If all of the territory of the district is included within more than one incorporated city, then such proportionate part of the money paid into the lighting fund shall be paid to the treasurer of each such city as the area of that portion of the district included in that city bears to the total area of the district. The funds so paid shall be administered by the legislative body of each such city, and each of the said legislative bodies shall have all the powers and perform all of the duties, so far as that portion of the district within its boundaries is concerned, granted to or imposed upon the board of supervisors of the county in which the district is located and shall carry out this part as to the portion of the district included within its boundaries to the same purpose and extent as if orginally constituted the governing body thereof.