See Full Constitution
Part Second

Article 34. Vacancies in Senate, How Filled

And in case there shall not appear to be a senator elected, by a plurality of votes, for any district, the deficiency shall be supplied in the following manner, viz. The members of the house of representatives, and such senators as shall be declared elected, shall take the names of the two persons having the highest number of votes in the district, and out of them shall elect, by joint ballot, the senator wanted for such district; and in this manner all such vacancies shall be filled up, in every district of the state and in case the person receiving a plurality of votes in any district is found by the Senate not to be qualified to be seated, a new election shall be held forthwith in said district. All vacancies in the senate arising by death, removal out of the state, or otherwise, except from failure to elect, shall be filled by a new election by the people of the district upon the requisition of the governor and council, as soon as may be after such vacancies shall happen.