance at balls because couples traded couples several times, so everyone got the chance to dance with different members of the opposite sex. Austen references this dance because it was a dance done during her time and adding in cultural context clues gives the novel a more realistic sense about it.