Section 17. If a city determines that the public ways or any portion thereof shall be sprinkled in whole or in part at the expense of the abutters, such expense for a municipal year, and the proportion thereof to be borne by abutters, and the rate to be assessed upon each linear foot of frontage upon such ways, shall be estimated and determined by the board of aldermen and assessed upon the estates abutting on such ways in proportion to the number of linear feet of each estate upon such ways or portion thereof sprinkled. The amount of such assessments upon each estate shall be determined by said board, or, if said board so designates, by the board of public works, board of street commissioners, superintendent of streets or other officer; and such board or officer shall, as soon as may be after the first day of January, cause a list of such ways or portions thereof to be made, specifying each estate and the number of linear feet thereof abutting thereon, the amount per linear foot, and the amount on each estate of such assessment, and certify and commit said list to the assessors of taxes. In a town such assessment shall be made by the assessors.