Final answer:
Bilbo is difficult to find because he uses the One Ring, and it is urgent to find him because he has the Arkenstone. The eagles arrive to help the dwarves, and they bury the Arkenstone with Thorin. Dain Ironfoot becomes the King under the Mountain, and Bilbo takes some treasure when he leaves, accompanied by Thranduil.
Step-by-step explanation:
1. When the battle ends, a man comes looking for Bilbo. It is difficult to find him because Bilbo put on the One Ring, which makes him invisible to others. The man can't see Bilbo and has trouble locating him.
2. It is urgent they find Bilbo because he has the Arkenstone, a precious gem that Thorin desires. The outcome of the battle and the resolution of the conflict hin_ges on the possession of the Arkenstone.
3. The eagles arrive to help the dwarves in their fight. They swoop down from the sky and attack the goblins and wargs, providing much-needed assistance.
4. They bury the Arkenstone and Thorin's sword, Orcrist, with Thorin under the mountain as a mark of respect and to honor his role as a leader.
5. After Thorin is buried, Dain Ironfoot becomes the King under the Mountain. He assumes the throne and takes charge of Erebor and its riches.
6. When Bilbo leaves, he takes some treasure from the dragon hoard. He takes a small chest of coins and a beautiful necklace made of emeralds that he finds in Smaug's lair.
7. The elf king, Thranduil, escorts Bilbo for part of his way home. Thranduil wants to ensure that Bilbo keeps his promise of sharing the treasure with him.