Well, Gollum did not accidentally fall... we are not told what happened to him after he got that little goblin.
But otherwise I suppose you meant that Gollum both carried and lost the Ring, as Dori both carried and "lost that burglar". We can also say it happened almost at the same place.
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"Should the story say 'he ate bread,' the dramatic producer can only show 'a piece of bread' according to his taste or fancy, but the hearer of the story will think of bread in general and picture it in some form of his own." -On Fairy-Stories
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