In case such aid shall not be voted by the people as provided in section 15-501 or in case the system of waterworks shall prove inadequate for the needs of the city of the primary class, both public and private, then the mayor and city council may contract with and procure individuals or corporations to construct and maintain a system of waterworks in such city of the primary class for any time not exceeding twenty years from the date of the contract, and with a reservation to the city of the right to purchase such waterworks at any time after the lapse of ten years from the date of the contract, upon payment to such individuals or corporation of an amount to be determined by the contract not exceeding the cost of construction of such waterworks. In other respects such contracts may be upon such terms as may be agreed upon by a two-thirds vote of the mayor and city council, recorded in the minutes, except that no such contract shall be made unless authorized by a majority vote of the legal voters at a special election called for such purpose.