"The Princess Royal" would not be listed in Playford because that title
relates to Carolan's tune--whether or not he actually wrote it (as I believe
he did). The princess in question was the eldest daughter in Carolan's time
of The MacDermot of Coolavin. The tune is in Donal O'Sullivan's book, but
under the alternative title, "Miss MacDermott"; it's no. 87. O'Sullivan
gives a fairly lengthy examination of the various incarnations of the tune,
including the one which I suspect Atsushi is asking about, "The Arethusa":
this is a once-famous song by William Shield (1748-1829) to the air of "The
"The Princess Royal" is said to be the air Eoghan Rua O Suilleabhain
intended for his song "Rodney's Glory". I used to think it was a bit of a
stretch from TPR to the tune played nowadays as "Rodney's Glory"; but
Goodman has a tune called "An Samhra Cruaig" (no. 233 in "Tunes of the
Munster Pipers") which may be seen as the "missing link" between the two.
O'Farrell has a setting of the tune, no. 135 in vol. 4 of the "Pocket
Companion", called simply "Air by Carolan".
All the preceding tunes are in minor mode. I've heard the morris tune known
as "The Princess Royal"; it's a very nice tune too, and what's perhaps
surprising is that it's just the Carolan tune played "as if" notated in a
major key. Another example of creative misreading.