The tune identified as "One Eye'd Riley" in Septimus Winner's "Excelsior Collection for the Flute" (N.Y., 1864), p.36, is (unsurprisingly) an unmistakable variant of "Paddy Will You Now?" in P.M. Haverty's "One Hundred Irish Airs," III (N .Y., 1859), p.142.

Winner's is the earliest printing I have found of a tune with this title. Both his and Haverty's are reminiscent of the "Riley" tune used by T.S. Eliot in his 1949 play "The Cocktail Party."

Several sources mention "One-Eyed Riley" as a marching tune in America beginning in the 1850s, but no one in the 19th century offered any lyrics to accompany it.

The modern song unquestionably descends from "Paddy, Will You Now?"

In Eliot's version, by the way, Riley is the narrator, not the father.