The returns of every election for the officers of the executive department shall be sealed up and transmitted by the returning officers to the Secretary of State, directed to the Speaker of the Legislature, who shall immediately after the organization of the Legislature, and before proceeding to other business, open and publish the same in the presence of a majority of the members of the Legislature. The person having the highest number of votes for each of said offices shall be declared duly elected; but if two or more have an equal and the highest number of votes, the Legislature shall choose one of such persons for said office. The conduct of election contests for any of said offices shall be in such manner as may be prescribed by law.