kashefian wrote:I know that this one is correct:

That dog keenly hunting the ducks must be a thoroughbred.

"Keenly hunting the ducks " modifies "dog". "Hunting" is the participle.

Now by the same token, is this one correct?

He went to the hospital believing himself fatally ill.

Whenever you have verb-ing modifier without a comma, it modifies the preceding noun (which happens to be the subject in the case of "dog" sentence, but in the case of "hospital" sentence, it is not the subject but the hospital).

1: That dog keenly hunting the ducks must be a thoroughbred. - CORRECT
2: He went to the hospital believing himself fatally ill. - INCORRECT

It does not make sense for "believing himself..." to modify the "hospital". This this sentence is incorrect. It can be corrected by placing the modifier prior to the entity that it should modify "he" in this case.

So note the key difference:
verb-ing separated by comma modifies the preceding clause.
verb-ing not separated by comma modifies the preceding noun.

In sentence 1, the verb-ing modifier is placed after a noun and not after a clause. (notice the verb appears after the modifier)
In sentence 2, the verb-ing modifier is placed after the clause. (notice the verb appears prior to the modifier)

Furthermore, in sentence 2 "believing himself fatally ill" appears to be idiomatically incorrect.

Thanks,