The returns of every election for said Executive Officers, until otherwise provided by law, shall be made out, sealed up, and transmitted by the returning officers prescribed by law, to the seat of Government, directed to the Secretary of State, who shall deliver the same to the Speaker of the House of Representatives, as soon as the Speaker shall be chosen, and the said Speaker shall, during the first week of the session of the Legislature, open and publish them in the presence of both Houses of the Legislature. The person, voted for at said election, having the highest number of votes for each of said offices respectively, and being constitutionally eligible, shall be declared by the Speaker, under sanction of the Legislature, to be elected to said office. But, if two or more persons shall have the highest and an equal number of votes for either of said offices, one of them shall be immediately chosen to such office by joint vote of both Houses of the Legislature. Contested elections for either of said offices, shall be determined by both Houses of the Legislature in joint session.