2005 California Government Code Sections 25060-25062 Article 4. Vacancies

25060.  Whenever a vacancy occurs in any board of supervisors, the
Governor shall fill the vacancy.  The appointee shall hold office
until the election and qualification of his successor.
25061.  The election of a supervisor to fill the vacancy for the
unexpired term shall be held at the next general election, unless the
term expires on the first Monday after January 1st succeeding the
election.
25062.  When a vacancy occurs from the failure of the person elected
to file his oath or bond as provided by law, and the person elected
is appointed to fill the vacancy, he shall hold office for the
unexpired term.