The mayor shall appoint two (2) alternate members to the board, which members shall meet the qualifications of Class IV members of the board. Alternate members shall be designated at the time of appointment by the mayor as "Alternate No. 1" and "Alternate No. 2." The terms of the alternate members shall be for two (2) years, except that the terms of the alternate members shall be such that the term of not more than one (1) alternate member shall expire in any one (1) year; provided, however, that in no instance shall the terms of the alternate members first appointed exceed two (2) years. A vacancy occurring otherwise than by expiration of term shall be filled by the mayor for the unexpired term only.