From the recording Set-dance: The Princess Royal

Set dances don’t get played too often in sessions, but they have a wonderful stateliness that reflects their original use as exhibition pieces by traveling dancing masters. Dance was massively popular in the countryside--an economical and participatory entertainment--and is mentioned in accounts by travelers going all the way back to the Middle Ages. Itinerant teachers were common characters whose stock in trade was the importation or creation of new, often Continental dances. Such masters would sometimes dance on a rain barrel or half-door, as a kind of “live infomercial.”
This is one of my favorites—A/K/A “Brian the Brave” and “The Gaelic League March”—and appears in O Neill’s 1850 as #641. A number of 19th-century sources attribute it to Carolan, presumably because of its alternate title “Miss McDermot’s,” a dedication to Mary McDermot, the “Princess Royal” or eldest daughter of one of his patrons, known as the Prince of Coolavin. On the other hand, it also shows up in English country dance collections under the correct title in the 1730s, and is closely related to the melody of “The Raggle-Taggle Gypsy.” The tune was also used for a song by Eoghain Rua Ó Súilleabháin called “Rodney’s Glory” (the 1850’s #1784), dedicated to the commanding officer under whom Red Owen served in a massive engagement with French ships in the Caribbean in 1782. A second nautical application comes from William Shield, who called the tune “The Arethusa” after an English ship of the American Revolution, and used it in his 1796 ballad opera Lock and Key. I honestly don’t remember where I first learned it.
Great harmonic stuff here from Roger.