Whenever the board shall contemplate the making of such improvements outside the corporate limits of any such city, it shall notify the county surveyor, whose duty it shall be to make an examination of the proposed improvement and report an estimate of the cost thereof to the board. If upon the consideration of such report, the county board determines to make the improvement, it shall cause personal notice to be served on the owners of property abutting on said road outside the corporate limits of such city of its intention to make such improvements, and if the owner is a nonresident, then by personal service upon the agent of such nonresident, if he has one residing in the county, and in case he has no such agent, by publishing such notice in a newspaper published in and of general circulation in such county. Upon the proof of service or publication of said notice, and after giving such owner an opportunity to be heard, the board shall decide upon the material to be used in such improvement and enter an order upon its records for the construction thereof; Provided, however, whenever the street, avenue, boulevard or road upon which improvements are contemplated, lies adjacent to and outside, or partly inside and partly outside, the corporate limits of any such city, the examination of the proposed improvement, report and estimate of the cost thereof, shall be made jointly by the county surveyor and the city engineer of such city, and after the county board shall determine to make such improvement and decide upon the material to be used therein, nothing further shall be done toward the completion of such improvement until such city by and through its proper officers shall agree in writing, a copy of which shall be filed with the board, to construct the one-half of said improvement lying next to or within the corporate limits of the city and pay the cost of said one-half. After such agreement shall have been filed with the county board, the board shall proceed to construct the other one-half of such improvement in the manner provided herein. The county shall pay two-thirds of the cost of the other one-half of such improvement, and the other one-third shall be paid by special assessment of all the real estate abutting on or adjacent to said one-half as provided in section 23-341; and provided further, wherever any such city shall have improved any portion, equal to one-half or more, of any such street, avenue, boulevard or road lying adjacent to and wholly outside, or partly outside and partly inside the corporate limits of any such city, and either paid or provided for the payment of the cost of the same, the county board may proceed to improve in like manner the remaining portion of said street, avenue, boulevard or road, and of the cost thereof the county shall pay two-thirds and the other one-third shall be paid by special assessment of all the real estate abutting on or adjacent to said portion as provided for in section 23-341.