I know that the rule for my/mine used to the same as a/an: if the noun that follows starts with a vowel sound, the pronoun ends in a 'n' sound, hence "mine eyes have seen the glory" in the Battle Hymn of the Republic. We don't use that construct anymore though, instead using "my" in front of all nouns, regardless of pronunciation. When did English make this shift?
You may want to ask this in /r/linguistics if you haven't yet.