If the City Commissioner or City Commissioners or any of them, designated in the petition, file with the Clerk of the City Commission within 5 days after the last mentioned notice from the City Treasurer his or their written resignation, the Clerk of the City Commission shall at once notify the election authorities of that fact and such resignation shall be irrevocable, and the City Commission shall proceed to fill the vacancy. In the absence of any such resignation the election authorities shall forthwith order and fix a day for holding a recall election for the removal of those not resigning. Any such election shall be held not less than 30 nor more than 60 days after the expiration of the period of 5 days last mentioned, and at the same time as any other general or special election held within such period; but if no such election be held within such period the election authorities shall call a special recall election to be held within the period aforesaid.