(a) All contests of the election of a Senator, and all questions affecting or relating to the qualification or eligibility of a Senator, or the right of any Senator to a seat or membership in the Senate, shall be referred to the Committee on State, Veterans, and Military Affairs, which shall proceed to hear and determine said matters, and each of them, in such manner, at such times, in such places, and under such rules, as said committee may adopt, for the purpose of facilitating such hearing.