Whenever any person elected or appointed a member or president of the town council of any incorporated town who has been notified of his election or appointment shall because of death or other reasons refuse or neglect to qualify as such member or president of such town council within ten days next succeeding the beginning of his term of office, unless prevented by sickness or prevented by necessary absence from the town, the town council acting without such person may declare his office as member or president of the town council vacant and such vacancy shall be filled by the town council, by appointing, by resolution, a registered elector of the town, to hold such office, if the term thereof continues so long, until the first Monday in January after the first municipal election occurring more than sixty days after the vacancy occurs, at which election an eligible person shall be elected to the office for the remainder of the term. For such actions a majority of the remaining members of the council shall constitute a quorum.