The “new Princess Royal” was published in D. Wright’s “Country Dances”, 1735. An earlier “Princess Royal” in Walsh’s Dancing Master was set to a completely different tune.

Edward Bunting ((1773-1843) noted the tune from the playing of the harpist, Arthur O’Neill, who told him it had been composed by Turlough O’Carolan (1670-1738). This attribution was first published in 1810. Bunting published the tune in his 3rd volume of Irish tunes 1840. (1)
O’Carolan is reputed to have called the tune “Miss MacDermott, or The Princess Royal”. The MacDermott princes traditionally presided at the inauguration of the kings of Connacht. Hence the reference to the “Princess Royal”.

William Shields adapted the tune for his song “The Arethusa,” in his opera The Lock and Key, performed in 1796.

In the Cotswolds it is used as a solo morris dance from several different villages, and in Abingdon as a set dance. O’Carolan’s composition is in the minor mode, as is Wright’s tune. Some of the versions found in the tradition are in the major.

For more detail see: Menteith C, 2018 Oct, ‘The Princess Royal’ tune, Folklife Traditions 59, p 46

Reference
1. Bunting, Edward, 1840,The Ancient Music of Ireland, Dublin, Hodges & Smith, tune 45.