My mother’s skipping rhyme. I don’t recall when I learned it, and I never skipped with it, but it popped into my mind. I went Googling. Here I found alternative versions: “My ball’s down the airy” or “Up in the airy”.
“Down the airy” may be the original: the airy is the “light well” between the pavement and the basement flat of a tenement. It let a small amount of light to that flat’s windows. If you go to the coffee shop in the National Portrait Gallery, and look up, you will see you are in the airy, which has been glassed over at sidewalk level. Sorry if that’s a spoiler: I got a shock of delight and surprise, seeing it.
I don’t know if my mother saw airies, or light wells, in Putney in the 1930s. “Down the airy” makes sense, “Down the dairy” doesn’t, though it is a little easier to sing and it alliterates. If you did not know what an airy was, you might change that to dairy, or “up in the airy”, because they seemed to make more sense.
“My ball’s down the dairy” is a Googlewhack.
Reading the explanation, it seems “airy” is the original, though stated as coming from East London in the 50s, not Putney (South west London) in the 30s.
The first link has rude rhymes too. I learned The Derby Ram at school, a bowdlerised version for singing in the Music class, and a rude version for singing with the Scouts. My precise version of Aunty Mary isn’t there:
Aunty Mary had a canary
up the leg of her drawers,
it widnae come doon for half a croon,
only for Santa Claus.
Perhaps there is no “right” version.