Any third-class city which is the county seat of a county having a population of more than two thousand five hundred (2,500) and less than six thousand (6,000) and an assessed valuation of tangible taxable property of more than nineteen million dollars ($19,000,000) and less than twenty-three million dollars ($23,000,000) and which city has been or shall be ordered by the secretary of health and environment to enlarge, improve or rebuild the city’s sewage disposal plant or construct a new sewage disposal plant shall have authority and is empowered to assess not more than thirty percent (30%) of the total cost of such improvement against other taxing subdivisions whose territory includes land within the city limits of said city if the governing bodies of such overlapping taxing subdivisions give their consent to such assessments, which they are hereby authorized and empowered to do, and if no more than ten percent (10%) of the total cost of such improvement shall be charged and assessed against the county and not more than ten percent (10%) against any school district.