Fiddler's Companion has an entry for this tune, which is a double jig as well as a song. Here is the text from the FC:

"The title comes from a song set to the tune. James Joyce gives a few lines in Ulysses:

***I gave it to Kitty because she was pretty,
The leg of the duck, the leg of the duck.
I gave it to Molly because she was jolly,
The leg of the duck, the leg of the duck.
***
Irish uilleann piper and music editor Terry Moylan (Johnny O'Leary) shared a car journey in 1970 with Sean Reid and Willie Clancy, who were entertaining each other with similar rhymes. Clancy gave the one above and also had:
***
I gave it to Nelly to stick in her belly
The leg of the duck, the leg of the duck.
She has it, she's got it, wherever she put it,
The leg of the duck, the leg of the duck.
***
Moylan says: "According to Johnny O'Leary, Denis Murphy had scores of these things. Willie could have got them from Denis, as they were close friends." Simple rhymes were helpful in the days of itinerant musicians when people would employ them to describe the tune they wanted to dance to when a visiting musician did not know the local name."

Anybody have some more examples of this? They'd be easy enough to construct, of course.

Breandan Breathnach related a story about a piper, name of Ned, who wanted something particular for his meal. He played this jig and the lady of the house asked him what the name of it was.

"Oh, praties and butter
Would make a good supper
For Neddie the piper,
Neddie the piper!"

In books like O'Neill's this tune is sometimes called Dónall na Gréine, which means Daniel of the Sun. A friend of mine was told by an Irish speaking friend of his that "Of the Sun" refers to someone conceived that way - i.e., out of doors, thus illicitly.