George Petrie (The Petrie Collection of the Ancient Music of Ireland, 1855) comments:

As the melody of the charter song of that singular social union of wit and talent which existed in Dublin, from the year 1779 to the close of the year 1785, and was called "The Monks of the Order of St. Patrick," but commonly known as "The Monks of the Screw" few of the readers of this work will require to be informed that this well-known charter song was written for the society by its Prior, the late John Philpot Curran; but it has not been hitherto known that the music selected by the gifted poet, as a fit medium for his serio-comic verses, was a gay Irish melody, arrayed in a mock solemnity, and which, no doubt, he had learnt in his own loved county of Cork.

The melody, according to Andrew Kuntz's  The Fiddler's Companion,  is a variant of Ta Me I n-Eagmais Ach Iocfadh Me Fos (I'm in debt but I'll pay them yet).  A set of The Monks of the Screw appears in O'Neill, but the one given by Petrie should be more accurate, as it came directly from Curran's son.