If an appeal has been made in reference to the necessity of the improvement as provided in section 6117.09 of the Revised Code, and it appears from the transcript certified to the board of county commissioners by the probate judge that the proposed improvement is not necessary for the public health, convenience, and welfare, or that the cost thereof will exceed the benefits resulting from such improvement, the board shall abandon the improvement. If it appears from such transcript that the improvement is necessary for the public health, convenience, and welfare and that the cost thereof will not exceed the benefits resulting therefrom, the board may proceed with the improvement.
R.C. § 6117.19