The Circuit Court of the City of Richmond shall have jurisdiction to try cases of offenses committed in Capitol Square except as hereinafter provided. The district court of the City of Richmond shall have jurisdiction to try misdemeanor cases arising under § 18.2-122, and all other offenses committed in the Capitol Square of which it would have jurisdiction if committed within the corporate limits and jurisdiction of the city; and the Capitol Police, or any member thereof, shall have the same authority to arrest and to swear out warrants for offenses committed on the Capitol Square as policemen of the City of Richmond have to arrest or to swear out warrants for offenses committed within the jurisdiction of the city.