Autograph (?)

OK, it's another miserable day - that wonderful Scots word "dreich" applies.
So, a little family history. A signature of Walter Scott, 1777 appears on the fly-leaf of an old book which was part of my late father's possessions. This book, entitled "A Dunse Spaw" was published in 1751 and is a treatise on the healing virtues of spa ("spaw") waters to be found near to the beautiful border town of Duns, Berwickshire, where I was born and where my father was parish minister from 1942 to 1952. I know that the great novelist Sir Walter Scott was a sickly child and, as a boy, was taken to several spas to seek a cure but can find no confirmation of his ever visiting Duns. If the date is correct, Walter would have been six/seven years old at the time so I have a double question for all you blippers:
Could this be a genuine signature of Sir Walter Scott?
Did young Walter ever visit Duns to take the waters?
Answers please on the back of a £10 note bearing the portrait of Sir Walter!!