Whenever the board of trustees of any town or village organized under the laws of the state of Missouri as such town or village shall, by resolution, declare that a necessity exists for the sprinkling, oiling or otherwise treating of any particular street, avenue, alley or streets, avenues, alleys and such public places within such town or village and whenever thereafter two-thirds of the resident taxpayers owning property adjoining and abutting on said street, or streets, avenues, alleys and such public places shall file with the board of trustees of such town or village a petition requesting that said street or streets, avenues, alleys and such public places so designated and set aside by resolution be sprinkled, oiled or otherwise treated, the board of trustees shall have power by ordinance to cause such sprinkling, oiling or treating to be done and the cost thereof to be provided for and defrayed by a special tax to be assessed in favor of town or village on the adjoining property fronting or bordering on the streets, avenues, alleys, and such public places, where such sprinkling, oiling or treating is proposed to be done in the proportion that the linear feet of each lot fronting or bordering on the street, avenue, alley and public place so to be sprinkled, oiled or treated bears to the total cost of such sprinkling, oiling or treating.