You have probably already observed this, and I might be wrong, but it appears as though the words "thy" and "thine" follow the same pattern as "a" and "an" in that "thy" precedes words beginning with consonants and "thine" precedes words beginning with vowels (e.g. Mark 9:42,45,47 "thy hand" "thy foot" "thine eye"). Let me know if this is right, or if you find any counter-examples.
Persevering with God's preserved words,
Patrick