In book 3 of "The Odyssey," what does Athena do as she departs?
This is actually kind of a strange part of the story. Athena starts by sort of making up a story about why she has to leave:
"I am the only older person among them; the rest are all young men of Telemachus' own age, who have taken this voyage out of friendship; so I must return to the ship and sleep there."
This is a perfectly plausible excuse for ditching early on the party. She goes a little further to explain why they won't be seeing her for a while:
"to-morrow I must go to the Cauconians where I have a large sum of money long owing to me."
This gives her a great excuse to disappear back to her own godly business for a while. She even gets a bit pushy before departing:
"As for Telemachus, now that he is your guest, send him to Lacedaemon in a chariot, and let one of your sons go with him. Be pleased to also provide him with your best and fleetest horses.”
So she has really set herself up to depart...she says why she is leaving, why she won't be around for a while, and what the King can do to set up Telemachus for his trip.
But then she does something amazing, which seemingly ruins all that prep-work:
"When she had thus spoken, she flew away in the form of an eagle, and all marveled as they beheld it. Nestor was astonished, and took Telemachus by the hand. “My friend,” said he, “I see that you are going to be a great hero some day, since the gods wait upon you thus while you are still so young."
That, I think, really answers the question you are asking. She reveals herself to be a goddess by turning into an eagle and flying away. Why she bothered with all that subterfuge before, I don't know, if she was just going to give herself away. But hey, she's a goddess...I guess she can do what she wants.