The registers date from 1784 to 1803. They're arranged alphabetically (after a fashion) and record those who were examined in the RCSI for Letters Testimonial, Mateship, Army Surgeon, Surgeoncy, Army Mate, Navy Mate, Assistant Surgeon and to be a ranked mate to a ranked officer.
When first established by Royal Charter in 1784, the College did not offer its own training for surgeons. It was set up to control and regulate standards and act as a type of accreditation board. Candidates for the examination would have learned their craft, typically as apprentices, from third parties and could then be tested by the RCSI to see if they met the required standard for the Army or Navy, one of the principal employers for those with such skills. You can see the information recorded in the sample below.
From being a regulator of standards, the RCSI quickly moved into the provision of education. Those who studied at the College in Dublin and successfully passed its examinations were licenced to work as surgeons, not just for the Army and Navy, but also in private practice. The licence gave them the freedom to chose where they used their skills. All graduates signed the Roll of Licentiates, and added their current 'term time' address in Dublin (a few gave their home address). The Rolls from 1828 to 1890 are available for viewing here,
RCSI Archivist Meadhbh Murphy has told Irish Genealogy News that the Rolls for 1891 to 1950 will be added in early 2015.