The Collaborative International Dictionary of English v.0.48:

Regius \Re"gi*us\ (r?l"?*?s), a. [L. regius, from rex, regis, a
   king.]
   Of or pertaining to a king; royal.
   [1913 Webster]

   Regius professor, an incumbent of a professorship founded
      by royal bounty, as in an English university.
      [1913 Webster]

WordNet (r) 3.0 (2006):

Regius professor
    n 1: holder of a British professorship created by a royal patron