§ 61. Payment by counties of a portion of the cost of construction under repair contracts. Whenever in the maintenance and repair of state highways the department of transportation shall have determined upon the necessity of resurfacing, reconstructing or repairing such highway, the county wherein the highway is located may by resolution provide that not to exceed fifty per centum of the cost of such resurfacing, reconstructing or repairing shall be borne by the county. The provisions of this chapter shall apply to such resurfacing, reconstructing and repairing of state highways in the same manner as to the original construction thereof in so far as the same may be applicable thereto.