In any city which contracts for the assessment of property by the county or township assessor as authorized by section 70.220 and does not elect or appoint an assessor, the county or township clerk shall deliver to the mayor, on or before the first day of October of each year, a certified abstract from his assessment books of all property within the city subject to taxation by the state and the assessed value thereof as agreed upon by the board of equalization. The mayor shall immediately transmit the abstract to the council which shall establish by ordinance the rate of taxes for the year; and for the purpose of giving cities of the third class representation on the county board of equalization, when said board is sitting for the purpose of equalizing the assessment on such city property, the mayor and one other person to be selected by the legislative body of the city shall sit with the county board of equalization when the board is passing upon the assessment of such city property, and shall each have a vote in said board, and they shall be paid for such service the same amount per day and out of the same fund as other members of the board of equalization.