A book well bound and indexed, to be denominated the "Docket" shall be kept in the office of the Mayor, in which shall be noted each case, brought or tried by him, together with the proceedings thereon, including a statement of the complaint, the summons, the return, the fact of appearance, the defense, the hearing, the judgement, the costs, and in case the judgment be one of conviction, the action taken to enforce the same. The record of each case shall be signed by the Mayor and the original papers thereof, if no appeal with writ of error or certiorari be taken, shall be kept together and preserved in his office.