about Bilbo's treasures

[lathspell]

In The Hobbit Bilbo gets his Mithril coat and we all know what happened to it, but now I reread the Hobbit and I saw that it was not the only thing he got at that time.
This is a quote from The Hobbit - Not at home. The one speaking is Thorin Oakenshield.

"Mr. Baggins!" he cried. "Here is the first payment of your reward! Cast off your old coat and put on this!"
With that he put on Bilbo a small coat of mail, wrought for some young elf-prince long ago. It was of silver-steel, which the elves call mithril, and with it went a belt of pearls and crystals. A light helm of figured leather strengthened beneath with hoops of steel, and studded about the brim with white gems, was set upon the hobbit's head.

Now, we know what happened to the mithril coat and to Sting. Also we know that he always was rich for his other treasures. But I can't find anywhere what happened to the belt and the helm. Later in the Hobbit it says that he hung his sword on the mantlepiece, and he had his coat on a stand in the hall, and later he lend it to a Museum. His Ring he kept a secret.

Does anyone know what happened to the belt and the helm?

greetings,
lathspell

[Birdland]

I suppose they either stayed at the museum at Michel Delving, or remained with him at Rivendell.

[Arwen Imladris]

Perhaps he passed them onto Frodo, and perhaps after that they passed onto Sam?

[Frodo Baggins]

When Bilbo is giving me the coat in the book, the belt of pearl and crystal is mentioned. Read it again, Lathspwll, you will see.

[lathspell]

hmm... I should read things better