Miss De Bourgh exerted herself so far as to curtsey and hold out her hand to both
Curtsying is a sign of respect typically used by those of lower status to address those of higher status. In this case, Anne De Bourgh is showing deference to Elizabeth and Lady Catherine curtsying first, which is somewhat strange, considering she is of higher status. By holding her hands out, she is also expecting a similar show of respect back.
Image of a light phaeton (Two Nerdy History Girls).

