Here's what Glostrad has to say on the tune:

Princess Royal was composed by Irish harper Turlough O’Carolan (1670-1738), who called it “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 Shield 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. It occurs in both minor (as the original) and major versions.

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