Also known as "Jackson's Rambles", "Turasa Blarnaig", "Paddy O'Brien's Jig" and "The Parish Girl". In the movie Titanic it was paired with "John Ryan's Polka" even though that is a polka while this is a jig.
This is a three part jig. The first and third parts seem to be in D Mixolydian with the flattened seventh note while the second part appears to be in G Major using a pentatonic scale.
It appears in O'Neill's Music of Irelandd (1850) and in Miles Krassen's more recent O'Neill's Music of Ireland. There are many recordings and it is popular with many Irish fiddle, flute and pennywhistle teachers.