The senate shall be the final judge of the elections, returns and qualifications of their own members, as pointed out in the constitution; and shall, [on the said last Wednesday in May] [annually,] determine and declare who are elected by each district, to be senators [by a majority of votes; and in case there shall not appear to be the full number of senators returned elected by a majority 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 such persons as shall be found to have the highest number of votes in such district, and not elected, amounting to twice the number of senators wanting, if there be so many voted for; and out of these shall elect by ballot a number of senators sufficient to fill up the vacancies in such district; and in this manner all such vacancies shall be filled up in every district of the commonwealth; and in like manner all vacancies in the senate, arising by death, removal out of the state, or otherwise, shall be supplied as soon as may be, after such vacancies shall happen.] [See Amendments, Arts. X, XIV and XXIV.]