The trial and determination of contested elections of electors of President and Vice-President, members of the General Assembly, and of all public officers, whether State, judicial, municipal or local, and contests involving questions submitted to the electors at any election shall be by the courts of law, or by one or more of the law judges thereof. The General Assembly shall, by general law, designate the courts and judges by whom the several classes of election contests shall be tried, and regulate the manner of trial and all matters incident thereto; but no such law assigning jurisdiction, or regulating its exercise, shall apply to any contest arising out of an election held before its passage.