Hi Roberta,
I could not get you exactly, so I have to analyze your rewritings carefully. Excuse me for doing so.
Yes, I found the term "dangling participle" has properly depicted the mistake, thanks. But using the other term "mis-related participle" to indicate such a mistake is invented by Mr.Echersley who, if I guessed correctly, was an English professor in Britain.
Now let's come to the point.
(1) The first rewriting:
The middle one of the three windows was halfway open; and while sitting close beside it, taking the air with an infinite sadness of mien like some disconsolate prisoner, Utterson saw Dr Jekyll.
I think you have rewritten the present participle clauses as a time-type enlargement; and this enlargement is modifying the subject Utterson. Am I on the right track?
(2) The second rewriting:
The middle one of the three windows was halfway open; and sitting close beside it, taking the air with an infinite sadness of mien like some disconsolate prisoner, was Utterson, who then saw Dr Jekyll.
For convenience, let's leave out the second present participle clause "taking the air with an infinite sadness of mien like some disconsolate prisoner", thus the sentence becomes:
...and sitting close beside it, was Utterson, who then saw Dr Jekyll.
The subject is "and sitting close beside it", the predicate is "was Utterson", while "who then saw Dr Jekyll" is the enlargment of "Utterson", am I on the right track?
Now leave out the enlargement, the sentence means:
Utterson was sitting close beside it.
Am I on the right track?
(3) The third writing:
... Utterson, while sitting close beside it, taking the air with an infinite sadness of mien like some disconsolate prisoner, saw Dr Jekyll.
Apparently, the middle part -- "while sitting close beside it, taking the air with an infinite sadness of mien like some disconsolate prisoner" is the time-type enlargement of the subject "Utterson". Am I on the right track?
I am sorry I was so wordy, Roberta!