The first of six supplemental volumes containing material which is not in the Bulaq edition Burton translated in the ten volumes of the Book of the Thousand Nights and One Night (which I have just finished reading); some may be included in various MSS of the Nights and some represent the kind of independent story which the Nights were formed from.
According to Burton's Foreword, the first two volumes of the supplement are actually a revision of the three volumes of John Payne's Tales from the Arabic, which contain stories from the printed Breslau edition and the Calcutta edition which are not in the Bulaq edition.
The first tale, "The Sleeper and the Waker", in which a poor man called Abu Hassan is fooled into thinking he is the Caliph (and which may be the source for Shakespeare's Christopher Sly at the beginning of The Taming of the Shrew), is one of the eleven stories added by Galland in his translation to the core stories (the only one for which an Arab source had been found when Burton wrote his notes; I don't know whether others have been found since.) Several of the shorter tales which follow were translated in the continuation of Galland by Dom Chavis and M. Cazotte.
The two longest parts of the book are "The Ten Wazirs, or the History of King Azadbakht and his Son" (derived from the Persian Bakhtyar Nameh) and "King Shah Bakht and his Wazir Al-Rahwan", both collections of tales with frame stories obviously derived from Persian originals. Some of the tales in these collections seem to be abridged or truncated versions of tales which were included at greater length in the Bulaq edition; there are obvious non sequiturs and one tale, "The Tale of the Falcon and the Locust", makes no sense whatever.
Mostly two sets of ransom stories with extensive footnotes. Burton being Burton the footnotes include a long and eye-watering one about castration that I would rather forget.