In all special municipal separate school districts where the district embraces less than the entire area of the county and where the majority of the educable children of such district reside outside the limits of the municipality, unless the governing authorities of the municipality and the county provide for one of the alternative methods of organization as set out in Sections 37-7-715 and 37-7-717, the said special municipal separate school district shall be governed by a board of trustees consisting of five members, to be elected by the qualified electors of such municipal separate school district from the district at large in the manner provided by Sections 37-7-209 through 37-7-219, and all duties imposed upon the county superintendent of education by said sections with reference to such elections shall be imposed upon and performed by the superintendent of the municipal separate school district. However, the first board of trustees of such special municipal separate school district shall be appointed in the following manner. The governing authorities of the municipality shall appoint three trustees, and such appointments shall be made so that one trustee shall be appointed to serve until the first Saturday of March following such appointment, one for two years longer, and one for four years longer. The board of education of the county shall appoint two trustees, such appointments to be made so that one trustee shall be appointed to serve until the first Saturday of March of the second year following such appointment, and one trustee for two years longer. After such original appointments the trustees of such a special municipal separate school district shall be elected for a term of five years, as herein provided. All such members of said board of trustees shall be residents and qualified electors of such school district. All vacancies which may occur during a term of office shall be filled by appointment by the remaining members of the board of trustees, such appointee to have the same qualifications as other members of the board. Such appointment shall be made within thirty days after the vacancy occurs. The person so appointed shall serve only until his successor shall have qualified. The successor to serve the remainder of the unexpired term shall be elected on the first Saturday of March next following the occurrence of such vacancy in the same manner as provided for by Sections 37-7-209 through 37-7-219.