Whenever the landowners of one (1) or more adjoining townships in any county or in a certain described portion thereof, amounting to three-fourths (¾) of the number of landowners in the territory described in the petition, shall file with the county court of the county a written statement and agreement signed by the landowners wherein they represent to the court that they have voluntarily agreed among themselves to improve certain county roads or parts thereof in the township or in the area described in the agreement, by contributing in cash the necessary funds therefor, and that the costs of the proposed improvements shall not exceed a total sum to be stated in the agreement, and that the costs shall be paid pro rata by the signers of the agreement in the proportion that the assessed value of their land shall bear to the total cost of the improvements, then the statement and agreement shall be spread in full upon the records of the county court. Thereafter, each and every owner and signer of the agreement shall be bound by the terms of the agreement and shall be personally liable for his proportionate part of the costs of the improvement as set out in this section.