If any member of the Board, being the Mayor for the time being of the City of Christchurch, or the Chairman for the time being of the Selwyn County Council, or the Chairman for the time being of the North Canterbury Hospital Board or other the institution for the time being having control of the Christchurch Hospital, or the Chairman for the time being of the Board of Governors of Canterbury College, shall refuse or be unwilling to act as a member of the Board, then the City Council, or the Selwyn County Council, or the North Canterbury Hospital Board or other the institution having control of the Christchurch Hospital, or the Board of Governors of Canterbury College, as the case may be, may, by resolution at any meeting of such Councils or Boards respectively, appoint a new member of the Board of Governors of the Institute in place of the Mayor or Chairman, as the case may be, refusing or being unwilling to act, for the residue of the term of office of such Mayor or Chairman, as the case may be; and the certificate under the hand of such Mayor or of any such Chairman, as the case may be, shall be conclusive evidence that such new member has been duly appointed.