In case of a vacancy, during the recess of the Senate, in any office which is not elective, the governor shall, by appointment, fill such vacancy, until the next meeting of the Senate, when he shall make a nomination for such office, and the person so nominated, when confirmed by the Senate, (a majority of all the senators elected concurring by yeas and nays) shall hold his office during the remainder of the term, and until his successor shall be appointed and qualified. No person, after being rejected by the Senate, shall be again nominated for the same office, during the same session, unless at the request of the Senate; nor shall such person be appointed to the same office during the recess of the Senate.